Work-energy theorem | PhysicsRank

You can lift a bag, hold it still, or carry it horizontally. All three can feel tiring, but the work done by a particular force depends on displacement in that force’s direction. Energy accounting starts by naming exactly what moves and which interaction transfers energy.

We will derive the work–energy theorem, build gravitational and spring potential energies, state the conditions for mechanical-energy conservation and distinguish energy from power. Potential-energy graphs then bring force, accessibility and stability into one picture.

What you will learn to do

Calculate signed work for constant and variable forces.
Apply net work to a kinetic-energy change.
Define potential-energy differences and choose a consistent zero.
Account for conservative, dissipative and driven transfers.
Derive Hooke-law spring energy and distinguish turning points from equilibrium.
Use power as an energy-transfer rate.
Read force, turning points and stability from potential graphs.

Useful before this chapterRates and accumulation: calculus for physics →Connected bodies, sliding friction and accelerating frames →Forces that turn a moving object →

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: Work, stored energy and power
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.

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.

Can a force act while doing zero work on a particle?

Compare your reasoning

Yes. Zero displacement gives zero work, and so does a force perpendicular to the instantaneous displacement. Force and energy transfer are related, but are not interchangeable quantities.

Revisit the explanation →

01 / Work, stored energy and power

Work measures force acting through a displacement

Holding a heavy object still can feel tiring, but the object’s displacement is zero and the holding force does no mechanical work on it during that stationary interval. Work in mechanics is a particular energy-transfer quantity. It combines a force with the displacement of its point of application, not simply with elapsed time or personal effort.

For a particle under a constant force, W = F·Δr = FΔr cos θ. A force component along the displacement contributes positive work; an opposite component contributes negative work; a perpendicular component contributes zero. The signs tell you whether that force adds to or subtracts from the particle’s kinetic energy as part of the net force.

For a curved path or variable force, divide the path into small displacements and add F·dr. The path integral retains both the changing force and the changing direction. Replacing a variable force by its final value usually loses the accumulated history. Use a force–position area only when the plotted force is the component along that position coordinate.

W=\int\mathbf F\cdot d\mathbf r,\qquad[W]=\mathrm{N\,m}=\mathrm J

Work is a scalar. It can be negative even though force magnitude and travelled distance are nonnegative.

Explain this equation

Read it in words

Work adds the component of a force along each tiny displacement of the point where that force acts.

Symbols and units

$W$: Work done by the specified force along the specified path.Joules (J).
$\mathbf F$: The force vector, which may change along the route.Newtons (N).
$d\mathbf r$: Tiny displacement of the force’s point of application.Metres (m).
$\int,\ \cdot$: The integral sums contributions; the dot product selects the force component along the displacement.Together the product contributes N·m = J.
$[W]$: Square brackets here identify the dimensions/units of work, not another variable.1 J = 1 N·m.

Signs and operations

A force component along motion contributes positive work; an opposing component contributes negative work. A perpendicular force contributes zero instantaneous work even if its magnitude is large.

When it applies

For a particle, use its actual path. For an extended body, use the actual displacement of the point of application; replacing it blindly by centre-of-mass displacement can miss energy transfers. Constant-force W = Fd cosθ is a special case.

Make it concrete

A constant 10 N force at 60° to a 2 m straight displacement does 10 J of work, because only 5 N acts along that displacement. Doubling a perpendicular component would not change this work.

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.

Equation (LaTeX): W=\int\mathbf F\cdot d\mathbf r,\qquad[W]=\mathrm{N\,m}=\mathrm J
Read it as: Work adds the component of a force along each tiny displacement of the point where that force acts.
Symbols:
W: Work done by the specified force along the specified path.; Joules (J).
\mathbf F: The force vector, which may change along the route.; Newtons (N).
d\mathbf r: Tiny displacement of the force’s point of application.; Metres (m).
\int,\ \cdot: The integral sums contributions; the dot product selects the force component along the displacement.; Together the product contributes N·m = J.
[W]: Square brackets here identify the dimensions/units of work, not another variable.; 1 J = 1 N·m.
Signs and operations: A force component along motion contributes positive work; an opposing component contributes negative work. A perpendicular force contributes zero instantaneous work even if its magnitude is large.
Conditions: For a particle, use its actual path. For an extended body, use the actual displacement of the point of application; replacing it blindly by centre-of-mass displacement can miss energy transfers. Constant-force W = Fd cosθ is a special case.

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

Only the force component along the motion contributes

A 10 N force acts at 60° above the horizontal while a particle moves 4 m horizontally to the right. Find the work done by this force.

Choose an approach

Project the force onto the actual displacement. The vertical component has no vertical displacement to act through.

The horizontal force component is 10 cos 60°.
$F_x=5\ \mathrm N$
Multiply this component by the horizontal displacement.
$W=F_x\Delta x=5(4)=20\ \mathrm J$

What this gives usThe force does +20 J of work. Its vertical component contributes zero during this horizontal displacement.

Check the result another way

The full product FΔr is 40 J, an upper bound attained only for perfect alignment. The cosine factor halves it, consistent with the 60° geometry.

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: Only the force component along the motion contributes
Problem: A 10 N force acts at 60° above the horizontal while a particle moves 4 m horizontally to the right. Find the work done by this force.

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

A particle moves at constant speed in a circle under a force directed exactly toward the centre. What work does this force do during the motion?

Positive, because the particle movesZero, because the force is perpendicular to each small displacementNegative, because the force points inward

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: Work, stored energy and power / Work measures force acting through a displacement
Problem: A particle moves at constant speed in a circle under a force directed exactly toward the centre. What work does this force do during the motion?
Choices:
1. Positive, because the particle moves
2. Zero, because the force is perpendicular to each small displacement
3. Negative, because the force points inward

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: Work, stored energy and power
Focus: Work measures force acting through a displacement
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
W=\int\mathbf F\cdot d\mathbf r,\qquad[W]=\mathrm{N\,m}=\mathrm J
Meaning: Work is a scalar. It can be negative even though force magnitude and travelled distance are nonnegative.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

02 / Work, stored energy and power

Derive the kinetic-energy change from Newton’s law

For a constant-mass particle in an inertial frame, dot Newton’s equation Fnet = m dv/dt with velocity v. The left side is the rate at which net work is done. The right side is the derivative of ½mv². Integrating over the motion gives the work–kinetic-energy theorem.

This theorem uses net work: add the work done by every included force on the particle. Work by one applied force equals ΔK only if all other forces contribute zero net work over that interval. A rough surface, gravity along a slope or a spring can change the balance.

Kinetic energy depends on speed squared. An energy equation can determine a speed magnitude but generally cannot determine the velocity direction or the travel time by itself. When direction, a force component or a time history is requested, combine energy with geometry, momentum or Newton’s equations.

\mathbf F_{\mathrm{net}}\cdot\mathbf v=\frac{d}{dt}\left(\frac12mv^2\right)\quad\Longrightarrow\quad W_{\mathrm{net}}=K_f-K_i

The derivation assumes a constant-mass particle in an inertial frame. Extended deformable systems require additional internal-energy accounting.

WORKED EXAMPLE

Applied work and net work are different totals

A 2 kg block initially moves at 3 m/s on a horizontal track. Over a 4 m displacement, an applied force does +30 J of work and friction does −14 J. Vertical forces do no work. Find the final speed.

Choose an approach

Sum the works before applying ΔK = Wnet. Include the initial kinetic energy.

The total work is the signed sum.
$W_{\mathrm{net}}=30-14=16\ \mathrm J$
The starting kinetic energy is nonzero.
$K_i=\frac12(2)(3^2)=9\ \mathrm J$
Add the kinetic-energy change and solve for speed.
$K_f=9+16=25\ \mathrm J=\frac12(2)v_f^2\quad\Longrightarrow\quad v_f=5\ \mathrm{m/s}$

What this gives usThe final speed is 5 m/s. The applied force transferred 30 J, but only 16 J appeared as an increase of the block’s kinetic energy.

Check the result another way

The kinetic-energy difference is 25 − 9 = 16 J. Equating the applied 30 J alone to ΔK would omit the frictional transfer.

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: Applied work and net work are different totals
Problem: A 2 kg block initially moves at 3 m/s on a horizontal track. Over a 4 m displacement, an applied force does +30 J of work and friction does −14 J. Vertical forces do no work. Find the final speed.

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: Work, stored energy and power
Focus: Derive the kinetic-energy change from Newton’s law
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
\mathbf F_{\mathrm{net}}\cdot\mathbf v=\frac{d}{dt}\left(\frac12mv^2\right)\quad\Longrightarrow\quad W_{\mathrm{net}}=K_f-K_i
Meaning: The derivation assumes a constant-mass particle in an inertial frame. Extended deformable systems require additional internal-energy accounting.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Work, stored energy and power

Accumulate a varying force over the actual path

A linearly changing force has a trapezoidal area under its force–position graph over a finite interval. Its average over position is the arithmetic mean of its endpoint values only because the graph is straight. A general curved force law requires an integral or a suitable numerical estimate.

A signed force–position graph can have positive and negative areas. Their sum is the work for motion in the positive coordinate direction. Reversing the path reverses dx, but the force itself may also change if it depends on the direction of motion, as friction does. Therefore you must reconsider the interaction before simply reversing a previously calculated work.

Work–energy is especially efficient when force varies with position and time is not needed. It avoids solving the entire differential equation of motion. A later check should still ask whether the calculated kinetic energy stays nonnegative and whether the particle can reach every part of the assumed path.

WORKED EXAMPLE

A growing force over a finite interval

A 2 kg particle moves from x = 1 m to x = 3 m along a smooth track under the only horizontal force Fₓ = (4 N/m)x. Its speed at x = 1 m is 2 m/s. Find the final speed.

Choose an approach

Integrate the position-dependent force and add its work to the initial kinetic energy.

Integrate over the given endpoints, not from the origin.
$W=\int_1^3 4x\,dx=[2x^2]_1^3=16\ \mathrm J$
The starting kinetic energy is 4 J.
$K_f=4+16=20\ \mathrm J$
Convert the resulting kinetic energy to the speed magnitude.
$20=\frac12(2)v_f^2\quad\Longrightarrow\quad v_f=\sqrt{20}\ \mathrm{m/s}$

What this gives usThe final speed is √20 m/s, approximately 4.47 m/s. The outward force is already nonzero at the starting position.

Check the result another way

The force rises from 4 N to 12 N over 2 m. The trapezium area is (4 + 12)/2 × 2 = 16 J, independently checking the integral.

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 growing force over a finite interval
Problem: A 2 kg particle moves from x = 1 m to x = 3 m along a smooth track under the only horizontal force Fₓ = (4 N/m)x. Its speed at x = 1 m is 2 m/s. Find the final speed.

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

A constant net force of 3 N acts along a particle’s 5 m displacement. By how many joules does its kinetic energy increase?

Your answer (J)Use a decimal, a fraction such as 1/3, or scientific notation.

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: Work, stored energy and power / Accumulate a varying force over the actual path
Problem: A constant net force of 3 N acts along a particle’s 5 m displacement. By how many joules does its kinetic energy increase?
Requested answer unit: J

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: Work, stored energy and power
Focus: Accumulate a varying force over the actual path
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
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.

Back to chapter contents ↑

04 / Work, stored energy and power

Potential energy stores the work of a conservative interaction

A force is conservative when its work between two positions is independent of the path within the domain being considered. Equivalently, its work around any closed path in that domain is zero. For such an interaction, define a potential-energy change as the negative of its work: ΔU = −Wconservative.

The minus sign makes energy accounting convenient. If gravity does positive work while an object falls, gravitational potential energy decreases. If a spring pushes an attached block toward its natural length and does positive work, spring potential energy decreases. The lost potential energy can appear as kinetic energy when no other energy transfers intervene.

Only differences in U affect these balances. Adding a constant to every potential-energy value changes neither force nor motion. Choose a zero that simplifies the geometry, state it and keep it consistent. Potential energy belongs to the interaction of a system—such as an object and Earth—even when elementary notation writes it as the object’s U.

U_f-U_i=-\int_i^f\mathbf F_c\cdot d\mathbf r,\qquad U_g=mgy+C\quad\text{for uniform gravity}

For upward y and constant downward gravity, raising the object increases U by mgΔy. The additive constant C sets the reference.

WORKED EXAMPLE

Change a reference without changing an energy difference

A 2 kg object moves from y = 1 m to y = 4 m in uniform gravity with g = 10 m/s². Find ΔU if zero potential is chosen first at y = 0 and then at y = 2 m.

Choose an approach

Calculate endpoint values in each convention, then compare the difference.

With zero at y = 0, potential is 20y joules.
$U_i=20\ \mathrm J,\quad U_f=80\ \mathrm J,\quad\Delta U=60\ \mathrm J$
With zero at y = 2 m, potential is 20(y − 2) joules.
$U_i=-20\ \mathrm J,\quad U_f=40\ \mathrm J,\quad\Delta U=60\ \mathrm J$

What this gives usBoth conventions give the same +60 J change. A negative potential value is permitted; it reflects the chosen zero.

Check the result another way

Gravity does −mgΔy = −20(3) = −60 J in either convention, equal to −ΔU.

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: Change a reference without changing an energy difference
Problem: A 2 kg object moves from y = 1 m to y = 4 m in uniform gravity with g = 10 m/s². Find ΔU if zero potential is chosen first at y = 0 and then at y = 2 m.

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: Work, stored energy and power
Focus: Potential energy stores the work of a conservative interaction
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
U_f-U_i=-\int_i^f\mathbf F_c\cdot d\mathbf r,\qquad U_g=mgy+C\quad\text{for uniform gravity}
Meaning: For upward y and constant downward gravity, raising the object increases U by mgΔy. The additive constant C sets the reference.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Work, stored energy and power

Conservation follows from an explicit work balance

Separate the forces into conservative interactions represented by U and the remaining forces whose work is Wother. The work–energy theorem gives ΔK = −ΔU + Wother, or Δ(K + U) = Wother. Mechanical energy K + U is conserved when this remaining work is zero over the interval.

This does not require the total force to be zero. During a fall, gravity accelerates the object and transfers energy between U and K while their sum can remain constant. Conversely, a constant-speed object can receive positive applied work that is balanced by negative frictional work; its unchanged K does not imply that no energy has been transferred.

State the system and the forces before invoking conservation. A time-dependent driven spring, a motor or a sliding contact can exchange energy with the chosen system. Total energy conservation remains a broader accounting principle, but mechanical energy alone may decrease or increase.

K_i+U_i+W_{\mathrm{other}}=K_f+U_f

Use Wother only for forces or transfers not already represented in the chosen potential energy; do not count gravity or spring work twice.

WORKED EXAMPLE

A curved smooth track need not require a path-by-path solution

A small bead starts at 4 m/s and descends by 3 m along a stationary smooth track. The track remains in contact with the bead and its normal force is perpendicular to motion. Take g = 10 m/s². Find the speed after the descent.

Choose an approach

The normal does no work and gravity is conservative. Use the vertical height change rather than the track length.

Set the loss of potential energy equal to the gain in kinetic energy.
$\frac12mv_f^2=\frac12m(4^2)+mg(3)$
Cancel mass and solve for the speed squared.
$v_f^2=16+2(10)(3)=76\ \mathrm{m^2/s^2}$

What this gives usThe speed is √76 m/s, approximately 8.72 m/s. Under the stated contact and friction assumptions, only the endpoint height difference is needed for this speed calculation.

Check the result another way

The kinetic energy per unit mass rises by 38 − 8 = 30 J/kg, exactly equal to gΔh. Energy alone does not verify that a particular track shape can maintain contact; that condition requires a separate normal-force check.

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 curved smooth track need not require a path-by-path solution
Problem: A small bead starts at 4 m/s and descends by 3 m along a stationary smooth track. The track remains in contact with the bead and its normal force is perpendicular to motion. Take g = 10 m/s². Find the speed after the descent.

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: Work, stored energy and power
Focus: Conservation follows from an explicit work balance
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
K_i+U_i+W_{\mathrm{other}}=K_f+U_f
Meaning: Use Wother only for forces or transfers not already represented in the chosen potential energy; do not count gravity or spring work twice.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Work, stored energy and power

A mechanical-energy loss needs an accounted destination

For a block sliding along a fixed rough horizontal surface, kinetic friction does negative work on the block. In the simple model with constant N and μk, that work is −μkNs over a sliding distance s. The block’s mechanical energy decreases. For a broader system containing both contacting bodies, the dissipative interaction increases internal energy, and some energy may leave as sound or heat.

The expression μkNs is a positive dissipated-energy amount in this stationary-surface model. The work on one body is signed and depends on that body’s displacement in the chosen frame. On a moving belt, friction can do positive work on a block even while the contact dissipates energy. “Friction always does negative work” is therefore too broad.

For a sequence containing a rough patch and a spring, write the whole energy balance or split it at the boundary. If the block stops before reaching the spring, a calculation assuming later spring compression is not physically reachable. Check whether the energy available is enough for every intervening loss.

WORKED EXAMPLE

Check whether a rough patch can be crossed

A 2 kg block enters a 3 m horizontal rough patch at 5 m/s. The coefficient is μk = 0.20 and g = 10 m/s². No other horizontal force acts. Does it emerge, and if so with what speed?

Choose an approach

Compare the initial kinetic energy with the work required against friction over the full patch.

Calculate the initial energy and friction magnitude.
$K_i=\frac12(2)(25)=25\ \mathrm J,\qquad f_k=0.20(2)(10)=4\ \mathrm N$
The full patch would remove 12 J.
$W_f=-4(3)=-12\ \mathrm J$
The remaining kinetic energy is positive.
$K_f=25-12=13\ \mathrm J\quad\Longrightarrow\quad v_f=\sqrt{13}\ \mathrm{m/s}$

What this gives usIt emerges at √13 m/s, approximately 3.61 m/s. There is enough initial energy to cross the patch.

Check the result another way

The stopping distance on a longer identical surface would be Ki/fk = 25/4 = 6.25 m, which exceeds 3 m. This confirms reachability independently.

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 whether a rough patch can be crossed
Problem: A 2 kg block enters a 3 m horizontal rough patch at 5 m/s. The coefficient is μk = 0.20 and g = 10 m/s². No other horizontal force acts. Does it emerge, and if so with what speed?

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: Work, stored energy and power
Focus: A mechanical-energy loss needs an accounted destination
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
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.

Back to chapter contents ↑

07 / Work, stored energy and power

Derive spring energy from the growing restoring force

For an ideal Hooke-law spring, let x be the signed extension from its natural length. The spring force on an attached particle is F = −kx along the extension axis, with k > 0. The force changes with position and points back toward x = 0. Integrating the negative of this force gives U = ½kx² when zero potential is chosen at natural length.

The square means the stored elastic energy is positive for both compression and extension. The force still has opposite signs in the two cases because it is the negative derivative of U. Do not use kx² without the factor one half: the force rises linearly from zero during a quasistatic stretch, so the average applied force over the stretch is half its final value.

This model assumes the spring remains within its linear elastic range and that hysteresis, permanent deformation and spring mass are negligible for the chosen calculation. A measured force–extension curve that is not straight requires integrating the actual curve; the familiar half-k-x-squared formula is not a definition for every elastic object.

F_s=-kx,\qquad U_s(x)-U_s(0)=-\int_0^x(-kx^{\prime})\,dx^{\prime}=\frac12kx^2

The units are (N/m) × m² = N·m = J. The coordinate x is deformation from natural length, not an arbitrary position coordinate.

WORKED EXAMPLE

Stretching from an already stretched state

An ideal spring has k = 200 N/m. Calculate the increase in its elastic energy when its extension changes quasistatically from 0.10 m to 0.20 m.

Choose an approach

Calculate the difference of squared extensions. Do not treat the 0.10 m additional stretch as though it began at natural length.

Evaluate the two elastic energy values.
$U_i=\frac12(200)(0.10)^2=1\ \mathrm J,\qquad U_f=\frac12(200)(0.20)^2=4\ \mathrm J$
Subtract to obtain the change.
$\Delta U=4-1=3\ \mathrm J$

What this gives usThe spring gains 3 J. A formula using only ½k(Δx)² would give 1 J and omit the initial loading.

Check the result another way

The quasistatic applied force rises from 20 N to 40 N over 0.10 m. Its trapezium work is (20 + 40)/2 × 0.10 = 3 J, confirming the energy difference.

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: Stretching from an already stretched state
Problem: An ideal spring has k = 200 N/m. Calculate the increase in its elastic energy when its extension changes quasistatically from 0.10 m to 0.20 m.

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

An ideal spring has k = 100 N/m and is compressed by 0.20 m. What elastic energy is stored relative to natural length?

Your answer (J)Use a decimal, a fraction such as 1/3, or scientific notation.

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: Work, stored energy and power / Derive spring energy from the growing restoring force
Problem: An ideal spring has k = 100 N/m and is compressed by 0.20 m. What elastic energy is stored relative to natural length?
Requested answer unit: J

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: Work, stored energy and power
Focus: Derive spring energy from the growing restoring force
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
F_s=-kx,\qquad U_s(x)-U_s(0)=-\int_0^x(-kx^{\prime})\,dx^{\prime}=\frac12kx^2
Meaning: The units are (N/m) × m² = N·m = J. The coordinate x is deformation from natural length, not an arbitrary position coordinate.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

08 / Work, stored energy and power

Use turning points without assuming equilibrium

For a block attached to a horizontal ideal spring on a smooth fixed track, K + ½kx² is constant. At a turning point the instantaneous speed is zero, so all this mechanical energy is elastic. At x = 0 the spring energy is smallest, so the speed is greatest.

A turning point is not generally an equilibrium point. At maximum extension, the spring force has its largest magnitude and points back toward the centre. The zero velocity lasts only an instant. Equilibrium instead concerns zero resultant force, which for this horizontal unforced model occurs at x = 0.

A vertical spring adds gravitational potential energy. If extension x is measured downward from natural length, Us + Ug = ½kx² − mgx plus a constant. Completing the square gives ½k(x − mg/k)² plus another constant. The equilibrium position shifts to xeq = mg/k. Oscillation energy measured around that equilibrium uses the displacement from equilibrium, but the spring’s own elastic energy still uses extension from natural length.

WORKED EXAMPLE

An attached block slows before the turning point

A 1 kg block attached to an ideal horizontal spring of k = 100 N/m is released from rest at extension 0.30 m on a smooth track. Find its speed when the extension is 0.10 m.

Choose an approach

Compare the loss of spring energy with the gain of kinetic energy.

The initial energy is entirely elastic.
$E=\frac12(100)(0.30)^2=4.5\ \mathrm J$
At extension 0.10 m, only 0.5 J remains elastic.
$K=4.5-\frac12(100)(0.10)^2=4\ \mathrm J$
Convert kinetic energy to speed.
$v=\sqrt{\frac{2K}{m}}=\sqrt8\ \mathrm{m/s}$

What this gives usThe speed is √8 m/s, approximately 2.83 m/s. On the first passage the block moves toward decreasing extension; energy alone gives the magnitude, and the release history gives the direction.

Check the result another way

At natural length, the same balance predicts vmax = 3 m/s. The speed at 0.10 m is smaller than this maximum, as expected.

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: An attached block slows before the turning point
Problem: A 1 kg block attached to an ideal horizontal spring of k = 100 N/m is released from rest at extension 0.30 m on a smooth track. Find its speed when the extension is 0.10 m.

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.

TRY THE IDEA

Follow the energy through one oscillation

Move the phase control slowly. Find both turning points and the central passage. Compare the spring force, speed and two energy values; identify why zero speed and zero force occur at different positions.

MOVE THROUGH A CYCLESpring + cart

Where does the energy go?

1 kg cart · 100 N/m spring · 10 cm amplitude · no friction

Position in the cycle 0%

Spring energy0.500 J

Kinetic energy0.000 J

Displacement from equilibrium10.0 cm

Velocity (right positive)0.00 m/s

Spring force (right positive)-10.00 N

At a turning point the cart pauses. All 0.500 J is stored in the spring.

Phase playback is slowed for inspection. Physical period: 0.628 s.

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: Work, stored energy and power
Focus: Use turning points without assuming equilibrium
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
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.

Back to chapter contents ↑

09 / Work, stored energy and power

Power measures how quickly work is transferred

Work describes an amount of energy transfer. Power describes its rate. Average power over an interval is W/Δt; instantaneous power is dW/dt. Two machines can do the same work with different powers if they take different amounts of time.

For a force acting on a particle, dW = F·dr gives P = F·v. The same force produces greater power when its point of application moves faster along its direction. A large holding force at zero velocity has zero mechanical power on the stationary load, though the machine or person may consume internal energy.

Power can be negative for a force removing kinetic energy. Always identify whose power is being reported: input electrical power, output mechanical power, work by gravity or dissipative power in a resistance are different quantities. Efficiency connects useful output to input for a specified process; it does not create an extra energy source.

P_{\mathrm{avg}}=\frac{W}{\Delta t},\qquad P=\mathbf F\cdot\mathbf v,\qquad[P]=\mathrm{J/s}=\mathrm W

A watt is a rate unit. A kilowatt-hour is an energy unit: 1000 W × 3600 s = 3.6 MJ.

WORKED EXAMPLE

Calculate useful power before accounting for efficiency

A hoist raises a 30 kg load vertically at constant speed 0.50 m/s. Take g = 10 m/s². If the overall efficiency from electrical input to useful lifting is 75%, find the useful mechanical power and the input power.

Choose an approach

Constant speed fixes the upward force on the load at mg. Multiply by load speed, then divide useful output by efficiency.

The useful lifting power is the rate of gravitational energy gain.
$P_{\mathrm{useful}}=mgv=30(10)(0.50)=150\ \mathrm W$
Efficiency is output divided by input.
$P_{\mathrm{input}}=\frac{150}{0.75}=200\ \mathrm W$

What this gives usThe useful output is 150 W and the electrical input is 200 W. The remaining 50 W is not useful lifting power under the stated efficiency definition.

Check the result another way

In 4 s the load rises 2 m, gaining 600 J. The useful rate 150 W supplies exactly 600 J; the input supplies 800 J.

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: Calculate useful power before accounting for efficiency
Problem: A hoist raises a 30 kg load vertically at constant speed 0.50 m/s. Take g = 10 m/s². If the overall efficiency from electrical input to useful lifting is 75%, find the useful mechanical power and the input power.

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: Work, stored energy and power
Focus: Power measures how quickly work is transferred
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
P_{\mathrm{avg}}=\frac{W}{\Delta t},\qquad P=\mathbf F\cdot\mathbf v,\qquad[P]=\mathrm{J/s}=\mathrm W
Meaning: A watt is a rate unit. A kilowatt-hour is an energy unit: 1000 W × 3600 s = 3.6 MJ.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

10 / Work, stored energy and power

Constant power does not mean constant force or acceleration

If a positive constant mechanical power P goes entirely into a particle’s kinetic energy, then K = Ki + Pt. Therefore speed squared grows linearly with time, while speed itself does not. The force needed along the velocity is P/v, so it decreases as speed rises.

The ideal constant-power relation becomes singular at zero speed because P/v would require an unbounded force. A real motor has force, torque and operating limits near startup. Apply this model only over a stated range with nonzero speed where the constant-power assumption is plausible.

For lifting or climbing, the input power may be split between increasing kinetic energy, increasing gravitational potential and overcoming resistance. Constant engine power does not imply that all of it increases speed. Write the rate balance for the actual model before using v² = u² + 2Pt/m.

WORKED EXAMPLE

Use the energy rate when the force is not constant

A 2 kg cart initially moves at 3 m/s on a smooth horizontal track. A drive supplies constant mechanical power 8 W entirely to its translation for 2 s. Find the final speed and compare drive force at the start and end.

Choose an approach

Add Pt to kinetic energy, then use F = P/v at each endpoint.

The initial kinetic energy is 9 J and the added work is 16 J.
$K_f=\frac12(2)(3^2)+8(2)=25\ \mathrm J$
Convert the final kinetic energy into the speed magnitude.
$v_f=5\ \mathrm{m/s}$
The same power requires different forces at different speeds.
$F_i=\frac83\ \mathrm N,\qquad F_f=\frac85=1.6\ \mathrm N$

What this gives usThe cart reaches 5 m/s. The drive force decreases from about 2.67 N to 1.60 N, so its acceleration is not constant.

Check the result another way

The kinetic-energy gain is 25 − 9 = 16 J, equal to 8 W × 2 s. Applying a constant-force kinematic formula without deriving an appropriate average would miss the changing force.

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 energy rate when the force is not constant
Problem: A 2 kg cart initially moves at 3 m/s on a smooth horizontal track. A drive supplies constant mechanical power 8 W entirely to its translation for 2 s. Find the final speed and compare drive force at the start and end.

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: Work, stored energy and power
Focus: Constant power does not mean constant force or acceleration
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
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.

Back to chapter contents ↑

11 / Work, stored energy and power

Read force from slope and motion from the energy gap

In one dimension, a conservative force obeys Fₓ = −dU/dx. A graph of U against x therefore supplies the force through its negative slope. A rising potential has a force toward decreasing x; a falling potential has a force toward increasing x. The height of the graph is not the force.

For a particle with fixed total mechanical energy E, kinetic energy at a position is K = E − U(x). Motion is classically allowed where U ≤ E. At U = E the speed is zero. If the potential continues above E and the local force reverses the motion, that boundary is a turning point.

An equilibrium occurs where the force is zero, meaning a stationary point of U when differentiable. A local minimum is stable against sufficiently small displacements in the one-dimensional conservative model because the force returns toward it. A local maximum is unstable. A flat region or vanishing second derivative needs further examination; zero slope alone does not classify stability.

F_x=-\frac{dU}{dx},\qquad v^2=\frac{2[E-U(x)]}{m}

The kinetic-energy relation gives speed squared where E − U is nonnegative. The sign of velocity follows from the motion history.

Locate allowed positions and turning points

Inspect a plotted point: Position (m) = -2.5

Potential U: 12.5
Total E: 8

For a 1 kg particle with U = 2x² joules and E = 8 J, the energy gap is kinetic energy. Inspect where the potential meets the energy line and where the gap is greatest. Lines connect the plotted samples.

WORKED EXAMPLE

One energy graph answers several different questions

A 1 kg particle moves in U(x) = (2 J/m²)x² with total energy 8 J. Find the allowed interval, turning points, maximum speed and force at x = 1 m.

Choose an approach

Use U ≤ E for accessibility, the largest energy gap for maximum speed and a derivative for force.

The allowed condition is 2x² ≤ 8.
$-2\ \mathrm m\leq x\leq2\ \mathrm m$
The endpoints have zero kinetic energy and are turning points.
$x_{\mathrm{turn}}=\pm2\ \mathrm m$
At x = 0 the potential is zero, making K = 8 J.
$v_{\max}=\sqrt{2(8)/1}=4\ \mathrm{m/s}$
The force follows from slope, not the value U(1).
$F_x=-4x\quad\Longrightarrow\quad F_x(1\ \mathrm m)=-4\ \mathrm N$

What this gives usThe particle oscillates between −2 m and +2 m in this ideal potential, with maximum speed 4 m/s at the stable equilibrium x = 0. At x = 1 m the force points toward decreasing x.

Check the result another way

This potential equals ½kx² with k = 4 N/m. A spring of amplitude 2 m has energy ½ × 4 × 4 = 8 J and restoring force −kx, independently confirming the graph interpretation.

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: One energy graph answers several different questions
Problem: A 1 kg particle moves in U(x) = (2 J/m²)x² with total energy 8 J. Find the allowed interval, turning points, maximum speed and force at x = 1 m.

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: Work, stored energy and power
Focus: Read force from slope and motion from the energy gap
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
Equations in this section (LaTeX):
F_x=-\frac{dU}{dx},\qquad v^2=\frac{2[E-U(x)]}{m}
Meaning: The kinetic-energy relation gives speed squared where E − U is nonnegative. The sign of velocity follows from the motion history.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

12 / Work, stored energy and power

Choose energy for endpoints, then add the missing dynamics

Energy is often the shortest route to speed after a known height change, spring deformation or position-dependent force. Start by writing the initial and final states explicitly: location, speed, stored energies and any work crossing the chosen system boundary. Cancel only terms whose equality follows from the model.

Then ask what energy has not told you. A speed does not fix direction. A possible energy state does not guarantee contact. A turning point does not guarantee equilibrium. A travel time usually requires integrating dx/v(x) or using additional equations. These are complementary questions, not failures of the energy method.

A strong solution checks dimensions, signs and limiting behaviour. With no height change and no external work, speed should remain unchanged if all other energy stores return to their initial values. Doubling a spring deformation should quadruple its stored energy in the Hooke-law range. Changing the potential zero must not change any predicted motion.

STOP & THINK

An energy calculation predicts a bead’s speed on a curved track. What additional condition may still need checking?

Whether the required normal force can maintain contactWhether kinetic energy must be negativeWhether the potential zero is the only possible one

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: Work, stored energy and power / Choose energy for endpoints, then add the missing dynamics
Problem: An energy calculation predicts a bead’s speed on a curved track. What additional condition may still need checking?
Choices:
1. Whether the required normal force can maintain contact
2. Whether kinetic energy must be negative
3. Whether the potential zero is the only possible one

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: Work, stored energy and power
Focus: Choose energy for endpoints, then add the missing dynamics
Learning goals:
- Calculate signed work for constant and variable forces.
- Apply net work to a kinetic-energy change.
- Define potential-energy differences and choose a consistent zero.
- Account for conservative, dissipative and driven transfers.
- Derive Hooke-law spring energy and distinguish turning points from equilibrium.
- Use power as an energy-transfer rate.
- Read force, turning points and stability from potential graphs.
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.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Net work changes kinetic energy for a constant-mass particle in an inertial frame.
A conservative interaction contributes ΔU = −Wc.
Mechanical energy changes by work not represented in the included potentials.
Spring energy uses deformation from natural length and is quadratic.
Power is a rate; constant power does not generally mean constant acceleration.
Energy gives possible speeds; directions, times and contact may need additional equations.

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 3 kg object rises by 2 m in uniform gravity with g = 10 m/s². What is the increase in gravitational potential energy?

Your answer (J)Use a decimal, a fraction such as 1/3, or scientific notation.

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: Work, stored energy and power
Problem: A 3 kg object rises by 2 m in uniform gravity with g = 10 m/s². What is the increase in gravitational potential energy?
Requested answer unit: J

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 machine transfers 900 J of useful work in 6 s. What is its average useful power?

Your answer (W)Use a decimal, a fraction such as 1/3, or scientific notation.

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: Work, stored energy and power
Problem: A machine transfers 900 J of useful work in 6 s. What is its average useful power?
Requested answer unit: W

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

At a spring oscillator’s turning point, which statement is generally true?

Velocity and restoring force are both zeroVelocity is zero but restoring force need not beKinetic energy is greatest

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: Work, stored energy and power
Problem: At a spring oscillator’s turning point, which statement is generally true?
Choices:
1. Velocity and restoring force are both zero
2. Velocity is zero but restoring force need not be
3. Kinetic energy is greatest

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

At a position where a one-dimensional potential U increases with x, which way does its conservative force point?

Toward increasing xToward decreasing xIt must be zero

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: Work, stored energy and power
Problem: At a position where a one-dimensional potential U increases with x, which way does its conservative force point?
Choices:
1. Toward increasing x
2. Toward decreasing x
3. It must be 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 5

A particle’s mechanical energy increases by 12 J. Under Δ(K + U) = Wother, what does this establish?

The work not already represented by the chosen potentials is +12 JEvery individual force does +12 JEnergy conservation has failed

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: Work, stored energy and power
Problem: A particle’s mechanical energy increases by 12 J. Under Δ(K + U) = Wother, what does this establish?
Choices:
1. The work not already represented by the chosen potentials is +12 J
2. Every individual force does +12 J
3. Energy conservation has failed

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.

COME 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

A spring is compressed twice as far within its linear range. How does stored energy change, and why?

Compare your reasoning

It becomes four times as large. The force rises with compression, and the area under the force–displacement graph is a triangle proportional to compression squared.

Revisit the explanation →

A few days later: change the situation

Two lifts raise equal masses through the same height, but one takes half the time. Compare useful work and average useful power.

Compare your reasoning

The useful gravitational energy increase is the same, mgh. The faster lift has twice the average useful power. Actual input energy can differ if their efficiencies differ.

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.

Back to chapter contents ↑