Torque, explained with examples
Torque is the turning effect of a force. It explains why a door is easier to push at the handle than near the hinges, and it is the rotational version of force in AP Physics.
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What torque is
A force can make an object move in a straight line, but it can also make it rotate. Torque, written τ (the Greek letter tau), measures how effectively a force causes rotation about a chosen point, called the pivot or axis. Torque depends on three things: how big the force is, how far from the pivot it acts, and the angle at which it is applied.
The formula
τ = r F sinθ
- r is the distance from the pivot to the point where the force acts, in metres;
- F is the size of the force, in newtons;
- θ is the angle between the force and the line from the pivot to that point.
Torque is measured in newton metres (N m). Although that is the same combination of units as the joule, torque is not energy, so we do not call it joules.
An equivalent way to think about it uses the lever arm (also called the moment arm): the perpendicular distance from the pivot to the line of action of the force. Then τ = F × (lever arm). Both methods give the same answer; use whichever is easier for the diagram.
Why angle matters
A force pointing straight at the pivot or straight away from it (θ = 0° or 180°) produces no torque at all, however large it is. A force at right angles to the line from the pivot (θ = 90°) produces the most torque. That is why you push a door at right angles to its surface, as far from the hinges as possible.
Direction and sign
Torques can turn an object counterclockwise or clockwise. In AP Physics, a common convention is to take counterclockwise as positive and clockwise as negative. Choose a convention at the start of each problem, mark it on your diagram, and stick to it.
Worked examples
Example 1: a wrench
A mechanic pulls with 80 N at right angles to a wrench, 0.25 m from the bolt. Find the torque.
- τ = rF sin 90° = 0.25 × 80 = 20 N m.
Example 2: an angled force
The same 80 N force is applied at 30° to the wrench handle. Find the torque.
- τ = 0.25 × 80 × sin 30° = 10 N m, half the previous value.
Example 3: a door
A 40 N push at right angles to a door acts 0.9 m from the hinges. What force at 0.3 m from the hinges gives the same torque?
- Torque = 40 × 0.9 = 36 N m. F = 36/0.3 = 120 N. Three times closer needs three times the force.
Rotational equilibrium
An object is in rotational equilibrium when the net torque on it is zero, so its angular velocity does not change. For a stationary object such as a balanced beam, both conditions of static equilibrium must hold: the net force is zero and the net torque about any point is zero. A smart choice of pivot removes unknown forces from the torque equation, because a force acting at the pivot has zero lever arm.
Example 4: a see-saw
A 300 N child sits 1.5 m from the pivot of a see-saw. Where must a 450 N adult sit on the other side to balance it?
- Counterclockwise torque = clockwise torque: 300 × 1.5 = 450 × d.
- d = 1.0 m.
Example 5: a beam on two supports
A uniform 6.0 m beam weighing 400 N rests on supports at each end. A 600 N load sits 2.0 m from the left end. Find the force from each support.
- Take torques about the left support to remove that force. The beam's weight acts at its centre, 3.0 m away.
- Right support × 6.0 = 400 × 3.0 + 600 × 2.0 = 2400, so the right support pushes up with 400 N.
- Forces: left + right = 1000 N, so the left support pushes with 600 N. It makes sense that the support nearer the load carries more.
Example 6: a hanging sign
A uniform 2.0 m horizontal rod of weight 50 N is hinged to a wall at one end and held by a cable from its far end at 30° to the rod. A 100 N sign hangs from the far end. Find the cable tension.
- Take torques about the hinge so the hinge force drops out.
- Cable: T × 2.0 × sin 30° = T. Clockwise: 50 × 1.0 + 100 × 2.0 = 250 N m.
- T = 250 N.
Torque and angular acceleration
When the net torque is not zero, the object's rotation speeds up or slows down. This is Newton's second law for rotation:
Στ = Iα
Here I is the rotational inertia, the rotational equivalent of mass, and α is the angular acceleration in rad/s². The same torque gives a smaller angular acceleration to an object whose mass is spread further from the axis.
Example 7: angular acceleration
A net torque of 6.0 N m acts on a wheel with rotational inertia 0.40 kg m². Find its angular acceleration.
- α = τ/I = 6.0/0.40 = 15 rad/s².
Common mistakes
- Using the distance along the force instead of the perpendicular distance to the pivot.
- Forgetting sinθ when the force is not perpendicular.
- Leaving out the weight of the beam itself, which acts at its centre if uniform.
- Choosing a pivot that keeps unknown forces in the torque equation.
- Mixing clockwise and counterclockwise signs.
Practice questions
Try each one before opening the answer.
Q1. A 50 N force acts at right angles 0.4 m from a pivot. Find the torque.
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Q2. A force acts directly towards the pivot. What torque does it produce?
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Q3. A 2 kg mass hangs 0.5 m from the pivot of a light rod. What mass 0.2 m on the other side balances it?
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Q4. A disc has I = 0.2 kg m² and angular acceleration 10 rad/s². Find the net torque.
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Frequently asked questions
Is torque on AP Physics 1?
Yes. Torque and rotational dynamics form a full unit in AP Physics 1, and torque also appears in AP Physics C: Mechanics using the cross product.
Is torque the same as a moment?
Yes. UK courses usually call it the moment of a force, measured in N m. The physics is identical.
Key takeaways
- τ = rF sinθ, or force times lever arm.
- Torque is greatest when the force is perpendicular.
- Equilibrium: net force zero and net torque zero.
- Choose the pivot to remove unknown forces.
- Στ = Iα links torque to angular acceleration.