Push gently on a heavy cupboard and it stays put. Push a little harder and it may still stay put. Friction has adjusted to both attempts; it has not immediately spent its entire allowance. The inequality is the physics here.
This chapter develops a repeatable method: assume no relative slipping, calculate the friction required, and compare it with the available limit. We will use the same method for a horizontal push, an incline and an accelerating platform. The models here describe ideal dry contact; real friction can depend on surface condition, speed and other details.
What you will learn to do
- Distinguish actual static friction from its maximum possible magnitude.
- Determine friction direction from impending relative slipping.
- Test a no-slip assumption using a force equation and an inequality.
- Account for changes in normal force on an incline or under an angled pull.
- Distinguish static and kinetic regimes without inventing a motion history.
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: Static friction: find what is needed Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. 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.
If a body stays at rest horizontally, must every horizontal force on it be zero?
Compare your reasoning
No. The net horizontal force must be zero in an inertial frame. A nonzero applied push can be balanced by static friction, provided the needed friction is within the available limit.
Revisit the explanation →01 / Static friction: find what is needed
Static refers to the contact, not to the room
Static friction acts when the contacting surfaces are not sliding relative to one another. Both bodies can still move through the room. A box carried without slipping on an accelerating truck is moving relative to the ground, yet its contact with the truck can involve static friction.
Friction opposes actual or impending relative sliding at the contact. This is more precise than saying it always opposes motion. If the truck accelerates forward, the unsupported tendency is for the box to lag behind the truck. Static friction on the box then points forward and speeds it up.
For a box at rest on a floor, a rightward push creates a tendency to slip right relative to the floor. Static friction on the box points left. If there is no horizontal push and no other tendency to slip, the static friction force can be zero.
STOP & THINK
A box accelerates rightward with a truck without sliding on its horizontal bed. What direction is static friction on the box, if it is the only horizontal force?
- Left, because friction opposes motion
- Right, because the box needs rightward acceleration
- Zero, because there is no sliding
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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: Static friction: find what is needed Focus: Static refers to the contact, not to the room Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. 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 / Static friction: find what is needed
The coefficient sets a limit
N is the compressive normal force. The product μsN is the largest available static friction in this ideal model, not automatically the force that acts.
The static coefficient μs is dimensionless. Multiplying it by N gives a force. Within the allowed range, the other force equations and the no-slip constraint determine the actual friction. Equality marks limiting equilibrium: the boundary at which a slightly greater demand cannot be supported.
For a box that remains at rest on a horizontal floor under a horizontal push F, horizontal balance requires friction of magnitude F. This solution is possible only while F ≤ μsN. Writing f = μsN first would wrongly predict acceleration under a small push, even though the contact could simply balance it.
WORKED EXAMPLE
A push below the limit
A 2 kg box rests on a horizontal floor. It is pushed horizontally with 5 N. Let μs = 0.40 and g = 10 m/s². Determine its friction and 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: Static friction: find what is needed Focus: The coefficient sets a limit Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. Equations in this section (LaTeX): |f_s|\leq\mu_s N Meaning: N is the compressive normal force. The product μsN is the largest available static friction in this ideal model, not automatically the force that acts. 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 / Static friction: find what is needed
When the rest assumption fails
If the required static friction exceeds its maximum, the assumed no-slip motion is impossible in this model. Once sliding occurs, use the kinetic-friction model specified by the problem. For a simple dry interface its magnitude is approximately μkN, directed opposite the relative sliding velocity.
Often μk is smaller than μs. Consequently the resistance may drop after sliding begins. You must be told, or otherwise justify, the kinetic coefficient; the static coefficient alone does not determine the subsequent acceleration. At exactly the static limit, rest is still an admissible ideal-model solution. A stated onset of slipping may select the next regime.
WORKED EXAMPLE
Increase the push until the box slides
For the same 2 kg box, let μk = 0.30. Apply a 10 N horizontal push from rest, exceeding the 8 N static limit. Find the acceleration once it is sliding rightward.
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INTERACTIVE LAB
Required force versus available friction
Start each trial at rest. Predict the friction at pushes of 3 N, 7 N and 8 N, then go just above 8 N. Next select “already sliding to the right” and compare the response at a 5 N push. Explain which assumption changed.
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STOP & THINK
A block has normal force 30 N and static coefficient 0.25. What is the maximum static friction magnitude?
Answer in N.
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: Static friction: find what is needed Focus: When the rest assumption fails Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. 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 / Static friction: find what is needed
Resolve weight before testing an incline
On an incline, choose one axis along the surface and one perpendicular to it. Weight points vertically downward. Its component into the surface is mg cos θ, and its component down the slope is mg sin θ, where θ is the angle of the incline above horizontal.
A stationary block tends to slip down the slope, so static friction points up it. If there are no other applied forces and no perpendicular acceleration, the normal force equals mg cos θ. Along the slope, rest requires f = mg sin θ. Comparing demand with capacity gives the condition for rest.
Mass cancels because both the downslope demand and the friction capacity scale with mass in this particular model.
WORKED EXAMPLE
Can the block remain on the ramp?
A 4 kg block is placed at rest on an incline with sin θ = 0.60 and cos θ = 0.80. Let g = 10 m/s² and μs = 0.50. Can it remain at rest?
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STOP & THINK
In this simple incline model, doubling the block’s mass changes the condition for remaining at rest how?
- It becomes twice as difficult to satisfy
- It stays the same
- Rest becomes possible at every angle
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: Static friction: find what is needed Focus: Resolve weight before testing an incline Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. Equations in this section (LaTeX): mg\sin\theta\leq\mu_smg\cos\theta\quad\Longrightarrow\quad\tan\theta\leq\mu_s\quad(0\leq\theta<90^\circ) Meaning: Mass cancels because both the downslope demand and the friction capacity scale with mass in this particular model. 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 / Static friction: find what is needed
No slipping can require acceleration
WORKED EXAMPLE
Carried by an accelerating platform
A 5 kg box is on a horizontal truck bed with μs = 0.35. The truck accelerates rightward at 2.0 m/s². Use g = 10 m/s². Can the box accelerate with it without slipping?
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: Static friction: find what is needed Focus: No slipping can require acceleration Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Static friction: find what is needed
A changed normal force changes the limit
An upward component of an applied pull reduces the support needed from the floor. It therefore reduces the friction capacity as well. A downward component increases the normal force. This is why finding N from the force diagram must come before substituting μsN.
For a block of mass m pulled with magnitude F at angle α above a horizontal floor, vertical balance gives N = mg − F sin α while contact is maintained. Rest horizontally requires f = F cos α. Combine these statements rather than silently replacing N with mg.
The first inequality tests horizontal no-slip motion; the second tests whether a compressive contact force is possible. Here 0 ≤ α < 90° and F ≥ 0.
At the threshold of horizontal slipping, equality gives F = μs mg/(cos α + μs sin α). This expression applies only to this geometry and maintained-contact regime. Before using a derived formula elsewhere, trace which diagram and constraints produced it.
A robust friction solution therefore has four parts: choose the body and frame; calculate N; calculate the friction demanded by the proposed no-slip motion; test the demand against the available limit. If the test fails, state that failure before changing to a sliding model. This explicit regime choice is often the central reasoning step in an exam problem.
STOP & THINK
A block stays on a horizontal floor while a hand pulls it partly upward. With no vertical acceleration, how does that upward component affect N?
- It reduces N below mg
- It leaves N equal to mg
- It increases N above mg
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: Static friction: find what is needed Focus: A changed normal force changes the limit Learning goals: - Distinguish actual static friction from its maximum possible magnitude. - Determine friction direction from impending relative slipping. - Test a no-slip assumption using a force equation and an inequality. - Account for changes in normal force on an incline or under an angled pull. - Distinguish static and kinetic regimes without inventing a motion history. Equations in this section (LaTeX): F\cos\alpha\leq\mu_s(mg-F\sin\alpha),\qquad mg-F\sin\alpha\geq0 Meaning: The first inequality tests horizontal no-slip motion; the second tests whether a compressive contact force is possible. Here 0 ≤ α < 90° and F ≥ 0. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Static friction is determined by the required no-slip motion, subject to a maximum.
- Its direction opposes relative slipping, which can mean accelerating a body forward.
- Compute the normal force from the diagram before calculating a friction limit.
- A failed static test requires a new motion model and appropriate kinetic data.
- At a stopping event, reconsider the regime instead of extending kinetic friction blindly.
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 stationary box has a maximum static friction of 12 N. A horizontal 7 N pull acts, with no other horizontal forces. What is its actual friction magnitude?
Answer in N.
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SELF CHECK 2
A calculation requires 9 N of static friction but the available maximum is 6 N. What is the valid conclusion?
- The actual static friction is 9 N
- The proposed no-slip motion is impossible
- The acceleration must be 3 m/s²
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SELF CHECK 3
A box rides on a horizontal accelerating platform. With μs = 0.20 and g = 10 m/s², what is the maximum acceleration for no slip in the stated dry-friction model?
Answer in m/s².
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SELF CHECK 4
A block slides right while its net force is leftward. What is happening at that instant?
- Its rightward speed is decreasing
- It must already be moving left
- It cannot have kinetic friction
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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.
Does friction always equal μN?Beginner · Hints and guided reasoningFind friction before the block slips.Intermediate · Hints and guided reasoningGrip sets a turning speedIntermediate · 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
A block remains at rest on a rough incline. How do you find friction before checking its maximum?
Compare your reasoning
Resolve weight along the slope and balance that component. The required friction is mg sin θ uphill in this simple setup. Then compare it with μs mg cos θ; equality is only the limiting case.
Revisit the explanation →A few days later: change the situation
A box rides without slipping on a truck accelerating forward. In which direction does friction act on the box, and why?
Compare your reasoning
Forward: the box needs a forward net force to share the truck’s acceleration. Static friction opposes the tendency to slip relative to the truck, not necessarily motion relative to the road.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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