Short on time? This is the recap for the calculus-based course, which treats torque as a vector (Physics 1 uses only its size). For diagrams and worked examples, use the full study guide.
Recap
- Torque is the turning effect of a force about a chosen axis. State the axis first.
- r runs from the axis to the point where the force acts. θ is the angle between r and F, tail to tail.
- Only the component of F perpendicular to r turns the object. The component along r gives no torque.
- The lever arm r⊥ is the perpendicular distance from the axis to the line of action of the force.
- A force diagram shows each force starting at the point where it acts, with distances from the axis.
- Weight acts at the centre of mass (uniform g).
- Net torque is the signed sum of torques about one axis.
Key relationships
| Relationship | Units | Meaning |
|---|---|---|
| τ = r × F | N·m | vector, perpendicular to the plane of r and F |
| |τ| = rF sin θ | N·m | size from r, F and the angle between them |
| τ = rF⊥, F⊥ = F sin θ | N·m | perpendicular component × distance |
| τ = r⊥F, r⊥ = r sin θ | N·m | lever arm × force |
| τ_z = xF_y − yF_x | N·m | component form for forces in the xy-plane |
| î × ĵ = k̂, ĵ × î = −k̂ | — | order matters: B × A = −(A × B) |
| τ_net = Στ (one axis) | N·m | counterclockwise +, clockwise − |
Right-hand rule
Fingers along r, curl towards F through the smaller angle: your thumb gives τ. For forces in the page, thumb out = counterclockwise, thumb in = clockwise.
Assumptions
- The system is rigid: every part turns through the same angle.
- Forces and r lie in one plane unless stated, so τ is along ±z.
- g = 9.8 m/s², uniform, so weight acts at the centre of mass.
Mistakes to avoid
- τ = rF for any angle. Include sin θ.
- Lever arm = distance to the point of application. It is the distance to the line of action.
- F × r instead of r × F. The sign flips.
- Hinge forces in the torque sum about the hinge. They give zero torque there.
- Adding torques about different axes.
- Dropping the minus sign on clockwise torques.
- Using the obtuse angle wrongly. sin(180° − θ) = sin θ, so both give the same size; the direction comes from the right-hand rule.
Quick self-check
- A 60 N force acts 0.25 m from an axis at 40° to r. Size of torque? (0.25 × 60 × sin 40° = 9.6 N·m)
- r = 0.30î m, F = −20ĵ N. Find τ. (τ_z = (0.30)(−20) − 0 = −6.0 N·m, so −6.0k̂ N·m, clockwise)
- A 45 N force acts 0.80 m from an axis; its line of action makes 30° with r. Lever arm and torque? (r⊥ = 0.80 sin 30° = 0.40 m; τ = 18 N·m)
Next: practice questions.