Dry friction and the friction angle

By the end of this topic you will be able to

Distinguish static from kinetic friction and apply F ≤ μsN.
Solve equilibrium problems for a body on a rough incline.
Calculate the angle of repose from the coefficient of static friction.

What friction does

When two surfaces are pressed together, friction opposes any tendency to slide. It is a reaction force: it only grows as large as it needs to be to prevent sliding, up to a limit.

Static friction $F$ holds the body at rest, and adjusts to whatever is needed, up to $F_{\max} = \mu_s N$ .
Kinetic friction acts once sliding starts, and is roughly constant: $F = \mu_k N$ .

Here $N$ is the normal reaction and $\mu_s \ge \mu_k$ are the coefficients of static and kinetic friction, which depend on the two materials and not on the contact area.

A block on an incline

For a block of weight $W$ on a slope of angle $\theta$ , resolve along and perpendicular to the slope:

N = W\cos\theta, \qquad F_{\text{needed}} = W\sin\theta - P

where $P$ is any force pulling up the slope. The block stays at rest as long as $|F_{\text{needed}}| \le \mu_s N$ .

With no pulling force, the block is on the point of slipping when $W\sin\theta = \mu_s W\cos\theta$ , that is

\tan\phi_s = \mu_s

The angle $\phi_s$ is the angle of repose: the steepest slope on which a heap of the material will stay put.

Once the block slides down, Newton's second law gives

a = g\left(\sin\theta - \mu_k\cos\theta\right)

Interactive modelTilt the slope and add a pulling force. See whether static friction can hold the block.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Worked example

A 100 N block rests on a 20° slope. $\mu_s = 0.4$ and $\mu_k = 0.3$ . Does it slip? What pulling force up the slope would make it just start to move up?

Normal force. $N = 100\cos20^\circ = 94.0$ N, so $F_{\max} = 0.4\times94.0 = 37.6$ N.

Friction needed. $F = 100\sin20^\circ = 34.2$ N, which is less than 37.6 N, so the block stays at rest. (Check: $\tan 20^\circ = 0.364 < 0.4$ .)

Pulling up. Just before it moves up, friction acts down the slope at its limit: $P = W\sin\theta + \mu_s N = 34.2 + 37.6 = 71.8$ N.

Practice problems

A crate on a flat floor weighs 500 N. $\mu_s = 0.35$ . What horizontal force is needed to start it moving?

Show answer

$N = 500$ N, so $F_{\max} = 0.35\times500 = 175$ N. A force slightly greater than 175 N is needed.
Wet sand has $\mu_s = 0.8$ . What is the steepest slope of a sand heap?

Show answer

$\phi_s = \tan^{-1} 0.8 = 38.7^\circ$ .
A block slides down a 30° slope with $\mu_k = 0.3$ . Find its acceleration.

Show answer

$a = 9.81(\sin30^\circ - 0.3\cos30^\circ) = 9.81(0.5 - 0.260) = 2.36$ m/s².
In the model, at what angle does a block with $\mu_s = 0.4$ begin to slide when P = 0?

Show answer

$\tan^{-1}0.4 = 21.8^\circ$ . Above this angle the state changes to sliding.

Key formulae

Limiting static friction: $F_{\max} = \mu_s N$
Kinetic friction: $F = \mu_k N$
Angle of repose: $\tan\phi_s = \mu_s$
Acceleration down a rough slope: $a = g\left(\sin\theta - \mu_k\cos\theta\right)$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (40 min). 5 min: demonstration with a book on a tilting board (students predict the angle); 10 min: static vs kinetic and the friction cone idea; 15 min: worked example; 10 min: model and angle of repose.

Activity. Bring a tilting board and measure the angle of repose for several materials, then check $\mu_s = \tan\phi$ with the model.

Discussion. Why does ABS braking help? (Static friction is larger than kinetic.)

Assessment idea. Add a second block connected by a rope over a pulley.