Measuring Friction

A plank, a hinge and a protractor are enough to measure a friction coefficient. Here are two classic laboratory methods, plus the reason wheels changed the world.

Method 1: the horizontal table#

Fix a plank horizontally on a frame, with a smooth pulley at one end. Place block B on the plank and tie it with a string over the pulley to a hanger H. Add standard weights to the hanger little by little until the block just begins to slide.

Let the block weigh $W_1$ and the loaded hanger $W_2$ at that moment. The string’s tension equals $W_2$, which equals the limiting static friction, while the normal force equals $W_1$. So

For kinetic friction, remove a little weight, give the block a gentle push, and adjust until it keeps moving at constant speed. Then tension = kinetic friction, and $\mu_k = W_2'/W_1$. Adding weights on the block raises $N$ and the friction rises in proportion, so the ratio stays fixed; turning the block onto a different face changes the contact area but not the friction. Both laws can be checked this way.

Method 2: the inclined plane#

No pulley needed. Hinge one end of the plank and slowly raise the other. With the block at rest on a slope of angle θ, three forces act: weight $mg$, normal force $N$, and static friction $f_s$ up the slope. Resolving along and perpendicular to the plank:

Raise the plank until the block just begins to slip. At that angle friction is at its limit, so

Lower the slope slightly, nudge the block, and find the angle at which it keeps sliding at constant speed; there kinetic friction balances $mg\sin\theta$ and $\mu_k = \tan\theta'$. On any steeper slope a sliding block accelerates at $a = g(\sin\theta - \mu_k\cos\theta)$.

Rolling friction#

Dragging a heavy iron chest across a rough floor is hard work. Fit it with four iron wheels and it moves easily over the very same floor. The wheel doesn’t slide: its point of contact is momentarily at rest on the floor while the centre moves ahead, so the surfaces don’t rub. The resistance to rolling is far smaller than kinetic friction, often small enough to ignore beside other forces.