Friction on an inclined plane

Physics · Senior school and first-year physics · about 10 min

A block on a ramp is one of the most common problems in mechanics, because it brings together free-body diagrams, resolving forces at an angle, and friction. Once you can do this one, most force problems follow the same pattern.

The key idea

Tilt your axes to line up with the slope. For a block of mass $m$ on an incline at angle $\theta$ :

N = mg\cos\theta \qquad F_{\parallel} = mg\sin\theta \qquad f_{max} = \mu N

It slides if $mg\sin\theta > \mu_s mg\cos\theta$ , that is, if $\tan\theta > \mu_s$ .

Try it yourself

Tilt the ramp and change the friction to see when the block starts to slide.

Why tilt the axes

The block can only move along the slope, so put one axis along it and one perpendicular to it. Then the normal force and friction each lie on an axis, and only gravity needs splitting:

along the slope, down: $mg\sin\theta$
into the slope: $mg\cos\theta$

Nothing moves perpendicular to the slope, so $N = mg\cos\theta$ .

A straight incline rising to the right, with its angle marked at the lower left. A rectangular block sits flat on it. From the block's centre: one dark red arrow straight down for weight; one dark blue arrow at right angles to the slope, pointing away from it, for the normal force; one dark blue arrow along the slope, pointing up the slope, for friction. Two thin dashed arrows show the parts of the weight: one along the slope pointing down the slope, and one into the slope. Mark the angle between the weight arrow and the dashed arrow going into the slope. — With tilted axes, only gravity needs splitting into two parts.

Static and kinetic friction

Static friction holds the block still. It can be anything up to $\mu_s N$ , just enough to stop it moving.
Once it slides, kinetic friction acts, $f_k = \mu_k N$ , opposite to the motion. $\mu_k$ is usually a bit smaller than $\mu_s$ .

So first check whether it moves at all ( $\tan\theta$ against $\mu_s$ ), then use $\mu_k$ for the acceleration.

Worked example: does it slide, and how fast does it speed up?

A $5$ kg block sits on a $30^\circ$ slope. The coefficients of friction are $\mu_s = 0.4$ and $\mu_k = 0.3$ . Take $g = 9.81$ m/s $^2$ . Does it slide? If so, what is its acceleration?

A straight incline rising to the right at 30 degrees above a thin horizontal reference line, with the angle marked at the lower left. A rectangular block sits flat on it. — The block in this example.

Show the solution

Does it slide? $\tan 30^\circ = 0.577$ , which is greater than $\mu_s = 0.4$ , so static friction can't hold it: it slides.

Normal force: $N = 5 \times 9.81 \times \cos 30^\circ = 42.48$ N

Kinetic friction: $f_k = 0.3 \times 42.48 = 12.74$ N, acting up the slope.

Acceleration along the slope:

a = g(\sin\theta - \mu_k\cos\theta) = 9.81\,(0.5 - 0.3 \times 0.866) = 2.36 \text{ m/s}^2

down the slope.

Common mistakes

Writing $N = mg$ on a slope

On an incline only the part of gravity into the slope is balanced: $N = mg\cos\theta$ .

Swapping $\sin$ and $\cos$

Check with an extreme: at $\theta = 0$ (flat) the force along the slope must be $0$ , so it's $mg\sin\theta$ .

Using the kinetic coefficient to decide whether it starts moving

Starting is decided by static friction, $\mu_s$ . Use $\mu_k$ once it's already sliding.

Check your understanding

1. A block rests on a $20^\circ$ slope. What is the normal force for a $10$ kg block ( $g = 9.81$ )?

2. A block just stays put on a slope with $\mu_s = 0.5$ . What is the steepest angle before it slides?

3. Which way does kinetic friction act on a block sliding down a slope?

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