Turning instead of moving
Wheels, doors, fans, planets and figure skaters all rotate: every part moves in a circle around anaxis of rotation. Points far from the axis move faster than points near it, but the whole object turns through the same angle in the same time. So we describe rotation with angles rather than distances.
Angle, angular velocity and acceleration
- Angular displacement θ is the angle turned, measured in radians (one full turn = 2π rad ≈ 6.28 rad).
- Angular velocity ω is how fast the angle changes: ω = Δθ/Δt, in rad/s. A point at distance r moves at speed v = rω.
- Angular acceleration α is how fast ω changes: α = Δω/Δt, in rad/s².
Counterclockwise rotation is usually counted as positive. In three dimensions these quantities point along the axis, using the right-hand rule: curl the fingers of your right hand in the direction of rotation and your thumb points along the axis. In the simulator, ⊙ means “out of the screen” (counterclockwise) and ⊗ means “into the screen” (clockwise).
Torque
A torque is the turning effect of a force. It depends on three things: how big the force is, how far from the axis it acts, and its direction:
τ = r F sin φ
Here τ (tau) is the torque in newton metres (N·m), r is the distance from the axis to the point where the force acts, F is the force in newtons and φ is the angle between the radius and the force. The quantity r sin φ is thelever arm r⊥: the perpendicular distance from the axis to the line along which the force acts. A force at right angles to the radius (φ = 90°) gives the most torque; a force pointing straight through the axis (φ = 0° or 180°) gives none.
Moment of inertia
The moment of inertia I measures how hard it is to change an object’s rotation. It depends on the mass and, even more, on how far that mass is from the axis: I = Σ m r². Doubling the distance of a mass from the axis quadruples its contribution.
| Object (axis through the centre) | Moment of inertia |
|---|---|
| Solid disk or cylinder | I = ½ M R² |
| Thin ring or hoop | I = M R² |
| Thin rod of length L | I = (1/12) M L² |
| Point mass at distance d | I = m d² |
Newton’s second law for rotation
A net torque gives an angular acceleration: τ = I α, just as F = m a for straight-line motion. With the same torque, a ring (all its mass at the rim) speeds up half as fast as a disk of the same mass and radius. With no torque, a spinning object keeps spinning at a constant ω: Newton’s first law for rotation.
Linear and rotational motion
| Linear | Rotational | Link |
|---|---|---|
| Displacement x | Angle θ | x = r θ |
| Velocity v | Angular velocity ω | v = r ω |
| Acceleration a | Angular acceleration α | a = r α |
| Mass m; force F | Moment of inertia I; torque τ | F = ma ↔ τ = Iα |
| Momentum p = mv | Angular momentum L = Iω | Both conserved if isolated |
Angular momentum
Angular momentum L = I ω (units kg·m²/s) is the rotational partner of momentum. A torque changes it: τ = ΔL/Δt. So when no outside torque acts, angular momentum is conserved. A person on a frictionless stool who pulls their arms in reduces I; L cannot change, so ω must increase. With I falling from about 5.0 to 1.4 kg·m² in the simulator, the spin rate rises about 3.5 times. The kinetic energy ½Iω² = L²/2I increases at the same time: the extra energy comes from the work the muscles do pulling the masses inwards against their tendency to move outwards.
Real-world applications
- Spanners, door handles and seesaws use a long lever arm to make a big torque from a small force.
- Figure skaters, divers and cats change their moment of inertia to spin faster or slower in the air.
- Flywheels store energy and smooth out engines; a large I resists changes in speed.
- Gyroscopes and spinning satellites keep their orientation because angular momentum is conserved.
- Collapsing stars spin faster as they shrink; neutron stars can rotate hundreds of times a second.
Common misconceptions
- “A bigger force always means more torque.” Torque also depends on the distance and the angle; a force through the axis gives none.
- “Objects with the same mass are equally easy to spin.” The distribution of the mass matters: a ring is harder to spin up than a disk.
- “A spinning object needs a torque to keep spinning.” Only changes in rotation need a torque.
- “If angular momentum is conserved, so is kinetic energy.” Pulling the arms in increases the kinetic energy.
What the model assumes
The objects are rigid and turn about a fixed, frictionless axle, with no air resistance. The force is attached to the object, like a small thruster, so it turns with the wheel and keeps the same angle to the radius; this keeps the torque constant. The ring’s spokes and the rod’s ends are idealised (massless spokes, point masses on the rod). The person on the stool is a simple model: the body and stool together have I = 1.2 kg·m², each arm is a 3 kg uniform rod from the axis, and each dumbbell is a 2 kg point mass. Real bodies are more complicated, but the conservation of angular momentum holds exactly. The wheel is stopped automatically if it spins faster than about 8 turns per second.
Key takeaways
- Torque τ = r F sin φ: more force, a longer lever arm or a more perpendicular push all give more torque.
- Moment of inertia depends on how far mass is from the axis: I = Σ m r².
- τ = I α is Newton’s second law for rotation; with no torque, ω stays constant.
- Angular momentum L = I ω is conserved when no outside torque acts, so lowering I raises ω.
- Kinetic energy is not conserved when a spinning body changes shape; work is done.