The formula

Magnitude of torque
τ = rF·sin θ
Using the lever arm
τ = r⊥F
Rotational form of Newton's second law
Στ = Iα

What the symbols mean

SymbolMeaningUnit
τTorque about the chosen axisN·m
rDistance from the axis to where the force is appliedm
FApplied forceN
θAngle between the position vector r and the force Fdegrees or radians
r⊥Lever arm, the perpendicular distance from the axis to the line of the forcem
IMoment of inertia about the same axiskg·m²
αAngular accelerationrad/s²

When it applies

  • You are analyzing rotation about a fixed axis, or checking whether an object stays in rotational equilibrium.
  • Measure r from the axis of rotation to the point where the force acts, and take θ between those two vectors.
  • Only the component of force perpendicular to r turns the object, so a force pointed straight at the axis produces no torque at all.

Worked example

Problem. You pull on a wrench 0.25 m from the bolt with a force of 80 N, at 60° to the handle. What torque does the bolt feel?

  1. Identify the three inputs: r = 0.25 m from the bolt to your hand, F = 80 N, and θ = 60° between the handle and the force.
  2. Use the magnitude form: τ = rF sin θ = (0.25 m)(80 N)(sin 60°).
  3. sin 60° = 0.866, so τ = (0.25)(80)(0.866) = 17.3 N·m.
  4. Round to two significant figures: τ = 17 N·m. Pulling perpendicular to the handle instead, at θ = 90°, would give the full 20 N·m.

Answer. 17 N·m, rising to 20 N·m if the same force were applied perpendicular to the handle.

Common mistakes

  • Measuring r to where you are standing rather than from the axis to the point where the force is applied.
  • Taking θ as the angle to the floor or to the horizontal instead of the angle between r and F.
  • Reporting torque in joules. Newton-meters of torque and joules of energy share dimensions but describe different quantities, so torque keeps N·m.
  • Forgetting that torque carries a sign. Choose one sense of rotation as positive and keep it when you add torques.

Related formulas

Sources