Torque and Work
Cranking a winch — work, but going in circles
- Turn a winch handle round and round and you clearly do work — your arm gets tired, a load rises.
- Yet nothing moves in a straight line; the handle just goes in circles.
- Work done by a torque turning through an angle is the rotational version of $W = Fd$.
- It feeds energy straight into a spinning object's rotational kinetic energy.
Work done by a torque
- A torque $\tau$ turning through an angle $\theta$ does work $W = \tau\,\theta$.
- It is the twin of $W = Fd$, with torque for force and angle (in radians) for distance.
- This work changes the object's rotational kinetic energy: $W = \Delta(\tfrac12 I\omega^2)$.
- Positive torque-work speeds the spin up; negative slows it down.
A constant torque of $5\ \text{N·m}$ turns a wheel through $4\ \text{rad}$. How much work is done, in joules?
$W = \tau\theta = 5 \times 4 = 20\ \text{J}$.
Work done by a torque turning through angle $\theta$ is:
$W = \tau\theta$ — the rotational twin of $W = Fd$.
The net work done by torques on a rigid body equals the change in its rotational kinetic energy.
This is the rotational work–energy theorem: $W_{\text{net}} = \Delta(\tfrac12 I\omega^2)$.
Rotational power
- The rate of doing rotational work is the power $P = \tau\,\omega$.
- It is the twin of $P = Fv$, with torque for force and angular speed for linear speed.
- A car engine's power is often quoted this way — torque times revs.
- More torque or a faster spin both mean more power delivered.
A torque of $5\ \text{N·m}$ turns a shaft at $\omega = 2\ \tfrac{\text{rad}}{\text{s}}$. What is the power, in watts?
$P = \tau\omega = 5 \times 2 = 10\ \text{W}$.
Rotational power is torque times ____ (the angular quantity).
$P = \tau\omega$ — torque times angular velocity.
Match each rotational energy quantity to its linear twin.
Each rotational energy formula matches a linear one by swapping $\tau\to F$, $\omega\to v$, $I\to m$.
The energy picture stays the same
- Rotational work and energy obey the same conservation rules as their linear cousins.
- The net work by all torques equals the change in rotational kinetic energy.
- Energy can flow between translation and rotation (a rolling object), but the total is conserved.
- Every linear energy tool has a rotational partner you already know how to use.
Work done by a torque
A constant torque turning through an angle does work in proportion to that angle.
Rotational work uses the angle in radians, not degrees, and the net torque. $W = \tau\theta$ only gives the right energy when $\theta$ is in radians — the same radian rule as the rest of rotation.
A constant torque of $5\ \text{N}\cdot\text{m}$ turns a wheel through $4\ \text{rad}$.
- $W = \tau\theta = 5 \times 4 = 20\ \text{J}$.
If it does this in $2\ \text{s}$ at $\omega = 2\ \tfrac{\text{rad}}{\text{s}}$, the power is $P = \tau\omega = 5 \times 2 = 10\ \text{W}$.
Work done by a torque is $W = \tau\theta$ (angle in radians), the twin of $W = Fd$. It changes the rotational kinetic energy. The rate of doing it is the rotational power $P = \tau\omega$, the twin of $P = Fv$.