Friction, Drag and Inclined Planes

Friction always acts along the surface, against the sliding, or against the way something would slide if friction weren't there. In Newton's second law it's just one more force to add up.

While an object is still, static friction matches your push exactly: push with 30 N and it pushes back with 30 N, so nothing moves. But it has a maximum. Push past that and the object starts to slide. Now kinetic friction takes over, and it's usually smaller, which is why a heavy box lurches when it finally gets going and is then easier to keep moving.

Friction comes from the tiny bumps on even smooth-looking surfaces catching and sticking to each other. Pressing the surfaces together harder makes more of them touch, so friction grows with the normal force.

For most dry surfaces the friction force is close to proportional to the normal force:

f_max = μ_sN (static, the most it can be)

f = μ_kN (kinetic, while sliding)

μ (the Greek letter mu) is the coefficient of friction. It has no units and depends on both surfaces. On flat ground with no other vertical forces, N = mg. Most dry pairs have μ between about 0.3 and 0.6. Rubber on dry concrete reaches about 1, and Teflon gets as low as 0.04. Water and oil act as lubricants and cut friction a lot.

On a ramp tilted at angle θ, the weight mg still points straight down. It's easiest to split it into components along the slope and at right angles to it:

along the slope: mg sin θ into the slope: mg cos θ

The slope pushes back with a normal force N = mg cos θ, smaller than on flat ground, so the maximum friction μ_smg cos θ shrinks as the ramp gets steeper while the downhill pull mg sin θ grows. If the block slides, the net force down the slope is mg sin θ − μ_kmg cos θ, so its acceleration is:

a = g(sin θ − μ_k cos θ)

The mass cancels: a heavy block and a light one of the same material slide down the same ramp with the same acceleration.

Tilt a board slowly with a block on it. The block starts to slip when the downhill pull equals the maximum friction: mg sin θ = μ_smg cos θ. Dividing both sides by mg cos θ gives:

tan θ = μ_s

That angle is the angle of repose. It's also a simple way to measure μ_s: tilt until it slips and take the tangent. Piles of sand, gravel and grain settle into cones whose sides sit at their own angle of repose, which is why engineers check it when designing slopes and storage heaps.

Once the block is moving, it keeps sliding on any slope steeper than tan θ = μ_k. Because μ_k is usually smaller, there's a band of angles where a block sits still but slides away if you nudge it.

Moving through air or water also costs a force, called drag. Unlike sliding friction, it depends strongly on speed. For everyday objects moving at everyday speeds it follows the drag equation:

F_D = ½ρC_dAv²

Here ρ is the density of the air (about 1.2 kg/m³ near sea level), A the frontal area, v the speed and C_d the drag coefficient, a number with no units that depends on shape. Because of the v², doubling your speed makes drag four times bigger, which is why cars use much more fuel at highway speeds.

When you first start to fall, drag is tiny and you speed up at almost g. As you go faster, drag grows, so the net force and your acceleration shrink. Eventually drag equals your weight and you stop speeding up. That steady speed is the terminal velocity. Setting ½ρC_dAv² = mg:

v_t = √(2mg ÷ (ρC_dA))

A skydiver falling belly-down reaches about 55 m/s (roughly 200 km/h). Diving head-first shrinks A and makes them much faster. Opening a parachute makes C_dA many times bigger, so the terminal velocity drops to a safe landing speed.

Brakes. Brake pads squeeze a spinning disc. Kinetic friction between pad and disc turns the car's kinetic energy into heat, which is why brakes get hot.

Tires. A rolling tire doesn't slide on the road; its contact patch is momentarily still, so the grip is static friction. On dry concrete that's up to about μ_s = 1.0, but water cuts it to about 0.30. Tread grooves squeeze water out of the way. Anti-lock brakes (ABS) stop the wheels locking, keeping them on the bigger static friction instead of skidding on kinetic friction, and letting the driver still steer.

Skis and skates. Steel on ice has a very low μ, about 0.03, so skaters and skiers glide far with little effort. Waxing skis lowers friction further.

Parachutes. A big canopy has a large area and high drag, so the terminal velocity is low enough to land safely.

People have used friction for as long as they've made fire by rubbing sticks, and fought it with rollers and grease. The rules for it took much longer to work out.