Friction

When a surface pushes on you, it doesn’t have to push straight out. Tilt that push even slightly and you have discovered friction.

The contact force#

Wherever two bodies touch, electromagnetic forces act between the charged particles at their surfaces. The upshot is that each body exerts a contact force on the other. By Newton’s third law the two contact forces are equal in size and opposite in direction.

The contact force on a body need not be perpendicular to the surface. So we split it into two perpendicular pieces:

Kinds of friction#

Friction acts between two solids, between a solid and a fluid, and between two fluids. Friction in fluids is called viscous force and gets its own chapter later. Here we stay with solid surfaces, where two cases matter:

• Kinetic friction: the surfaces slip over each other.
• Static friction: the surfaces are in contact but do not slip.

Building a reliable theory of friction from atomic forces is hard. Fortunately a wide range of experiments can be summarised by a few simple laws, which the next lessons develop.

Slide a phone across a table and it soon stops. Kinetic friction did that, and it follows two rules about direction and size that are easy to state and easy to misapply.

Direction#

When body A slides over body B, the friction on A acts so as to oppose the slipping. Precisely:

By the third law the friction on B (due to A) is opposite to that, which is the same as opposite to B’s velocity relative to A. The two rules agree.

Here is where intuition slips. A small block sits on a long wheeled trolley rolling to the right, and the block is sliding toward the back of the trolley. From the road, both move to the right (the block more slowly). But relative to the trolley the block moves left, so friction on the block points right, in the direction it is already moving over the road. Explore it:

Size#

The magnitude of kinetic friction is proportional to the normal force between the surfaces:

The constant $\mu_k$ is the coefficient of kinetic friction. It depends on the two materials and on how smooth or rough their surfaces are: small for polished surfaces, large for rough ones.

What it does not depend on

• Speed. Once sliding, the friction is $\mu_kN$ whatever the speed. This is a good approximation for modest relative speeds (up to roughly 10 m/s).
• Contact area. For a given normal force, a brick slid on its broad face and on its narrow end feels the same friction.

Three friends lean on a loaded steel cupboard and it doesn’t budge a millimetre. Something is pushing back exactly as hard as they push. That something is static friction.

A force that adjusts itself#

Surfaces in contact that are not slipping can still exert friction on each other. The cupboard is at rest, so the net force on it is zero: the floor must be pushing it back, along the floor, as hard as the friends push it forward. This is static friction.

Push gently and static friction is small; push harder and it grows to match. It is self-adjusting: it takes whatever size and direction are needed to keep the two surfaces at rest relative to each other. But only up to a point.

Once the applied force exceeds the limiting friction $\mu_sN$, static friction can no longer keep up, and slipping begins. From then on kinetic friction $\mu_kN$ takes over. Usually $\mu_s$ is a little larger than $\mu_k$, which is why getting something moving takes more effort than keeping it moving. Like kinetic friction, the limiting friction does not depend on the contact area for a given normal force.

Which way does static friction point?#

Use Newton’s laws. Put block A on block B on a table and pull B to the right. If nothing moves, A is in equilibrium and no other horizontal force acts on it, so the friction on A is zero. Pull harder so that both blocks accelerate together to the right: A now needs a net force to the right, and friction is the only candidate. So friction on A points right, in the direction of motion.

A second trick for the direction: imagine the surfaces were frictionless for a moment. Work out which way they would slip relative to each other; static friction points against that would-be slipping.

Everything we know about dry friction fits on an index card. The reasons behind it take us down to the atoms.

The laws of friction#

1. If the surfaces slip, friction has magnitude $f_k = \mu_kN$, where $N$ is the normal force and $\mu_k$ the coefficient of kinetic friction.
2. Kinetic friction on a body is opposite to its velocity relative to the body applying the friction.
3. If the surfaces do not slip, friction satisfies $f_s \le \mu_sN$; its direction and size are whatever is needed to prevent slipping.
4. For a given normal force, neither $f_k$ nor the limiting $\mu_sN$ depends on the area of contact.

Typical values#

Friction coefficients depend on smoothness, cleanliness, moisture and even the surrounding air, so tables give only rough guides. Approximate coefficients of static friction:

Why friction exists#

A polished steel plate looks flat to the eye, but at atomic scale its surface is a landscape of hills and valleys. Put two such surfaces together and they touch only at the tips of the highest bumps. The real area of contact is a tiny fraction of the apparent area.

At those contact points atoms come so close that molecular bonds form. To slide one surface over the other, these bonds must be broken; the material around them is deformed, and new bonds form further along. The deformation sends vibrations into both bodies, which die away as random thermal motion: the surfaces get warm. That is why a force is needed both to start sliding and to keep it going, and why rubbing your hands heats them.

This picture also explains the laws. Press the surfaces harder and more bumps touch, so the real contact area, and with it the friction, grows in proportion to the normal force. Spread the same load over a bigger apparent area and each bump is pressed less hard; the real contact area, and so the friction, stays about the same.

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.