Torque and Angular Momentum — The Calculations
Torque is the rotational cousin of force, and angular momentum is the rotational cousin of linear momentum — the two are linked exactly the way force and momentum are, through τ = dL/dt. This guide gives both formulas, a moment-of-inertia reference table for common shapes, and four worked examples ending in the classic spinning-skater problem that shows what conservation of angular momentum actually predicts.
Torque
where r is the distance from the pivot to the point where the force is applied, F is the force, and θ is the angle between r and F. Torque is maximum when the force is applied perpendicular to r (θ = 90°, sinθ = 1) and zero when the force is applied straight along r (θ = 0°, since a push directly toward or away from the pivot cannot make anything rotate). Unit: newton-metre (N·m).
Angular momentum
where I is the moment of inertia (the rotational equivalent of mass — how the body's mass is distributed about the axis) and ω is angular velocity in rad/s. Just as τ = dp/dt for linear motion, τ = dL/dt — torque is the rate of change of angular momentum. When the net external torque on a system is zero, L stays constant: this is the law of conservation of angular momentum.
Moment of inertia — common shapes
| Shape (axis through centre unless stated) | Moment of inertia I |
|---|---|
| Solid sphere | (2/5) M R² |
| Solid disc or cylinder | (1/2) M R² |
| Thin ring or hoop | M R² |
| Thin rod, axis through centre | (1/12) M L² |
| Thin rod, axis through one end | (1/3) M L² |
Worked example 1 — torque with a perpendicular force
A force F = 50 N is applied at r = 0.3 m from a pivot, perpendicular to the lever (θ = 90°).
τ = rF sinθ = 0.3 × 50 × 1 = 15 N·m
Worked example 2 — torque at an angle
A force F = 40 N is applied at the end of a 0.5 m wrench, at 60° to the wrench's length.
τ = rF sinθ = 0.5 × 40 × sin60°
0.5 × 40 = 20, and sin60° = 0.8660, so τ = 20 × 0.8660 = 17.3 N·m
Applying the same 40 N force perpendicular to the wrench instead would give 0.5 × 40 = 20 N·m — noticeably more torque for the same effort, which is why mechanics always tell you to pull a spanner perpendicular to its handle.
Worked example 3 — angular momentum of a spinning disc
A solid disc of mass M = 2 kg and radius R = 0.4 m spins at ω = 10 rad/s about its centre. Find its angular momentum.
I = (1/2)MR² = (1/2)(2)(0.4²) = (1/2)(2)(0.16) = 0.16 kg·m²
L = Iω = 0.16 × 10 = 1.6 kg·m²/s
Worked example 4 — conservation of angular momentum (the spinning skater)
A figure skater spinning with arms outstretched has moment of inertia I₁ = 4 kg·m² at angular speed ω₁ = 2 rad/s. She pulls her arms in, reducing her moment of inertia to I₂ = 1 kg·m². No external torque acts (ignoring friction with the ice), so angular momentum is conserved. Find her new angular speed.
I₁ω₁ = I₂ω₂ → ω₂ = I₁ω₁ / I₂ = (4 × 2) ÷ 1 = 8 rad/s
Cutting her moment of inertia to a quarter of its original value quadruples her spin rate — this is exactly why skaters pull their arms in to spin faster, and it is angular momentum, not angular velocity or moment of inertia individually, that stays constant.
Common mistakes that cost marks
- Forgetting the sinθ factor in torque — using τ = rF regardless of the angle between the force and the lever arm.
- Using the wrong moment-of-inertia formula for the shape given — a solid sphere, a disc and a ring all have different I even at the same M and R.
- Assuming ω alone is conserved when a shape changes — it is L = Iω that stays constant, not ω by itself, which is precisely why ω changes when I does.
- Confusing torque with work. Both carry the unit N·m, but torque (τ = r × F, a cross product) and work (W = F·s, a dot product) are entirely different physical quantities.
- Getting the rotation direction wrong. Use the right-hand rule consistently for the sign of torque and angular momentum in a given problem.
Where torque and angular momentum appear in exams
| Exam | Typical use |
|---|---|
| CBSE Class 11 | System of Particles and Rotational Motion — torque, moment of inertia, conservation of L |
| JEE Main & Advanced | Rigid-body rotation combined with translation (rolling without slipping) |
| NEET | Direct τ = rF sinθ and L = Iω substitution questions |
| GATE (Engineering Mechanics) | Torque and angular momentum in machine-design and dynamics numericals |
Once you've calculated a torque in N·m, use the built-in Torque converter to switch it into other torque units — useful for cross-checking engineering-style answers that ask for kgf·m or lb·ft instead of SI units.
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