Torque and angular momentum
The turning effect of a force
Push a door near its hinges and it hardly moves. Push at the handle with the same force and it swings open easily.
The turning effect of a force is called torque, or moment of force. It depends on the force and on where you apply it.
Let r be the position vector from the axis to the point where the force acts. Torque is defined as τ = r × F.
Its size is τ = rF sin θ, where θ is the angle between r and F. The unit is the newton metre, N m.
Extend the force into a line, its line of action. The perpendicular distance from the axis to that line is r sin θ, the moment arm. So torque is force times moment arm.
Push straight towards the hinge, θ = 180°, and the torque is zero however hard you push. Push at right angles, θ = 90°, and the torque is the biggest: rF.
Because τ is a cross product, it points along the axis. A force turning the door anticlockwise, seen from above, gives a torque pointing up.
Angular momentum
Linear momentum p = mv measures the motion of a particle. For turning, we need its rotational partner.
The angular momentum of a particle about a point O is l = r × p, where r is its position measured from O.
Its size is l = rp sin θ. For a particle going round a circle of radius r with speed v, r and p are at right angles, so l = mvr.
Now differentiate l = r × p with time. The result is dl/dt = r × F. So τ = dl/dt.
This is Newton’s second law for rotation: torque changes angular momentum, just as force changes linear momentum.
For a system of particles, internal torques cancel in pairs. Only the external torque counts: τ_ext = dL/dt.
If the total external torque is zero, L stays constant. This is the law of conservation of angular momentum.
Watch a puck on a smooth table, tied to a string through a hole. Pull the string down. The pull points to the hole, so it has no torque about it. As r shrinks, the puck speeds up, keeping mvr the same.
Explore torque in 3D
- r runs from the pivot (the origin) to where the force acts. F is the force.
- Turn F around and watch τ = rF sin θ change. When is it zero?
- Press “Top view” to look down the axis. Anticlockwise turning gives τ pointing up at you.
Work it out
You pull on a spanner with a force of 20 N, 0.25 m from the nut, at 30° to the spanner. What torque do you apply?
- 5 N m
- 2.5 N m
- 4.3 N m
- 80 N m
Answer
B. τ = rF sin θ = 0.25 × 20 × sin 30° = 5 × 0.5 = 2.5 N m. Pulling at 90° would give the full 5 N m.
Quick check
A particle moves in a straight line at constant velocity. The line does not pass through point O. What happens to its angular momentum about O?
- It is zero
- It increases as the particle moves away
- It stays constant
- It changes direction
Answer
C. l = r⊥ p, where r⊥ is the perpendicular distance from O to the line of motion. Both r⊥ and p stay the same, so l is constant. This fits τ = dl/dt: no force, no torque, no change.
Summary
- Torque:
τ = r × F, τ = rF sin θ - Angular momentum of a particle:
l = r × p, l = rp sin θ - Second law for rotation:
τ = dl/dt - System of particles:
τ_ext = dL/dt
- Torque is force times moment arm (perpendicular distance from the axis to the line of action). Unit: N m.
- A force along r, pointing at the axis, gives zero torque.
- Torque and angular momentum are cross products, so they point along the axis (right-hand rule).
- If the total external torque is zero, total angular momentum is conserved.