Two carts tied together look like two objects, but the string links their possible motions. The useful trick is to switch viewpoints: study the pair to find acceleration, then study one cart to uncover the tension hidden inside the pair.
This chapter develops that method from two carts to moving pulleys, rough contacts and an accelerating observer. Each worked solution checks whether the connection or contact assumed at the start is physically possible.
What you will learn to do
- Use separate-body and combined-system equations for different unknowns.
- Derive motion ratios from a fixed string length.
- Apply kinetic friction to relative sliding and calculate the actual normal force.
- Transform Newton’s equation into a translating accelerated frame.
- Explain apparent weight and test contact/tension constraints.
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: Connected bodies, sliding friction and accelerating frames Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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.
When two bodies pull on each other, why can their interaction disappear from the equation for the combined system?
Compare your reasoning
The internal force pair cancels when their momentum equations are added. External forces remain. The cancellation does not mean that tension vanishes on either individual body.
Revisit the explanation →01 / Connected bodies, sliding friction and accelerating frames
Decide which bodies an equation describes
A connection between bodies supplies both a force interaction and a motion constraint. A taut inextensible string may require two ends to have related accelerations; a contact can make two blocks move together while it persists. Newton’s second law alone does not determine that relationship. The geometry of the connection supplies additional information.
Begin with separate force diagrams. A string pulls each attached body along the local string direction; it cannot push. A surface supplies a normal force while the bodies remain in contact. Static or kinetic friction depends on the relative motion at that contact. These forces act on particular bodies, so label each diagram with its chosen body.
Next decide whether adding equations will help. For a combined system, equal and opposite internal forces cancel in the vector sum. External forces remain. A common acceleration lets you write a particularly simple total-mass equation. If different parts accelerate in different directions, use their component equations and constraints rather than assigning the whole system an unjustified single acceleration.
WORKED EXAMPLE
Use the system first, then recover the internal tension
Two carts of masses 2 kg and 4 kg are joined by a light taut horizontal string on a smooth level track. An 18 N horizontal force pulls the 4 kg cart away from the 2 kg cart. Find the acceleration and string tension.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
STOP & THINK
Why does the internal tension disappear when the two-cart equations are added?
- Because a moving string has no tension
- Because its two forces are equal and opposite within the chosen system
- Because tension is smaller than the applied force
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: Decide which bodies an equation describes Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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.
02 / Connected bodies, sliding friction and accelerating frames
What a light string and smooth pulley actually assume
A light string is treated as having negligible mass. A finite tension difference along a straight massless segment would require an unbounded acceleration in the elementary model, so its tension magnitude is uniform while the ideal conditions hold. A smooth massless pulley redirects the string without requiring a torque to spin a finite rotational inertia. These assumptions allow the same tension magnitude on both sides.
A real massive pulley can require different tensions to provide rotational acceleration. A massive string can also have tension that changes along its length. Do not carry the equal-tension result into those models without checking the corresponding force and torque balances.
A taut string of fixed length provides a length equation. For two hanging ends over one fixed pulley, an increase in one hanging length requires an equal decrease in the other. Their speeds and acceleration magnitudes are equal, but their directions are opposite. Choose one positive sense on each side to make that constraint explicit.
WORKED EXAMPLE
Let the constraint determine the acceleration signs
Masses 3 kg and 5 kg hang from an ideal light string over a fixed smooth massless pulley. The string is taut. Take g = 10 m/s². Find their acceleration and the tension after release.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: What a light string and smooth pulley actually assume Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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.
03 / Connected bodies, sliding friction and accelerating frames
Count changing string segments before assuming equal accelerations
A movable pulley can have two string segments supporting it. Moving that pulley by a distance changes both segment lengths. The free end must compensate for their combined length change, so its displacement can be twice the pulley’s displacement. Equal tension does not imply equal acceleration.
Write the total string length using coordinates measured in stated directions. Constant pieces around fixed pulley arcs can be included in one constant. Differentiate the length relation once for velocities and twice for accelerations. This procedure is safer than memorising a speed ratio from a similar-looking drawing.
For a massless movable pulley supported by two vertical portions of the same ideal string, the upward string forces sum to 2T. A load attached to that pulley responds to this combined support. The load’s acceleration equation still needs its actual mass, gravity and acceleration; 2T = mg holds only when its vertical acceleration is zero.
WORKED EXAMPLE
A length equation supplies the motion ratio
A string has one end fixed overhead, runs down around a movable pulley, returns up over a fixed pulley and then down to a free end. Let y be the movable pulley’s downward displacement and z the free end’s downward displacement from reference positions. The pulley arcs remain fixed in length. If the movable pulley accelerates upward at 0.6 m/s², find the free end’s acceleration.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: Count changing string segments before assuming equal accelerations Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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 / Connected bodies, sliding friction and accelerating frames
Kinetic friction follows relative sliding at the contact
When surfaces slide relative to each other, a common dry-friction model gives the magnitude fk = μkN. The coefficient μk is dimensionless and N is the actual normal force, not automatically mg. The force on each surface opposes its local relative sliding with the other surface.
This direction rule is more precise than saying friction always opposes the object’s ground-frame velocity. A stationary block placed on a moving conveyor belt can be accelerated forward by kinetic friction because its surface slides backward relative to the belt. The friction on the belt points the other way.
The constant-coefficient model is an approximation suitable for many elementary problems. Real friction can depend on speed, surface condition and temperature. In the ideal model, determine the sliding direction first, obtain N from the perpendicular dynamics, and then use μkN along the surface. At zero relative speed, reconsider static friction instead of forcing the same kinetic direction through the stopping instant.
WORKED EXAMPLE
A stopping calculation has an endpoint
A block slides at 6 m/s on a fixed horizontal rough surface with μk = 0.20. No horizontal applied force acts. Use g = 10 m/s². Find its stopping time and distance.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
STOP & THINK
A block is initially stationary on the ground while a belt beneath it moves right. Before the block matches the belt speed, which way does kinetic friction on the block act?
- Left
- Right
- It must be zero because the block initially has zero ground speed
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: Kinetic friction follows relative sliding at the contact Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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 / Connected bodies, sliding friction and accelerating frames
An angled force changes the friction through the normal force
For a block pulled upward at an angle on a horizontal surface, part of the pull supports its weight. If it remains in contact with zero vertical acceleration, N = mg − F sin θ. A downward push instead increases N. The horizontal pull and the changed friction must both appear in the horizontal equation.
Check N ≥ 0. A surface cannot supply a negative compressive normal force in this contact model. If the calculation demands N < 0, the assumed contact has failed and the block’s airborne motion must be analysed instead. Setting N to a negative value and multiplying by μ would create a fictitious attractive friction force.
A pull that reduces N can reduce friction, but its own horizontal component also changes with angle. Finding an optimal angle is therefore a complete force-balance problem, not simply a request to minimise N. The allowed angle, contact condition and specified sliding state all matter.
This version assumes upward-angled pulling, rightward sliding, persistent contact and zero vertical acceleration.
WORKED EXAMPLE
Use the normal equation before calculating kinetic friction
A 4 kg block slides right on a horizontal surface with μk = 0.25. A 20 N force pulls at an angle whose sine is 0.6 and cosine is 0.8. Take g = 10 m/s². Find the acceleration.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: An angled force changes the friction through the normal force Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. Equations in this section (LaTeX): N=mg-F\sin\theta,\qquad ma_x=F\cos\theta-\mu_kN Meaning: This version assumes upward-angled pulling, rightward sliding, persistent contact and zero vertical acceleration. 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 / Connected bodies, sliding friction and accelerating frames
Combine constraints and friction without guessing the result
A rough table connected to a hanging mass brings several ideas together. The hanging mass provides a driving weight, the table block contributes inertia and friction, and the taut string relates their accelerations. Treat the system as coupled equations with an explicit assumed direction.
If the system starts at rest, first ask whether static friction can hold it. The hanging weight may be balanced by a tension that the table contact can support within its static limit. If sliding is stated or established, use kinetic friction in the direction opposing that sliding. A negative acceleration relative to an assumed positive velocity means slowing down; it does not automatically mean the assumed velocity is impossible.
After solving, test tension positivity, normal-force positivity and the phase of motion. When the hanging mass hits the floor, the table block cannot continue under the same tension and common-acceleration model. Motion after a constraint changes is a new interval with new equations.
WORKED EXAMPLE
A moving hanging mass pulls a rough-table block
A 3 kg block on a horizontal table is connected over an ideal fixed pulley to a 2 kg hanging mass. The hanging mass is already moving downward and the table block slides toward the pulley. The table’s μk is 0.20. Take g = 10 m/s². Find the acceleration and tension while the string stays taut and both motions are unobstructed.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
STOP & THINK
A 2 kg block is already sliding right on a level surface with μk = 0.30. A horizontal 10 N force pulls right. With g = 10 m/s², what is its rightward acceleration?
Answer in m/s².
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: Combine constraints and friction without guessing the result Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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 / Connected bodies, sliding friction and accelerating frames
Translate the acceleration before writing the force equation
An observer in an accelerating vehicle can describe an object using coordinates fixed to that vehicle. For axes that translate without rotating, the object’s ground acceleration is its vehicle-relative acceleration plus the frame acceleration A. Substitute this relation into Newton’s second law to derive the equation appropriate to the moving frame.
The result is ΣFreal − mA = ma′. The extra term −mA is often called a fictitious or inertial force. It is a bookkeeping term caused by the accelerating coordinates; it is not an additional interaction with a new object. It has no Newton-third-law partner on some external body.
Use one frame consistently. In the ground inertial frame, include real forces and the ground acceleration. In the translating accelerated frame, include the inertial term and the relative acceleration. Including −mA in an inertial-frame equation would count the frame effect twice. Rotating frames require additional terms and are outside this purely translating treatment.
The frame axes remain parallel to the inertial axes. A prime labels quantities measured relative to that translating frame.
WORKED EXAMPLE
A smooth-floor block seen from an accelerating carriage
A carriage accelerates rightward at 2 m/s². A small block rests on its smooth horizontal floor and has no horizontal applied force. Find the block’s horizontal acceleration relative to the ground and relative to the carriage.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
STOP & THINK
In a nonrotating frame accelerating rightward, which way does the inertial force on a positive-mass particle point?
- Rightward
- Leftward
- Along the particle’s relative velocity
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames
Focus: Translate the acceleration before writing the force equation
Learning goals:
- Use separate-body and combined-system equations for different unknowns.
- Derive motion ratios from a fixed string length.
- Apply kinetic friction to relative sliding and calculate the actual normal force.
- Transform Newton’s equation into a translating accelerated frame.
- Explain apparent weight and test contact/tension constraints.
Equations in this section (LaTeX):
\mathbf a=\mathbf a^{\prime}+\mathbf A,\qquad\sum\mathbf F_{\mathrm{real}}-m\mathbf A=m\mathbf a^{\prime}
Meaning: The frame axes remain parallel to the inertial axes. A prime labels quantities measured relative to that translating frame.
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 / Connected bodies, sliding friction and accelerating frames
Apparent weight measures support, not gravitational attraction
A weighing scale measures the normal support force. In a vertically accelerating lift this can differ from mg even though the gravitational field has not changed. With upward positive, N − mg = ma for a passenger moving with the lift. An upward acceleration increases N; a downward acceleration reduces it.
Velocity direction alone does not determine apparent weight. A lift moving upward while slowing has downward acceleration and a reduced scale reading. The force balance depends on the change of velocity, not on whether the lift is currently above or below its starting floor.
In a nonrotating accelerated frame, combine gravity and the inertial term into an effective gravitational field geff = gvector − A. A freely hanging thread at rest relative to the frame aligns along this effective downward direction. This construction expresses two force components compactly; it does not mean Earth’s actual gravitational field has tilted.
WORKED EXAMPLE
Read the acceleration from the force balance
A 50 kg passenger remains on the floor of a lift accelerating downward at 2 m/s². Take g = 10 m/s². Find the scale reading.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
WORKED EXAMPLE
A thread tilts because its tension must supply horizontal acceleration
A small mass hangs at rest relative to a carriage accelerating horizontally at 7.5 m/s². Take g = 10 m/s². Find the thread’s angle from the vertical.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Connected bodies, sliding friction and accelerating frames Focus: Apparent weight measures support, not gravitational attraction Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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 / Connected bodies, sliding friction and accelerating frames
Finish by testing the model you assumed
A result is not complete until its assumed constraints are checked. A negative string tension indicates that a string would need to push, so the taut-string model must be replaced. A negative normal force indicates loss of ordinary contact. A required static friction larger than μsN indicates slipping, after which μk may be relevant.
Distinguish a negative acceleration from an impossible force. With a chosen positive direction, negative acceleration can simply describe slowing or acceleration the other way. By contrast, the sign restriction on a string tension magnitude or a compressive contact normal is a physical restriction on the interaction itself.
For a multi-stage problem, write the event ending each stage: speed reaches zero, a hanging mass reaches the floor, a string goes slack or a body leaves a surface. Carry the endpoint position and velocity into the next stage, then redraw forces and revise the constraint. Keeping one equation across a changed contact is a common reason otherwise correct algebra gives an impossible story.
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: Connected bodies, sliding friction and accelerating frames Focus: Finish by testing the model you assumed Learning goals: - Use separate-body and combined-system equations for different unknowns. - Derive motion ratios from a fixed string length. - Apply kinetic friction to relative sliding and calculate the actual normal force. - Transform Newton’s equation into a translating accelerated frame. - Explain apparent weight and test contact/tension constraints. 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
- Internal forces cancel only when both members of the interaction are inside the system.
- Equal string tension does not imply equal endpoint acceleration.
- Kinetic friction opposes relative slipping at the contact.
- An accelerating translating frame requires the inertial term −mA.
- Check contact, tension and phase boundaries after solving.
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
Two carts of masses 1 kg and 3 kg are connected by an ideal string on a smooth level track. A 12 N force pulls the 3 kg cart. What is the common acceleration?
Answer in m/s².
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
SELF CHECK 2
A 60 kg passenger accelerates upward with a lift at 1 m/s². With g = 10 m/s², what support force does the scale exert?
Answer in N.
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
SELF CHECK 3
A taut-string model gives a negative value for the tension magnitude. What should you do?
- Keep the negative value and let the string push
- Reassess the constraint because an ordinary string cannot push
- Reverse gravity
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
SELF CHECK 4
After a sliding block reaches zero relative speed on a surface, what is the next step?
- Continue the same kinetic-friction direction indefinitely
- Reassess the forces and whether static contact can persist
- Set every force to zero
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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.
Two hanging masses share an accelerationIntermediate · Free sample · Hints and guided reasoningA puck slows on a rough tableIntermediate · Free sample · Hints and guided reasoningBalance in an accelerating frameIntermediate · Hints and guided reasoningThe string does not read either weightIntermediate · Hints and guided reasoningHow much force crosses the contact?Intermediate · Hints and guided reasoningWhere did the kinetic energy go?Intermediate · Hints and guided reasoningSliding down a rough slopeIntermediate · Hints and guided reasoningA pull partly lifts the loadIntermediate · 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
Why can two bodies in a pulley arrangement have different acceleration magnitudes even if the string is inextensible?
Compare your reasoning
The string-length constraint depends on how many changing segments support each body. A moving pulley can link one displacement to twice another. Derive that relation from geometry before using forces.
Revisit the explanation →A few days later: change the situation
A hanging thread tilts in an accelerating vehicle. How could you explain it using either the ground frame or the vehicle frame?
Compare your reasoning
In the inertial ground frame, horizontal tension produces acceleration. In the vehicle frame, include the inertial force opposite the vehicle acceleration and use equilibrium. Each account must keep its frame assumptions consistent.
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 ↑