Short on time? This page is the recap. For the derivation, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A torque transfers energy into or out of a rigid system only when the system turns while the torque acts.
- A tangential force F at radius r moves its point of application along an arc s = rΔθ, so W = Fs = (rF)Δθ = τΔθ.
- A torque in the same sense as the rotation does positive work (energy in). An opposing torque, such as axle friction, does negative work (energy out).
- A force through the axle exerts no torque and does no work on the rotation.
- The net work done by all torques equals the change in rotational kinetic energy.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| W = τΔθ | τ in N·m, Δθ in rad, W in J | work by a constant torque |
| s = rΔθ | s, r in m | arc length moved by the point where a force acts |
| τ = rF⊥ | N·m | torque from the perpendicular force component |
| W = area under τ–θ graph | J | work by a changing torque |
| W_net = ½Iω² − ½Iω₀² | J | find a final angular velocity, or a stopping angle |
| 1 rev = 2π rad; 90° = π/2 rad | rad | convert angles first |
Factors of change: τ × 2 at the same Δθ → W × 2. From rest, W × 2 → ω × √2. A stopping angle against a constant friction torque grows with ω₀², so doubling ω₀ multiplies it by 4.
Assumptions behind the numbers
- The object is rigid and turns about a fixed axis in one plane.
- W = τΔθ needs a constant torque; otherwise use the graph area.
- Friction torques are treated as constant unless a graph says otherwise.
Mistakes to avoid
- Degrees or revolutions in W = τΔθ. Convert to radians.
- Calling torque “energy” because both use N·m. Write work in J.
- Positive work for friction. It opposes the rotation, so its work is negative.
- Largest torque × whole angle for a changing torque. Use the area.
- Work done by a torque that does not turn anything. It is zero.
- Forgetting the starting kinetic energy when the object is already spinning: W_net is the change in K.
Quick self-check
- A constant torque of 5.0 N·m turns a wheel through 4.0 rad. How much work is done? (20 J)
- A constant torque of 2.0 N·m acts through 3.0 revolutions. How much work is done? (3.0 × 2π = 18.8 rad, so 37.7 J)
- Torque rises steadily from 0 to 4.0 N·m as a wheel turns from 0 to 3.0 rad. How much work is done? (Triangle: ½ × 3.0 × 4.0 = 6.0 J)
Next: practice questions.