IB Physics HLRigid Body MechanicsHL onlyPaper 1 & 2~11 min read
Torque & Couples
Every time you push a door, tighten a bolt, or turn a steering wheel, you’re producing a turning effect — and how strong that turning effect is depends on more than just how hard you push. It matters where you push and at what angle. This turning effect is the moment of a force, and when we’re talking about rotation it’s called torque. Get comfortable with it here, because torque is to rotation what force is to straight-line motion — the thing that makes rotating objects speed up or slow down.
📘 What you need to know
A moment is the turning effect of a force about a pivot
Moment = force × perpendicular distance from the pivot, in newton metres (N m)
Torque is the change in rotational motion due to a turning force: τ = Fr sin θ
When the force is perpendicular (θ = 90°) this simplifies to τ = Fr
A couple is a pair of equal, opposite, non-aligned forces that produce rotation only
A couple produces no net force but a net turning effect (a torque)
The torque of a couple = Fr sin θ for each force, giving a net torque of 2Fr sin θ
The moment of a force
A moment is the turning effect a force has when it acts at a distance from a pivot. The further from the pivot you apply the force, and the larger the force, the greater the turning effect — which is exactly why a longer spanner makes a stubborn bolt easier to shift.
Moment of a force
Moment = Force × perpendicular distance from the pivot
The moment is measured in newton metres (N m). The crucial word is perpendicular: it’s the distance measured at right angles to the line of the force that counts.
A perpendicular force F acting a distance r from the pivot produces a moment of F × r.
Non-perpendicular forces
Forces don’t always act at a convenient right angle. When a force acts at an angle θ to the lever, only the part of it that acts perpendicular to the lever produces a turning effect. We handle this with the torque equation:
Torque of a forceτ = Fr sin θ
where τ is the torque in newton metres (N m), F is the applied force in newtons (N), r is the distance from the axis of rotation to where the force is applied in metres (m), and θ is the angle between the force and the axis of rotation. When the force acts perpendicular to the lever (θ = 90°), sin 90° = 1, and the equation neatly simplifies to τ = Fr.
When the force acts at an angle θ, only the perpendicular distance r sin θ contributes to the turning effect, so τ = Fr sin θ.
WE 1
A mechanic applies a 15 N force perpendicular to a spanner, a distance of 0.20 m from the bolt. Calculate the moment about the bolt.
Step 1 — force is perpendicular, so use moment = F × r
moment = F × r
Step 2 — substitutemoment = 15 × 0.20moment = 3.0 N mA longer spanner (bigger r) would give a larger moment for the same push.
WE 2
A 60 N force is applied at the end of a 0.40 m lever, at an angle of 35° to the lever. Calculate the torque produced about the pivot.
Step 1 — force is at an angle, so use τ = Fr sinθ
τ = F × r × sinθ
Step 2 — substituteτ = 60 × 0.40 × sin 35°τ = 24 × 0.574τ = 13.8 N m (3 s.f.)Only the perpendicular part of the force turns the lever — the sin 35° picks it out.
The effect of angle on torque
Because of the sin θ term, the same force produces different torques depending on its angle. The turning effect is greatest when the force is perpendicular (θ = 90°, sin 90° = 1) and drops to zero when the force is in line with the lever (θ = 0°, sin 0° = 0). This is why you instinctively push a wrench at right angles — push along it and nothing turns at all.
The same force gives the most torque when perpendicular to the wrench, less at a reduced angle, and almost none when nearly in line with it.
Couples
A couple is a special arrangement of forces: a pair of forces that are equal in size, opposite in direction, and not acting along the same line. Think of the two hands turning a steering wheel, or your finger and thumb twisting a tap. A couple has a distinctive property — it produces rotation only, with no overall push in any direction.
A couple: two equal, opposite forces separated by a perpendicular distance. There’s no net force, but there is a net turning effect.
Unlike the moment of a single force, the turning effect of a couple doesn’t depend on a pivot — it’s the same about any point. The turning effect a couple produces is called a torque, and because both forces contribute, the net torque of a couple is:
Torque of a coupleτ = 2Fr sin θ (= 2Fr when perpendicular)
Each force of the couple produces a torque Fr sin θ about the centre, and the two add together (they turn the same way), giving double the torque of a single force.
Force 1 Fr sinθ
+
Force 2 Fr sinθ
=
Couple 2Fr sinθ
A couple gives you zero resultant force but a real torque — so the object doesn’t move off in any direction, it just spins up. A steering wheel is the classic example: your two hands push opposite ways, the wheel stays put on its column, but it rotates. Zero linear acceleration, non-zero angular acceleration.
WE 3
A ruler of length 0.30 m is pivoted at its centre. Two equal and opposite 4 N forces are applied at its ends, each at 30° to the ruler, forming a couple. Calculate the magnitude of the torque of the couple.
Step 1 — torque of a couple uses the perpendicular component of the force
τ = force × perpendicular distance
Step 2 — find the perpendicular component of each forceF⊥ = 4 × sin 30° = 2.0 NStep 3 — multiply by the distance between the forces (0.30 m)τ = 2.0 × 0.30τ = 0.60 N mTaking the 0.30 m as the full distance between the two forces already accounts for both — no need to double again.
🛠️ Working out a torque
Identify the pivot (or axis of rotation) and the point where the force acts.
Measure r — the distance from the pivot to the force.
Check the angle. Perpendicular force → τ = Fr; at an angle → τ = Fr sin θ.
For a couple, use the force × the perpendicular distance between the two forces.
Keep units in metres so the torque comes out in N m.
💡 Top tips
Perpendicular is key. It’s always the perpendicular distance (or the perpendicular part of the force) that produces the turning effect.
Maximum at 90°. A force gives the most torque when applied at right angles, and none when in line with the lever.
Couples need no pivot. Their torque is the same about any point, so you don’t need to be told where the pivot is.
Mind the “distance”. For a couple, using the full distance between the forces already counts both — don’t double it a second time.
Quick recap: A moment is the turning effect of a force, force × perpendicular distance, in N m. Torque generalises this as τ = Fr sin θ, greatest when perpendicular and zero when in line. A couple is two equal, opposite, non-aligned forces that give zero net force but a net torque, producing rotation only.
⚠ Common mistakes
Using the full distance instead of the perpendicular distance (or forgetting the sin θ)
Thinking a moment is a “turning force” — it’s the effect of a force applied at a distance
Assuming a couple produces a net force — it produces only a turning effect
Doubling the couple torque twice when the distance already spans both forces
Mixing up the angle — check whether it’s measured from the lever or from the axis of rotation
You’ve now got the cause of rotation — torque — nailed down. The next question is what happens when all the torques on a body balance out. Just as balanced forces give translational equilibrium, balanced torques give rotational equilibrium, and that’s exactly where we head next.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
