Rotation and Torque: Levers, Spin and Angular Momentum

To describe something turning, measure the angle θ it has turned through. Physicists use radians: the angle in radians is the arc length s traveled by a point divided by its distance r from the axis,

θ = s ÷ r

A full turn moves a point once around the circumference, 2πr, so one turn is 2π rad, and 1 rad is about 57.3°.

Angular speed is the angle turned per second, ω = Δθ ÷ Δt. Every point on a spinning wheel has the same ω, but points farther out move faster along their bigger circles: v = rω. That's why the outside of a merry-go-round feels wilder than the middle. Earth spins once a day, so its ω is about 7.3 × 10⁻⁵ rad/s, and a spot on the equator moves at about 465 m/s.

Angular acceleration is how fast ω changes, α = Δω ÷ Δt, in rad/s². A fan speeding up after you switch it on has an angular acceleration; once it reaches a steady speed, α is zero.

A force makes something turn only if it's applied away from the axis and not straight toward or away from it. Torque measures this turning effect:

τ = rF sin θ

Here r is the distance from the axis to where the force acts, F is the size of the force and θ is the angle between the force and the line from the axis. Torque is largest when you push at right angles (θ = 90°, sin θ = 1) and zero when you push straight along the handle (θ = 0°).

That's why door handles sit far from the hinges, and why a plumber slips a pipe over a wrench handle to free a stuck bolt: doubling r doubles the torque for the same push. Torques can turn things counterclockwise or clockwise, and if they cancel, nothing starts to turn.

A lever is a rigid bar that turns about a pivot, called the fulcrum. A seesaw is a lever, and so are crowbars, scissors, wheelbarrows and your forearm. A lever balances when the torques on the two sides are equal:

F₁d₁ = F₂d₂

Here d₁ and d₂ are the distances from the pivot when the forces act at right angles to the bar. A heavy child sitting close to the middle of a seesaw can balance a lighter child sitting farther out. Archimedes stated this law of the lever in the 200s BC.

A lever can turn a small force into a big one, but it doesn't give you something for nothing. If the long end moves 3 times as far as the short end, the force at the short end is 3 times bigger, and the work done on each side is the same.

For an object to stay completely still, the forces on it must cancel and the torques must cancel too. Engineers check both conditions when designing bridges, cranes and shelves.

Every object has a center of mass, the balance point where you can treat all its mass as if it were gathered together. Gravity acts as if it pulls there. Throw a hammer spinning through the air, and its center of mass follows a smooth curve even while the rest of it tumbles around that point.

For masses along a line, the center of mass is the weighted average of their positions:

x_cm = (m₁x₁ + m₂x₂ + …) ÷ (m₁ + m₂ + …)

For a uniform shape, like a ruler or a ball, it's at the geometric center. For an uneven object, it sits closer to the heavier part. An object rests stably when its center of mass is above its base of support. Lean too far and the center of mass moves past the edge of the base, gravity's torque tips you over, and you fall. Racing cars are built low and wide to keep the center of mass low and over a wide base.

Newton's second law has a turning version. A net torque gives an angular acceleration:

τ = Iα

The rotational inertia I plays the role of mass. It depends on both the mass and how far that mass is from the axis: for a single small mass m at distance r, I = mr². Because r is squared, mass far from the axis counts for much more than mass near it.

A hoop of mass M and radius R has all its mass at the rim, so I = MR². A solid disk of the same mass and radius has I = ½MR², because much of its mass is near the axle. Roll the two down the same ramp and the disk wins every time. A tightrope walker carries a long pole for the same reason: its large rotational inertia means a small wobble doesn't quickly turn into a fall.

A spinning object has angular momentum:

L = Iω

If no outside torque acts, L stays constant. This is the conservation of angular momentum, the turning version of the conservation of momentum. If I gets smaller, ω must get bigger to keep the product the same.

A figure skater starts a spin with her arms out, then pulls them in. Her rotational inertia drops, so she spins faster, sometimes several times faster. Divers and gymnasts tuck to spin quickly and stretch out to slow down before landing. A collapsing star can shrink into a neutron star about 20 km across, spinning up to hundreds of times a second.

The direction of the spin axis is conserved too. A spinning top stays up, and spinning wheels help a moving bicycle stay steady. A spinning gyroscope keeps pointing the same way, which is why spacecraft use spinning gyroscopes to sense and control how they're turning. (Phones use tiny vibrating sensors also called gyroscopes.) Kepler's second law from Unit 11 is another example: a planet speeds up near the Sun because its angular momentum about the Sun stays the same.

The law of the lever is one of the oldest results in physics. The rules for spinning bodies took much longer: they grew out of work on pendulum clocks, then Euler's mathematics of rigid bodies, and finally experiments with spinning wheels that could show Earth turning.