Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which does not integrate angle-dependent torques). For the derivations, graphs and worked examples, use the full study guide.
Recap
- A torque transfers energy into or out of a rigid body only if the body turns while the torque acts.
- When the body turns through dθ, the point where a force acts moves ds = r dθ along the tangent. Only F sin φ does work, so dW = rF sin φ dθ = τ dθ.
- State a positive sense of rotation. Torque with the rotation: positive work. Against it: negative work. τ = 0 or Δθ = 0: no work.
- Every point of a rigid body turns through the same dθ, so the works of several torques add to ∫τ_net dθ.
- With α = ω dω/dθ, τ_net = Iα integrates to W_net = ½Iω² − ½Iω₀².
Key relationships
| Relationship | When to use it | Graph meaning |
|---|---|---|
| W = ∫τ dθ | Any torque, θ in rad | Signed area under τ–θ graph |
| W = τΔθ | Constant torque only | Rectangle |
| W_net = ∫τ_net dθ = ΣWᵢ | Several torques | Add signed areas |
| W_net = ½Iω² − ½Iω₀² | Rigid body, fixed axis | — |
| α = ω dω/dθ | Chain rule step | — |
| ω greatest where τ_net = 0 | Torque changes sign | Where τ_net graph crosses the axis |
Assumptions behind the numbers
- The body is rigid and turns about a fixed axis, so I is constant.
- Angles are in radians; 1 rev = 2π rad.
- Axles are frictionless unless a friction torque is given.
Mistakes to avoid
- Work from a torque that does not turn anything.
- Revolutions or degrees in W = τΔθ.
- W = τΔθ for a torque that changes with angle.
- Adding sizes of areas above and below the axis.
- Leaving out friction’s negative work.
- Writing a torque in J or work in N·m.
- Equal work means equal ω. Equal work means equal ΔK.
Quick self-check
- A constant 5.0 N·m torque turns a wheel through 4.0 rad in its direction of rotation. Work? (20 J)
- A friction torque of 0.30 N·m acts while a wheel turns 10 rev. Work by friction? (−0.30 × 20π = −19 J)
- τ(θ) = 6.0 − 2.0θ from 0 to 3.0 rad, in the direction of rotation. Work? (18 − 9.0 = 9.0 J)
Next: practice questions.