Q.What is the angle of friction? Prove that the tangent of the angle of friction is equal to the coefficient of limiting friction. OR Derive the expression for the motion of a car on a banked road.
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The Static Friction Limit: Why a Heavy Box Won't Move Until You Really Push
Imagine you're trying to push a heavy wooden crate across a rough floor. You lean into it gently — nothing happens. You push a little harder — still nothing. The crate stays perfectly still, as if glued to the spot. Then, at some point, you push just a bit more, and suddenly the crate lurches forward.
That invisible "sticking point" — the exact moment the crate finally gives way — is the static friction limit.
The Intuition: Friction as a "Smart" Force
Friction between two surfaces that aren't sliding is called static friction. What makes it special is that it's self-adjusting. It doesn't have a fixed value. Instead, it automatically grows to match whatever force you apply — up to a point.
Think of it like a tug-of-war where your opponent (static friction) matches your pull exactly, but only until you exceed their maximum strength. As long as you pull less than their limit, you both stay in equilibrium and nothing moves. The moment you exceed that limit, you win — and the crate starts sliding.
Static friction only exists when there is no relative motion between the surfaces. Once sliding begins, it's replaced by kinetic friction, which is usually weaker.
The Precise Statement
The static friction limit (also called limiting friction) is the maximum possible value of static friction that can act between two surfaces in contact before they start sliding relative to each other.
Mathematically:
fs≤μsN
Where:
- fs = static friction force (the actual value, which can be anything from 0 up to the limit)
- μs = coefficient of static friction (a constant that depends on the two materials — rubber on concrete is high, ice on steel is low)
- N = normal reaction force (the force pressing the surfaces together, usually equal to weight on a horizontal surface)
The static friction limit is the equality case:
fs,max=μsN
This is the maximum static friction the surfaces can provide. Apply a force less than this, and the object stays put. Apply a force equal to this, and the object is on the verge of moving (impending motion). Apply a force greater than this, and the object accelerates.
A Concrete Example
A 10 kg block rests on a horizontal floor. μs=0.4 between the block and floor. Take g=10 m/s2.
Normal reaction: N=mg=10×10=100 N
Static friction limit: fs,max=0.4×100=40 N
Now, what happens as you push?
| Applied Force | Static Friction | Result |
|---|---|---|
| 10 N | 10 N (matches) | Block stays still |
| 25 N | 25 N (matches) | Block stays still |
| 40 N | 40 N (matches) | Block is just about to move |
| 45 N | 40 N (cannot exceed limit) | Block accelerates forward |
The angle of friction is defined using the resultant of the normal reaction and the limiting friction force, and comparing it to the definition of the friction coefficient shows the two must be related by a simple identity. …
The angle of friction is the angle the resultant contact reaction makes with the normal at the point of sliding, and tan of that angle works out to be exactly the coefficient of limiting friction.
Consider a body resting on a rough horizontal surface, on the verge of sliding under an applied horizontal force. At this limiting condition, two forces act on it from the surface: the normal reaction N (perpendicular to the surface) and the limiting friction force f = mu_s N (along the surface, opposing the tendency to slide), where mu_s is the coefficient of limiting (static) friction.
The resultant R of N and f makes some angle theta with the normal N. By definition, this angle theta is called the angle of friction. From the geometry of the force triangle (N and f are perpendicular components of R):
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Showing the 12 most recent of 31 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Which of the following is the greatest?(a) Static friction(b) Kinetic friction(c) Rolling friction(d) Fluid friction
›Reveal solutionSolution
Experimentally, (limiting) static friction > kinetic friction > rolling friction, and fluid friction is generally the smallest of all for comparable conditions.
Friction between two solid surfaces has three regimes: static friction (opposes the start of relative sliding, up to a maximum/limiting value), kinetic friction (opposes actual sliding, once motion has started), and rolling friction (opposes rolling motion, arising mainly from small deformation at the contact patch). Experimentally:
fstatic, max>fkinetic>frolling …
- CBSE 2026Set ANNUAL1 markQ.By using ball bearings, sliding friction can be converted into ............. friction.
›Reveal solutionSolution
Ball bearings replace sliding contact with rolling contact, and rolling friction is much lower than sliding (kinetic) friction.
When two surfaces slide directly against each other, they experience sliding (kinetic) friction, which is relatively large because of interlocking surface irregularities scraping past one another. Ball bearings are small hard spheres placed between two surfaces (e.g. inside a wheel's axle housing) so that, instead of sliding, the surfaces roll over the balls. Rolling friction is much smaller than sliding friction because the contact area deforms only slightly and there is far less relative scraping motion at the contact points. This is why machines, whe …
- CBSE 2026Set ANNUAL1 markQ.Define coefficient of friction. OR Define angle of friction.
›Reveal solutionSolution
μ=f/N: the coefficient of friction is how many times larger the maximum frictional force is compared to the normal (perpendicular) contact force.
When two surfaces are in contact and one tends to slide (or slides) over the other, a frictional force f opposes this relative motion, and it is found experimentally to be directly proportional to the normal reaction force N pressing the two surfaces together:
f=μN
The constant of proportionality μ is called the coefficient of friction:
μ=Nf
…
- CBSE 2026Set ANNUAL1 markMCQQ.Match the column - select the correct definition (from Column B) for the term 'Friction' (Column A):(a) change in linear momentum(b) motion opposing force(c) loss of energy(d) rate of change of momentum(e) ability of doing work(f) rate of doing work
›Reveal solutionSolution
Friction is the force that opposes relative motion between contacting surfaces.
Friction arises at the point of contact between two surfaces and always acts in a direction that opposes the relative sliding motion (kinetic friction) or the tendency toward such motion (static friction) between the surfaces. It never aids motion; its defining role i …
- CBSE 2026Set ANNUAL1 markQ.Write the answer in one word/one sentence: Write the dimensional formula for coefficient of friction mu.
›Reveal solutionSolution
Coefficient of friction has no dimensions — it is a pure (dimensionless) number.
The coefficient of friction is defined as:
μ = f / N
where f is the frictional force and N is the normal reaction, both measured in newtons (dimensions [M L T⁻²]). Since μ is the ratio of two quantities with identical dimensions, all dimensions cancel:
μ = [M L T⁻²] / [M L T⁻²] = [M⁰ L⁰ T⁰]
…
- CBSE 2025Set ANNUAL1 markMCQQ.If normal force is doubled then coefficient of friction will be (A) doubled (B) four times (C) equal (D) half
›Reveal solutionSolution
The coefficient of friction μ does not depend on the normal force; it stays the same when N is doubled.
The friction force is related to the normal reaction by:
f=μN
…
- CBSE 2025Set ANNUAL1 markMCQQ.Which frictional force is the lowest?(a) Sliding friction(b) Rolling friction(c) Static friction(d) Both (A) and (B)
›Reveal solutionSolution
The general ordering of friction magnitudes (for the same pair of surfaces) is: static friction > sliding (kinetic) friction > rolling friction -- so rolling friction is the lowest.
Static friction: opposes the start of relative motion; its maximum value is usually the largest of the three.
Sliding (kinetic) friction: acts once surfaces are sliding over each other; smaller than maximum static friction.
…
- CBSE 2025Set ANNUAL1 markMCQQ.The coefficient of friction between the tyres and road is 0.1. The maximum speed with which a cyclist can take a circular turn of radius 3 m without skidding is (Take g = 10 ms^-2)(a) sqrt(15) ms^-1(b) sqrt(3) ms^-1(c) sqrt(30) ms^-1(d) sqrt(10) ms^-1
›Reveal solutionSolution
For a cyclist taking a flat circular turn, friction alone supplies the centripetal force; equating them gives v_max = sqrt(mu g r) = sqrt(3) m/s.
On a flat (unbanked) circular road, the maximum speed without skidding is set by the condition that the required centripetal force does not exceed the maximum available friction force:
…
- CBSE 2025Set ANNUAL1 markMCQQ.Match the following - Column A item: Coefficient of friction. Pick the matching relation from Column B.(a) F.V (Force . Velocity)(b) T is proportional to sqrt(l)(c) eta is proportional to 1/(dv/dx)(d) mu_s = tan(theta)(e) Y is proportional to 1/l(f) v^2/r
›Reveal solutionSolution
Coefficient of friction (mu_s) is related to the angle of friction/repose theta by mu_s = tan(theta).
When a body rests on an inclined surface, it is on the verge of sliding when the angle of incline equals the angle of friction theta, at which point the maximum static friction force just balances the component of gravity along the incline. This condition gives mu_s = tan(theta) — the stand …
- CBSE 2025Set sz1 markMCQQ.The maximum value of static friction when the body is at the verge of starting motion is known as: (A) Impending motion (B) Angle of repose (C) Static friction (D) Limiting friction
›Reveal solutionSolution
The maximum static friction, reached at the verge of motion, is called limiting friction.
Static friction adjusts itself to balance the applied force and can take any value up to a maximum.
The instant the body is on the verge of sliding, static friction attains this maximum value, known as limiting friction (fsmax=μsN). …
- CBSE 2024Set ANNUAL1 markMCQQ.If normal reaction becomes double then coefficient of friction will be (A) doubled (B) four times (C) equal (D) halved
›Reveal solutionSolution
Doubling the normal reaction leaves μ unchanged; only the friction force doubles.
The coefficient of friction is defined by f=μN, i.e. μ=f/N. It depends only on the nature and roughness of the two surfaces in contact, not on the magnitude of the normal reaction. If N is doubled, the frict …
- CBSE 2024Set SET-NDP60001 markMCQQ.The maximum value of static friction is called:(a) Rolling friction(b) Kinetic friction(c) Limiting friction(d) None of the above
›Reveal solutionSolution
The maximum value of static friction, reached just before sliding begins, is called limiting friction.
When an applied force tries to slide one surface over another, static friction adjusts itself to exactly balance the applied force, up to a maximum value. Below this maximum, the body stays at rest (static friction is self-adjusting). Once the applied force exceeds this maximum, the body starts to slide, and friction drops to (and stays close to) kinetic friction. This maximum value of static friction — the threshold value just before motion begins — is called limiting friction, fs,max=μsN, where μs is the coefficient of static friction and N is th …
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