Friction, Inclined Planes and Belt Friction

Friction is the force that resists sliding between two surfaces. GATE asks friction questions on blocks, inclines, ladders and belts; all follow from a few rules.

Laws of dry (Coulomb) friction

• Friction acts along the surface, opposite to the motion (or the tendency of motion).
• Before sliding, friction takes whatever value equilibrium needs, up to a limit: F ≤ μ_s N.
• At the point of sliding, F = μ_s N (limiting friction). While sliding, F = μ_k N, with μ_k slightly less than μ_s.
• Friction does not depend on the contact area.

Angle of friction φ: the angle between the normal reaction and the total reaction at limiting friction. tan φ = μ.

Angle of repose: the steepest incline on which a block stays at rest. It equals the angle of friction.

Block on an inclined plane

For a block of weight W on an incline at angle θ:

• Normal reaction N = W cos θ.
• Component pulling it down the slope = W sin θ.
• It stays at rest if W sin θ ≤ μ W cos θ, that is tan θ ≤ μ.

The force P parallel to the incline needed to push it up at constant speed: P = W(sin θ + μ cos θ). To just stop it sliding down: P = W(sin θ − μ cos θ).

Ladder against a wall

A ladder of length L and weight W leans on a rough floor (μ₁) and a wall (μ₂). Draw the FBD with N and F at both ends, take moments about the foot, and use F ≤ μN at each surface. For a uniform ladder with a smooth wall, the smallest angle θ with the floor at which it can stand is given by tan θ_min = 1/(2μ₁).

Belt friction

For a flat belt wrapped through angle θ (radians) on a pulley, at the point of slipping:

T₁/T₂ = e^(μθ), where T₁ is the tight-side tension.

Power transmitted = (T₁ − T₂) v. For a V-belt with groove angle 2β, replace μ by μ/sin β.

Worked example

A 100 N block rests on a 30° incline with μ = 0.25. Will it slide, and what force parallel to the slope pushes it up at constant speed?

• tan 30° = 0.577 > 0.25, so it slides if released.
• P = 100 (sin 30° + 0.25 cos 30°) = 100 (0.5 + 0.2165) = 71.65 N.

Practice