A broad tray and a narrow tube can contain water at the same depth and exert the same pressure at their bottoms. The shapes still affect total forces. Separating pressure at a point from force on a surface resolves the apparent puzzle.
We will develop the pressure gradient, distinguish gauge and absolute readings, follow manometer columns and check hydraulic force multiplication. Then we use the same pressure physics to understand floating, apparent weight and what a balance actually measures.
What you will learn to do
- Distinguish local pressure from net force across a surface.
- Derive the hydrostatic gradient and state its density assumptions.
- Use gauge/absolute references and follow layered manometer columns.
- Check hydraulic force multiplication against displaced volume and work.
- Derive buoyancy from pressure and use the displaced-fluid volume.
- Interpret floating and apparent weight with explicit system boundaries.
- Test floating geometry, load margins and water-level changes with a before-and-after volume ledger.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Why does the same pressure produce a greater force on a larger flat surface?
Compare your reasoning
For uniform pressure acting normally, force equals pressure times area. Pressure describes force per unit area; it is not the total force itself.
Revisit the explanation →01 / Pressure, floating and fluid support
Pressure is a local scalar; its force has a surface direction
A fluid at rest exerts contact forces normal to a surface. Pressure describes the normal force per unit area in the small-area limit. At a point in an ordinary fluid at rest, pressure is the same for every surface orientation, but the pressure force on each chosen face points along that face’s normal.
For uniform pressure over a flat face, the force magnitude is pA. If pressure varies across the face, divide it into small areas and add p dA. For a curved surface, the force directions also vary, so the vector contributions must be resolved before summing.
A pressure value by itself does not give the net force on a wall or piston. Fluid or air on the other side may also press. For a flat piston with uniform pressures p₁ and p₂ on opposite sides, the net pressure force is (p₁ − p₂)A. Keeping that difference explicit prevents atmospheric pressure from being counted on only one side.
The SI pressure unit is the pascal: 1 Pa = 1 N/m². Area conversions must be squared.
WORKED EXAMPLE
Use the pressure difference across a piston
A flat piston has area 4.0 cm². A gas exerts 130 kPa on one side while the other side is exposed to 100 kPa air. Find the net pressure force, neglecting the piston’s weight and friction.
Subtract the opposite pressures and convert the area to square metres.
The net pressure difference is 30 kPa.
Convert 4.0 cm² to 4.0 × 10⁻⁴ m² and multiply.
What this gives usThe net pressure force is 12 N toward the lower-pressure side. Using the full 130 kPa would omit the opposing atmospheric force.
Check the result another way
The two individual force magnitudes are 52 N and 40 N, whose difference is 12 N.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Use the pressure difference across a piston Problem: A flat piston has area 4.0 cm². A gas exerts 130 kPa on one side while the other side is exposed to 100 kPa air. Find the net pressure force, neglecting the piston’s weight and friction. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support
Focus: Pressure is a local scalar; its force has a surface direction
Learning goals:
- Distinguish local pressure from net force across a surface.
- Derive the hydrostatic gradient and state its density assumptions.
- Use gauge/absolute references and follow layered manometer columns.
- Check hydraulic force multiplication against displaced volume and work.
- Derive buoyancy from pressure and use the displaced-fluid volume.
- Interpret floating and apparent weight with explicit system boundaries.
- Test floating geometry, load margins and water-level changes with a before-and-after volume ledger.
Equations in this section (LaTeX):
p=\frac{dF_\perp}{dA},\qquad F_{\mathrm{net}}=(p_1-p_2)A\quad\text{for opposite uniform pressures}
Meaning: The SI pressure unit is the pascal: 1 Pa = 1 N/m². Area conversions must be squared.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Pressure, floating and fluid support
A pressure gradient supports the fluid’s weight
Take a small horizontal slice of fluid at rest with area A and thickness dy. Its bottom pressure force acts upward, its top pressure force downward and its weight downward. Static force balance requires pressure to decrease with upward height at the rate dp/dy = −ρg.
For approximately constant density and gravitational field, integrate this gradient to obtain p = p₀ + ρgh at depth h below a reference surface. Pressure at equal heights is equal within the same connected fluid in static equilibrium under these assumptions. Separate disconnected containers can have different surface pressures and therefore different pressures at the same absolute height.
The pressure difference depends on vertical depth, not on the sloping distance down a tube or the total amount of liquid above some imagined column. Container walls can carry forces; fluid equilibrium must be checked locally. A narrow deep tube can produce a larger bottom pressure than a wide shallow vessel.
The second form assumes constant ρ and g. In a compressible atmosphere or a strongly stratified fluid, density can vary and must remain inside the integration.
Explain this equation
Read it in words
Pressure decreases with upward height in a fluid at rest. At a depth h below a reference surface, a constant-density fluid has the surface pressure plus the weight-per-area of the fluid column.
Symbols and units
- Pressure at the point of interest.Pascals (Pa).
- Vertical coordinate chosen positive upward.Metres (m).
- Fluid mass per unit volume, treated as constant in p(h).kg/m³.
- Magnitude of gravitational acceleration.m/s².
- Depth below the reference surface, chosen positive downward.Metres (m).
- Pressure at that reference surface.Pascals (Pa).
Signs and operations
dp/dy is negative because pressure falls upward. The depth form has a plus sign because h increases downward. These are the same balance written with opposite coordinate directions.
When it applies
Use a fluid at rest in uniform gravity. The linear depth formula assumes constant density; compressible fluids or varying density require integration. Pressure differences use the same reference on both sides.
Make it concrete
At 0.30 m below a water surface, with ρ = 1000 kg/m³ and g = 10 m/s², pressure is 3000 Pa above the surface value. That is a gauge difference, not the full absolute pressure.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Equation (LaTeX): \frac{dp}{dy}=-\rho g,\qquad p(h)=p_0+\rho gh
Read it as: Pressure decreases with upward height in a fluid at rest. At a depth h below a reference surface, a constant-density fluid has the surface pressure plus the weight-per-area of the fluid column.
Symbols:
p: Pressure at the point of interest.; Pascals (Pa).
y: Vertical coordinate chosen positive upward.; Metres (m).
\rho: Fluid mass per unit volume, treated as constant in p(h).; kg/m³.
g: Magnitude of gravitational acceleration.; m/s².
h: Depth below the reference surface, chosen positive downward.; Metres (m).
p_0: Pressure at that reference surface.; Pascals (Pa).
Signs and operations: dp/dy is negative because pressure falls upward. The depth form has a plus sign because h increases downward. These are the same balance written with opposite coordinate directions.
Conditions: Use a fluid at rest in uniform gravity. The linear depth formula assumes constant density; compressible fluids or varying density require integration. Pressure differences use the same reference on both sides.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.WORKED EXAMPLE
Find a pressure difference without needing atmospheric pressure
Two points in water at rest lie 0.50 m and 2.0 m below the same free surface. Take ρ = 1000 kg/m³ and g = 10 m/s². Find the pressure difference between the deeper and shallower points.
Subtract the depth equations so that the common surface pressure cancels.
The vertical separation is 1.5 m.
Insert density, gravity and the depth difference.
What this gives usThe deeper point is at 15 kPa greater pressure. The result is independent of the common surface pressure.
Check the result another way
The units kg/m³ × m/s² × m reduce to kg/(m·s²) = N/m², confirming pressure units.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Find a pressure difference without needing atmospheric pressure Problem: Two points in water at rest lie 0.50 m and 2.0 m below the same free surface. Take ρ = 1000 kg/m³ and g = 10 m/s². Find the pressure difference between the deeper and shallower points. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
STOP & THINK
Two connected vessels contain the same liquid at rest, with equal free-surface pressure and height. Their shapes differ. How do pressures compare at equal depths?
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Topic: Pressure, floating and fluid support / A pressure gradient supports the fluid’s weight Problem: Two connected vessels contain the same liquid at rest, with equal free-surface pressure and height. Their shapes differ. How do pressures compare at equal depths? Choices: 1. They are equal 2. The wider vessel always has greater pressure 3. The narrower vessel always has greater pressure Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support
Focus: A pressure gradient supports the fluid’s weight
Learning goals:
- Distinguish local pressure from net force across a surface.
- Derive the hydrostatic gradient and state its density assumptions.
- Use gauge/absolute references and follow layered manometer columns.
- Check hydraulic force multiplication against displaced volume and work.
- Derive buoyancy from pressure and use the displaced-fluid volume.
- Interpret floating and apparent weight with explicit system boundaries.
- Test floating geometry, load margins and water-level changes with a before-and-after volume ledger.
Equations in this section (LaTeX):
\frac{dp}{dy}=-\rho g,\qquad p(h)=p_0+\rho gh
Meaning: The second form assumes constant ρ and g. In a compressible atmosphere or a strongly stratified fluid, density can vary and must remain inside the integration.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Pressure, floating and fluid support
Choose a pressure reference and keep it consistent
Absolute pressure is measured relative to vacuum. Gauge pressure is pressure relative to a stated ambient pressure, commonly the local atmosphere. A gauge reading of zero means equality with that reference, not absence of molecular pressure or contact forces.
For an open water tank with air pressure pₐ above it, the gauge pressure at depth h is ρgh and the absolute pressure is pₐ + ρgh. Pressure differences in a hydraulic calculation may let the same atmospheric contribution cancel. Gas laws and vapour-pressure comparisons usually require absolute pressure.
A negative gauge pressure is possible when the absolute pressure is below ambient. It does not automatically mean negative absolute pressure. For example, 80 kPa absolute in surroundings at 100 kPa corresponds to −20 kPa gauge. State which quantity an instrument displays before using it in an equation.
The pressure reference changes the offset, not the gradient
- Gauge
- 0
- Absolute
- 100
STOP & THINK
Water is open to air at 100 kPa. At depth 2 m, with ρ = 1000 kg/m³ and g = 10 m/s², what is the absolute pressure in kPa?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support / Choose a pressure reference and keep it consistent Problem: Water is open to air at 100 kPa. At depth 2 m, with ρ = 1000 kg/m³ and g = 10 m/s², what is the absolute pressure in kPa? Requested answer unit: kPa Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: Choose a pressure reference and keep it consistent Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Pressure, floating and fluid support
Follow pressure continuously through a manometer
A manometer relates an unknown pressure to a visible vertical column difference. The reliable procedure is to start at a point of known pressure and move through each fluid. Moving downward adds ρg times the vertical drop; moving upward subtracts it. At an ideal interface without surface-tension effects, pressure is continuous.
Within the same connected static liquid, two points at the same height have the same pressure. Across two different liquids, equal height alone is not sufficient unless the pressures are connected through an interface and the corresponding columns are accounted for.
A tilted tube changes the visible length of a column but hydrostatic pressure still depends on its vertical height. In layered liquids, each layer contributes its own density times thickness. Use the density belonging to the segment traversed, not an average density chosen without volume or height weighting.
WORKED EXAMPLE
Read which side has the greater pressure
An open U-tube manometer contains a liquid of density 800 kg/m³. A gas connects to the left arm, and the right arm is open to the atmosphere. The right liquid surface is 0.25 m above the left surface. Take g = 10 m/s² and neglect gas density and capillary effects. Find the gas gauge pressure.
The gas pushes the left surface lower, so it must exceed atmospheric pressure. Move down from the right surface to the left surface’s height.
Start at atmospheric pressure on the right surface.
At the same level in the connected manometer liquid, this equals the gas interface pressure.
What this gives usThe gas gauge pressure is +2.0 kPa. The lower surface is on the higher-pressure side.
Check the result another way
The column difference provides a sign check before arithmetic: a gas below atmospheric pressure would instead allow the atmosphere to depress its own side.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Read which side has the greater pressure Problem: An open U-tube manometer contains a liquid of density 800 kg/m³. A gas connects to the left arm, and the right arm is open to the atmosphere. The right liquid surface is 0.25 m above the left surface. Take g = 10 m/s² and neglect gas density and capillary effects. Find the gas gauge pressure. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
Add the pressure contributions of different layers
An open vessel contains 0.30 m of oil of density 800 kg/m³ above 0.50 m of water of density 1000 kg/m³. Take g = 10 m/s². Find the bottom gauge pressure, neglecting surface-tension effects.
Each layer adds pressure according to its own density and vertical thickness.
The oil adds 2400 Pa from its surface to the interface.
The water adds another 5000 Pa below the interface.
What this gives usThe bottom gauge pressure is 7.4 kPa. Pressure is continuous at the interface, but its depth gradient changes with density.
Check the result another way
Replacing the oil by water would give 8.0 kPa. The lower-density oil reduces the pressure by (1000 − 800) × 10 × 0.30 = 600 Pa, giving the same 7.4 kPa.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Add the pressure contributions of different layers Problem: An open vessel contains 0.30 m of oil of density 800 kg/m³ above 0.50 m of water of density 1000 kg/m³. Take g = 10 m/s². Find the bottom gauge pressure, neglecting surface-tension effects. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: Follow pressure continuously through a manometer Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Pressure, floating and fluid support
A hydraulic press trades displacement for force
An additional pressure applied to a confined static fluid is transmitted through it. This is Pascal’s principle. It concerns a pressure increment; hydrostatic pressure can still differ at different heights. In the simple press model, piston-height differences, piston weight, leakage and friction are neglected or separately accounted for.
If the same pressure increment acts on piston areas A₁ and A₂, the output-to-input force ratio is A₂/A₁. An incompressible fluid also requires equal displaced volumes: A₁d₁ = A₂d₂. The larger force therefore acts through a smaller displacement.
Multiplying those relations shows that ideal input and output works are equal. The device amplifies force but does not create energy. Real devices need more input work because of losses and may store energy in compressibility or elastic deformation.
WORKED EXAMPLE
Check a force multiplier using work
An ideal hydraulic press has piston areas 2 cm² and 50 cm². A 40 N input force moves the smaller piston by 10 cm. Neglect height changes, weights and losses. Find the output force and output displacement.
Use the common pressure increment for force and volume conservation for displacement.
The area ratio is 25.
The output displacement is reduced by that ratio.
What this gives usThe output force is 1000 N through 0.40 cm. The input moves farther to supply that amplified force.
Check the result another way
Input work is 40 × 0.10 = 4 J; output work is 1000 × 0.0040 = 4 J, consistent with the ideal energy balance.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Check a force multiplier using work Problem: An ideal hydraulic press has piston areas 2 cm² and 50 cm². A 40 N input force moves the smaller piston by 10 cm. Neglect height changes, weights and losses. Find the output force and output displacement. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: A hydraulic press trades displacement for force Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Pressure, floating and fluid support
A varying pressure needs an area integral
Pressure grows with depth, so a submerged vertical wall does not experience a uniform load. If atmospheric pressure acts on both sides and the liquid surface is at the top edge, the net pressure at depth h is ρgh. A horizontal strip of width b and thickness dh experiences force dF = ρghb dh.
Integrating from the surface to depth H gives F = ½ρgbH². The resultant does not act at the geometric mid-height because deeper strips carry larger forces. Its moment about the top edge gives the centre-of-pressure depth 2H/3 for this rectangular, surface-starting case.
The result changes if the top edge is submerged below the free surface, if the opposite side also contains liquid or if the wall is curved. Return to the pressure difference and actual area geometry rather than treating 2H/3 as a universal location.
WORKED EXAMPLE
Find both the load and where it acts
A vertical rectangular gate is 2 m wide and extends from a water surface to depth 3 m. Air at the same atmospheric pressure acts on the other side. Take water density 1000 kg/m³ and g = 10 m/s². Find the net water-pressure force and its depth of action.
Integrate the triangular gauge-pressure distribution and compare its moment with a single equivalent force.
Integrate force over horizontal strips.
The moment integral weights each strip by its depth.
Divide moment by force to locate the resultant.
What this gives usThe net force is 90 kN and acts 2 m below the surface. The gate’s own weight and hinge reactions would enter a separate structural balance.
Check the result another way
Average gauge pressure is half the bottom value: 15 kPa. Multiplying by gate area 6 m² gives 90 kN. The force lies below the 1.5 m geometric centre, consistent with greater pressure at depth.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Find both the load and where it acts Problem: A vertical rectangular gate is 2 m wide and extends from a water surface to depth 3 m. Air at the same atmospheric pressure acts on the other side. Take water density 1000 kg/m³ and g = 10 m/s². Find the net water-pressure force and its depth of action. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: A varying pressure needs an area integral Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Pressure, floating and fluid support
The pressure imbalance supplies an upward resultant
A submerged body experiences pressure over its wetted surface. Deeper fluid generally presses more strongly than shallower fluid, producing an upward resultant. To find that resultant for a body surrounded by hydrostatic fluid, imagine replacing the body’s submerged region with the same fluid that would otherwise occupy it.
That imagined fluid parcel would be in equilibrium: the surrounding pressure forces would balance its weight. Replacing it by the object does not change the surrounding hydrostatic pressure distribution in the ideal calculation. The same resultant pressure force therefore equals the weight of the displaced fluid. This is Archimedes’ principle.
For an incompressible fluid of uniform density, B = ρfluid g Vsubmerged. The density belongs to the fluid, and the volume is the displaced submerged volume. A partly floating object does not displace its full volume. Surface tension, moving-fluid effects or a sealed contact excluding fluid beneath the object can require a different pressure-force calculation.
The body’s own mass determines its weight, not the displaced-fluid density in the buoyancy expression.
WORKED EXAMPLE
Deeper pressure need not mean greater buoyancy
A rigid object of volume 500 cm³ is completely submerged in water of density 1000 kg/m³, with fluid surrounding it on all sides. Use g = 10 m/s². Find its buoyant force and state whether moving it deeper changes that value in the incompressible model.
Convert the full submerged volume and multiply by fluid density and gravity.
Convert cubic centimetres to cubic metres.
Use the displaced water’s weight.
What this gives usThe buoyant force is 5.0 N and stays the same when the rigid object moves deeper in this constant-density model. Absolute pressures grow, but their resultant difference depends on the unchanged displaced volume.
Check the result another way
The displaced water mass is 0.50 kg and its weight is 5.0 N. A compressible object or fluid could change volume or density with depth, so the conclusion has stated limits.
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Worked-example topic: Deeper pressure need not mean greater buoyancy Problem: A rigid object of volume 500 cm³ is completely submerged in water of density 1000 kg/m³, with fluid surrounding it on all sides. Use g = 10 m/s². Find its buoyant force and state whether moving it deeper changes that value in the incompressible model. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
STOP & THINK
Which density belongs in B = ρgVsub for an object immersed in a uniform liquid?
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Topic: Pressure, floating and fluid support / The pressure imbalance supplies an upward resultant Problem: Which density belongs in B = ρgVsub for an object immersed in a uniform liquid? Choices: 1. The object’s density 2. The liquid’s density 3. The average of the two densities Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support
Focus: The pressure imbalance supplies an upward resultant
Learning goals:
- Distinguish local pressure from net force across a surface.
- Derive the hydrostatic gradient and state its density assumptions.
- Use gauge/absolute references and follow layered manometer columns.
- Check hydraulic force multiplication against displaced volume and work.
- Derive buoyancy from pressure and use the displaced-fluid volume.
- Interpret floating and apparent weight with explicit system boundaries.
- Test floating geometry, load margins and water-level changes with a before-and-after volume ledger.
Equations in this section (LaTeX):
B=\rho_f gV_{\mathrm{sub}},\qquad\mathbf B\text{ points opposite gravity in the static uniform-fluid model}
Meaning: The body’s own mass determines its weight, not the displaced-fluid density in the buoyancy expression.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.08 / Pressure, floating and fluid support
A free floating body adjusts its displaced volume
A body freely floating at rest, with weight and buoyancy as its only vertical forces and negligible air buoyancy and surface tension, satisfies B = mg. It displaces a mass of liquid equal to its own mass. A hollow sealed vessel can float even when its solid material is denser than water because its mass is spread over a larger enclosed volume.
For a uniform solid body of density ρbody and total volume V, vertical equilibrium gives Vsub/V = ρbody/ρfluid if the ratio is less than one. If the body is fully submerged and has greater average density than the fluid, buoyancy alone cannot support it; an additional upward force is needed to keep it at rest.
The vertical force condition does not by itself guarantee stability against tilting. The positions of the centre of mass and the shifting centre of buoyancy matter for rotational stability. In simple displacement questions we assume a stable orientation and no flooding; a boat’s loading limit may be set by stability or freeboard before its entire volume is submerged.
WORKED EXAMPLE
Find how much volume is below the surface
A uniform block of density 600 kg/m³ and volume 3.0 × 10⁻³ m³ floats freely in water of density 1000 kg/m³. Assume a stable orientation, no absorption and negligible air buoyancy. Find its submerged volume and buoyant force using g = 10 m/s².
Use vertical equilibrium to equate displaced-water mass with block mass.
The submerged fraction is the density ratio.
Calculate the submerged volume.
The block mass and supported weight follow directly.
What this gives usSixty percent of the block’s volume is submerged, displacing 1.8 kg of water. Buoyancy is 18 N.
Check the result another way
Directly calculating ρwater g Vsub gives 1000 × 10 × 0.0018 = 18 N, matching the block’s weight.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Find how much volume is below the surface Problem: A uniform block of density 600 kg/m³ and volume 3.0 × 10⁻³ m³ floats freely in water of density 1000 kg/m³. Assume a stable orientation, no absorption and negligible air buoyancy. Find its submerged volume and buoyant force using g = 10 m/s². Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: A free floating body adjusts its displaced volume Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
09 / Pressure, floating and fluid support
A support force changes when buoyancy shares the load
For a dense body suspended at rest from an external spring balance and immersed in a static liquid, the upward tension and buoyancy together balance its weight. Thus T = mg − B. The spring balance reads this support force, often called apparent weight in the liquid.
If the object is less dense than the liquid, full submersion can require a downward tether rather than an upward support. The downward force then has magnitude B − mg. A negative result for an assumed upward tension signals that the chosen support direction or contact model is wrong, not that a string can push.
Comparing air and submerged support readings can determine a body’s volume and mean density when buoyancy in air is negligible and the object neither absorbs liquid nor traps extra bubbles. The reading difference is the buoyant force. It is not a change in the object’s mass.
WORKED EXAMPLE
Infer mean density from two support readings
An object reads 10 N on a spring balance in air and 6 N when fully submerged in water of density 1000 kg/m³. Take g = 10 m/s², neglect air buoyancy and assume no bubbles or absorption. Find the object’s volume and mean density.
The 4 N reduction is buoyancy. Use it to find displaced volume, then combine with the mass from the air reading.
Calculate the buoyant force from the support difference.
Solve B = ρwater g V for the fully submerged volume.
The mass remains 1 kg in both measurements.
What this gives usThe volume is 400 cm³ and mean density is 2500 kg/m³. The balance reduction reflects fluid support, not lost mass.
Check the result another way
The relative density formula gives ρbody/ρwater = 10/(10 − 6) = 2.5, agreeing with 2500/1000.
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Worked-example topic: Infer mean density from two support readings Problem: An object reads 10 N on a spring balance in air and 6 N when fully submerged in water of density 1000 kg/m³. Take g = 10 m/s², neglect air buoyancy and assume no bubbles or absorption. Find the object’s volume and mean density. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: A support force changes when buoyancy shares the load Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
10 / Pressure, floating and fluid support
The liquid receives the opposite force
If an object is suspended from a stand outside a beaker’s balance and lowered into the liquid without touching the beaker, the liquid pushes the object upward. The object pushes the liquid downward by the equal reaction. The beaker balance must support that extra downward load.
With no overflow and the object stationary, the increase in the beaker’s scale reading equals the buoyant force, while the external suspension reading decreases by that same amount. The sum of the two support readings accounts for the same total weight as before immersion.
If the object is instead supported entirely within the beaker-and-object system, internal buoyancy forces cancel when that whole system is considered. The scale then supports the total weight of the included contents, subject to any overflow or other external forces. Always draw the boundary around the system actually resting on the balance.
WORKED EXAMPLE
Track the force between the object and the water
A 1 kg object of volume 500 cm³ is held stationary fully under water by a string attached to an external stand. The water density is 1000 kg/m³, g = 10 m/s², and no water spills. The object does not touch the beaker. Find the string tension and the increase in the beaker’s balance reading.
Use the object’s force balance first, then Newton’s third law for its interaction with the fluid.
The buoyant force is the weight of 0.50 kg of displaced water.
The string supports the rest of the object’s 10 N weight.
The fluid receives the opposite downward reaction.
What this gives usThe string tension is 5 N and the beaker balance reading increases by 5 N. Together those supports still carry the object’s full 10 N weight.
Check the result another way
Before immersion the external string supports 10 N. Afterwards it supports 5 N while the beaker gains 5 N. The redistributed support is consistent with the unchanged total weight.
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Worked-example topic: Track the force between the object and the water Problem: A 1 kg object of volume 500 cm³ is held stationary fully under water by a string attached to an external stand. The water density is 1000 kg/m³, g = 10 m/s², and no water spills. The object does not touch the beaker. Find the string tension and the increase in the beaker’s balance reading. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
STOP & THINK
Why can the simple displaced-volume buoyancy formula fail for an object making a watertight seal against a tank bottom?
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Topic: Pressure, floating and fluid support / The liquid receives the opposite force Problem: Why can the simple displaced-volume buoyancy formula fail for an object making a watertight seal against a tank bottom? Choices: 1. Its mass becomes zero 2. The hydrostatic pressure field no longer acts over the assumed fully wetted surface 3. Water stops having density Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: The liquid receives the opposite force Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
11 / Pressure, floating and fluid support
From floating fraction to a useful loading limit
A floating platform is a weighing device in disguise. Add a kilogram to it and it must displace another kilogram of liquid. It usually does so by sinking a little farther, until the extra pressure support balances the added weight. The liquid has not become denser; the immersed geometry has changed.
Draft is the vertical distance from a chosen bottom to the free surface. Freeboard is the height left above that surface. For an upright rectangular platform of constant horizontal area A and height H, immersed volume is A times draft d. For a curved hull that relation changes with depth: integrate its horizontal cross-sectional area instead of treating it as constant.
A useful loading question specifies a minimum freeboard, a permitted draft or a flooding limit. “It can still just float” is a different criterion from “it retains the requested margin”. Here orientation is assumed stable and payload is dry and carried above the water. Real tipping and structural limits are additional checks.
M is unloaded mass, m is added payload and b is the required freeboard. The last expression is a vertical-support limit for the stated geometry, not a general boat safety rating.
WORKED EXAMPLE
Keep a margin above the water
An upright sealed rectangular platform has horizontal area 0.12 m², height 0.25 m and mass 18 kg. In water of density 1000 kg/m³, what dry payload can it carry while keeping at least 0.05 m above water? Assume stable orientation, no flooding, and negligible surface tension and air buoyancy.
Translate the freeboard requirement into a maximum immersed volume before balancing weights.
The permitted draft is the height minus the required freeboard.
This draft displaces enough water to support 24 kg in total.
The platform itself uses part of that support.
What this gives usThe payload limit is 6 kg for this freeboard requirement. Using the full external volume would instead predict 12 kg at zero freeboard, a different condition.
Check the result another way
Unloaded draft is 18/(1000 × 0.12) = 0.15 m. The extra 6 kg increases it by 0.05 m, leaving the required 0.05 m above water.
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Worked-example topic: Keep a margin above the water Problem: An upright sealed rectangular platform has horizontal area 0.12 m², height 0.25 m and mass 18 kg. In water of density 1000 kg/m³, what dry payload can it carry while keeping at least 0.05 m above water? Assume stable orientation, no flooding, and negligible surface tension and air buoyancy. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
A density reading is a volume change
A floating density instrument has mass 24 g. Its cylindrical stem has area 2.0 cm² across both waterlines. It moves from a liquid of density 1000 kg/m³ to one of density 800 kg/m³. How much farther does it sink? Assume it remains upright, only the stem crosses the surfaces, no liquid is absorbed and surface tension is negligible.
Find the two displaced volumes from the unchanged mass, then convert their difference into a stem length.
In the first liquid, the required volume is 24 cm³.
In the second liquid, the same mass needs 30 cm³ of displacement.
Only the stem supplies the extra 6 cm³.
What this gives usIt sinks 3 cm farther in the less dense liquid. The buoyant force stays equal to its weight while the displaced volume increases.
Check the result another way
Writing d’s change directly as (M/A)(1/ρ₂ − 1/ρ₁) gives 0.030 m. The reciprocal-density dependence also explains why equal density steps need not make equally spaced stem marks.
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Worked-example topic: A density reading is a volume change Problem: A floating density instrument has mass 24 g. Its cylindrical stem has area 2.0 cm² across both waterlines. It moves from a liquid of density 1000 kg/m³ to one of density 800 kg/m³. How much farther does it sink? Assume it remains upright, only the stem crosses the surfaces, no liquid is absorbed and surface tension is negligible. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support
Focus: From floating fraction to a useful loading limit
Learning goals:
- Distinguish local pressure from net force across a surface.
- Derive the hydrostatic gradient and state its density assumptions.
- Use gauge/absolute references and follow layered manometer columns.
- Check hydraulic force multiplication against displaced volume and work.
- Derive buoyancy from pressure and use the displaced-fluid volume.
- Interpret floating and apparent weight with explicit system boundaries.
- Test floating geometry, load margins and water-level changes with a before-and-after volume ledger.
Equations in this section (LaTeX):
\rho_f A d=M+m,\qquad d\le H-b,\qquad m_{\max}=\rho_f A(H-b)-M
Meaning: M is unloaded mass, m is added payload and b is the required freeboard. The last expression is a vertical-support limit for the stated geometry, not a general boat safety rating.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.12 / Pressure, floating and fluid support
One object can displace two different liquids
Imagine a block that is denser than oil but lighter than a lower liquid. It cannot rest with its full volume in either liquid under buoyancy alone: the upper liquid gives too little support and the lower one gives too much. Straddling their interface can supply just the required combination.
Replace the volume occupied by the object with the liquids that would otherwise be there. Some replacement volume belongs to the upper liquid and some to the lower liquid. Adding the weights of those portions gives the upward pressure resultant. Averaging the two densities without weighting by their displaced volumes would lose the geometry.
For a fully immersed object, the two liquid volumes add to its total volume. For an upright prism of constant area they can also be expressed as area times the respective heights. But before using that constraint, check that the upper layer is deep enough to cover the required upper part. A force equation can be correct while the assumed geometry is impossible.
The supported mass equals the mass of both displaced liquid portions. Gravity cancels from the vertical force balance; it has not disappeared from the buoyant force.
Explain this equation
Read it in words
For an object supported only by two static liquids, add the masses of the two displaced portions. Their total equals the object’s mass.
Symbols and units
- Mass densities of the upper and lower liquids. The lower liquid is denser in this stable layering.kg/m³.
- Volumes of the object actually immersed in the upper and lower liquids, respectively.Cubic metres (m³).
- Total mass supported by the liquids, including any attached load.Kilograms (kg).
Signs and operations
Subscripts u and ℓ label upper and lower liquids. Both density–volume products are positive masses. The plus sign adds their contributions; it does not average the liquid densities.
When it applies
Use vertical equilibrium, constant liquid densities, negligible air buoyancy and surface tension, and no other vertical support. If part of the object is in air, its two liquid-immersed volumes add to less than its total volume. This force balance alone does not test stability against tilting.
Make it concrete
For 100 cm³ immersed in an 800 kg/m³ liquid and 100 cm³ in a 1200 kg/m³ liquid, the supported mass is 0.080 + 0.120 = 0.200 kg. At g = 10 m/s² the total upward force is 2.0 N.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Equation (LaTeX): \rho_uV_u+\rho_\ell V_\ell=M Read it as: For an object supported only by two static liquids, add the masses of the two displaced portions. Their total equals the object’s mass. Symbols: \rho_u,\ \rho_\ell: Mass densities of the upper and lower liquids. The lower liquid is denser in this stable layering.; kg/m³. V_u,\ V_\ell: Volumes of the object actually immersed in the upper and lower liquids, respectively.; Cubic metres (m³). M: Total mass supported by the liquids, including any attached load.; Kilograms (kg). Signs and operations: Subscripts u and ℓ label upper and lower liquids. Both density–volume products are positive masses. The plus sign adds their contributions; it does not average the liquid densities. Conditions: Use vertical equilibrium, constant liquid densities, negligible air buoyancy and surface tension, and no other vertical support. If part of the object is in air, its two liquid-immersed volumes add to less than its total volume. This force balance alone does not test stability against tilting. Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
Let the density decide the division
An upright uniform prism is 20 cm tall with density 1000 kg/m³. It straddles two immiscible liquids of densities 800 kg/m³ above and 1200 kg/m³ below. The upper layer is 14 cm deep after equilibrium. Find the prism’s height in the lower liquid. Assume stable orientation, no contact, negligible surface tension and uniform gravity.
First assume full immersion. Use h in the lower liquid and H − h in the upper, then check the available upper-layer depth.
Cancel the common area and g from the vertical force balance.
The denser lower liquid supplies the extra support missing from an all-upper-liquid placement.
The upper part also occupies 10 cm, less than the available 14 cm.
What this gives usTen centimetres lies in each liquid; the prism’s top is 4 cm below the upper free surface. The full-immersion assumption is therefore consistent.
Check the result another way
Equal displaced volumes weight the two densities equally, giving an effective displaced-fluid density of (800 + 1200)/2 = 1000 kg/m³, matching this prism’s density.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Let the density decide the division Problem: An upright uniform prism is 20 cm tall with density 1000 kg/m³. It straddles two immiscible liquids of densities 800 kg/m³ above and 1200 kg/m³ below. The upper layer is 14 cm deep after equilibrium. Find the prism’s height in the lower liquid. Assume stable orientation, no contact, negligible surface tension and uniform gravity. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
Catch an impossible geometry before trusting the answer
Keep the same 20 cm prism and densities, but now the upper layer is only 4 cm deep in the final equilibrium. Find the height emerging into air. Assume the lower liquid is deep, the prism is upright and stable, and air buoyancy and surface tension are negligible.
The fully immersed result needs 10 cm of upper liquid, which is unavailable. The prism crosses the whole 4 cm upper layer, leaving a lower part h and possibly an exposed part.
Use the actual upper-liquid height, not H − h.
Solve for the lower-liquid portion.
Subtract both immersed lengths from the full height.
What this gives usThe prism protrudes 2 cm into air. Reusing the deep-interface split would hide 6 cm of liquid that is not actually present.
Check the result another way
Per square metre of prism area, the displaced masses are 32 kg above and 168 kg below, totalling 200 kg, equal to the prism’s mass per square metre. All three heights are nonnegative and sum to 20 cm.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Catch an impossible geometry before trusting the answer Problem: Keep the same 20 cm prism and densities, but now the upper layer is only 4 cm deep in the final equilibrium. Find the height emerging into air. Assume the lower liquid is deep, the prism is upright and stable, and air buoyancy and surface tension are negligible. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: One object can displace two different liquids Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): \rho_uV_u+\rho_\ell V_\ell=M Meaning: The supported mass equals the mass of both displaced liquid portions. Gravity cancels from the vertical force balance; it has not disappeared from the buoyant force. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
13 / Pressure, floating and fluid support
Compare displaced volume before and after
When something moves between a floating carrier and the liquid, total mass alone does not determine the new water level. While carried, its weight makes the carrier displace liquid. After sinking, the object occupies its own volume and the bottom supplies the support that buoyancy cannot. The same mass can therefore exclude different volumes of water.
Write a before-and-after ledger of what occupies space below the free surface. Keep the liquid amount fixed unless melting, pouring, spilling or absorption changes it. In a vertical-sided tank with horizontal area A_t, the change in the total volume below the surface is A_t times the level change. Do not subtract the floating carrier’s waterline area: its change of draft has already been included in its displaced-volume change.
These are settled-state comparisons. Ignore splashing and transient waves, state whether the carrier remains afloat, and check whether a material changes phase or density. A familiar slogan about melting ice can fail when the surrounding liquid is different.
WORKED EXAMPLE
A sinking cargo can lower the water
A 0.48 kg metal piece of density 8000 kg/m³ is initially carried dry on a floating platform. It is transferred into water of density 1000 kg/m³ and sinks. In a vertical-sided tank of area 0.14 m², find the settled water-level change. The platform stays afloat, no water spills, no air is trapped, and the metal does not dissolve.
Compare the displacement attributable to the metal: weight-equivalent volume while carried, physical volume after sinking.
The platform initially displaces extra water equal in mass to the metal.
After the transfer, the submerged metal occupies its own smaller volume.
Use the signed difference to find the level change.
What this gives usThe level falls by 3.0 mm. The metal still has the same weight, but the bottom now carries the portion that its buoyancy cannot support.
Check the result another way
The lost displacement is 420 cm³. Spread over a tank area of 1400 cm², that corresponds to a drop of 0.30 cm. If the object instead floated freely, its displacement would still equal its mass divided by liquid density and this change would vanish.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: A sinking cargo can lower the water Problem: A 0.48 kg metal piece of density 8000 kg/m³ is initially carried dry on a floating platform. It is transferred into water of density 1000 kg/m³ and sinks. In a vertical-sided tank of area 0.14 m², find the settled water-level change. The platform stays afloat, no water spills, no air is trapped, and the metal does not dissolve. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
Melting depends on which liquid supports the ice
A 0.12 kg piece of pure ice floats in a liquid of density 1200 kg/m³. It melts into water of density 1000 kg/m³, which forms an unmixed upper layer during the comparison. The vertical-sided vessel area is 0.020 m². Find the change of the top free-surface level. Neglect evaporation, thermal expansion of the supporting liquid, trapped gas and volume change from mixing.
Compare the support liquid displaced by the floating ice with the liquid volume supplied by its melted mass.
The floating ice initially displaces 100 cm³ of the denser supporting liquid.
Melting supplies 120 cm³ of water; the ice’s original geometric volume is not needed.
The extra 20 cm³ raises the upper free surface.
What this gives usThe top level rises by 1.0 mm in this model. For floating pure ice in water of the same final water density, the two volumes would be equal and the level would stay unchanged.
Check the result another way
The sign follows without arithmetic: a given mass occupies more liquid volume at 1000 kg/m³ than the volume it displaces while floating in a 1200 kg/m³ liquid. Mixing or temperature changes require their own volume information.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Melting depends on which liquid supports the ice Problem: A 0.12 kg piece of pure ice floats in a liquid of density 1200 kg/m³. It melts into water of density 1000 kg/m³, which forms an unmixed upper layer during the comparison. The vertical-sided vessel area is 0.020 m². Find the change of the top free-surface level. Neglect evaporation, thermal expansion of the supporting liquid, trapped gas and volume change from mixing. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Focus: Compare displaced volume before and after Learning goals: - Distinguish local pressure from net force across a surface. - Derive the hydrostatic gradient and state its density assumptions. - Use gauge/absolute references and follow layered manometer columns. - Check hydraulic force multiplication against displaced volume and work. - Derive buoyancy from pressure and use the displaced-fluid volume. - Interpret floating and apparent weight with explicit system boundaries. - Test floating geometry, load margins and water-level changes with a before-and-after volume ledger. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Static pressure acts normally and its force must be summed over a surface.
- For constant density, pressure increases by ρg times vertical depth.
- Gauge pressure is measured relative to ambient; absolute pressure is referenced to vacuum.
- Hydraulic force multiplication trades away displacement in the ideal device.
- Buoyancy equals displaced-fluid weight under the hydrostatic surrounding-fluid conditions.
- Scale and tension readings depend on which forces and bodies their supports carry.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A uniform body of density 750 kg/m³ floats freely in a liquid of density 1000 kg/m³. What percentage of its volume is submerged, neglecting air buoyancy and surface tension?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A uniform body of density 750 kg/m³ floats freely in a liquid of density 1000 kg/m³. What percentage of its volume is submerged, neglecting air buoyancy and surface tension? Requested answer unit: % Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 2
A gas has absolute pressure 85 kPa while the surrounding atmosphere is 100 kPa. What is its gauge pressure?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A gas has absolute pressure 85 kPa while the surrounding atmosphere is 100 kPa. What is its gauge pressure? Requested answer unit: kPa Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 3
A rigid fully submerged object is moved deeper in an incompressible liquid of uniform density. Under hydrostatic conditions with no contact, what happens to buoyancy?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A rigid fully submerged object is moved deeper in an incompressible liquid of uniform density. Under hydrostatic conditions with no contact, what happens to buoyancy? Choices: 1. It increases in proportion to absolute pressure 2. It remains unchanged 3. It becomes zero Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 4
An ideal hydraulic press multiplies force by 20. How does output displacement compare with input displacement?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: An ideal hydraulic press multiplies force by 20. How does output displacement compare with input displacement? Choices: 1. It is 20 times larger 2. It is one twentieth as large 3. It is unchanged Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 5
A tank wall has atmospheric air on one side and water open to the same atmosphere on the other. Which pressure gives the net water load at depth h?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A tank wall has atmospheric air on one side and water open to the same atmosphere on the other. Which pressure gives the net water load at depth h? Choices: 1. ρgh 2. Atmospheric pressure alone 3. Atmospheric pressure plus ρgh without subtracting the other side Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 6
A uniform object of density 950 kg/m³ is fully immersed across deep layers of density 750 kg/m³ above and 1250 kg/m³ below. It floats without other vertical forces. What percentage of its volume is in the lower liquid?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A uniform object of density 950 kg/m³ is fully immersed across deep layers of density 750 kg/m³ above and 1250 kg/m³ below. It floats without other vertical forces. What percentage of its volume is in the lower liquid? Requested answer unit: % Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 7
A 0.30 kg piece of density 6000 kg/m³ moves from a floating carrier into water of density 1000 kg/m³ and sinks. The carrier remains afloat in a vertical-sided 0.10 m² tank. No spill, trapped air or dissolution occurs. What is the water-level change in mm, positive upward?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: A 0.30 kg piece of density 6000 kg/m³ moves from a floating carrier into water of density 1000 kg/m³ and sinks. The carrier remains afloat in a vertical-sided 0.10 m² tank. No spill, trapped air or dissolution occurs. What is the water-level change in mm, positive upward? Requested answer unit: mm Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 8
An upright prism’s fully immersed calculation requires 8 cm of its height in an upper liquid, but that layer is only 3 cm deep at equilibrium. What should you do next?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Pressure, floating and fluid support Problem: An upright prism’s fully immersed calculation requires 8 cm of its height in an upper liquid, but that layer is only 3 cm deep at equilibrium. What should you do next? Choices: 1. Keep the answer because the force equation balances 2. Reconsider the geometry and include an exposed portion if needed 3. Replace the upper liquid density by the lower liquid density Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Pressure added by a water columnIntermediate · Free sample · Hints and guided reasoningSeparate depth from curvatureIntermediate · Hints and guided reasoningRead the sign at a depressed meniscusIntermediate · Hints and guided reasoningKeep room above the waterIntermediate · Hints and guided reasoningThe same weight needs more volumeIntermediate · Hints and guided reasoningShare support between two liquidsIntermediate · Hints and guided reasoningCheck the layer before the fractionAdvanced · Hints and guided reasoningSame mass, different displaced volumeAdvanced · Hints and guided reasoningTest the melting shortcutAdvanced · Hints and guided reasoningWhat is the net force on a submerged block?Beginner · Hints and guided reasoningTwo liquid layers add pressureIntermediate · Hints and guided reasoningHow much of the block is submerged?Intermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Explain buoyancy using pressure rather than quoting the displaced-fluid rule.
Compare your reasoning
Pressure generally increases with depth in a resting fluid. The upward force on lower surfaces exceeds the downward contribution on higher surfaces. Integrating those pressure forces gives the buoyant force.
Revisit the explanation →A few days later: change the situation
A floating block is moved from fresh water into a denser liquid. What happens to the displaced volume, if the block still floats?
Compare your reasoning
Its weight is unchanged, so the buoyant force must be unchanged. A denser liquid needs a smaller displaced volume to supply that force; less of the block is submerged.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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