AcademyRotational Dynamics
Academy
Work and Power in Rotation
Level 1 - Physics topic page in Rotational Dynamics.
Principle
Rotational work is torque accumulated over angular displacement.
Notation
\(W\)
rotational work
\(\mathrm{J}\)
\(\tau\)
torque along the rotation axis
\(\mathrm{N\,m}\)
\(\theta\)
angular position or displacement
\(\mathrm{rad}\)
\(P\)
power
\(\mathrm{W}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(K_{\mathrm{rot}}\)
rotational kinetic energy
\(\mathrm{J}\)
Method
Derivation 1: Build work from torque and angle
Linear work uses force times displacement. Rotational work uses torque times angular displacement, with angle measured in radians.
Small angular work
\[dW=\tau\,d\theta\]
Total work
\[W=\int_{\theta_i}^{\theta_f}\tau(\theta)\,d\theta\]
Constant torque
\[W=\tau\Delta\theta\]
The graph shows the interpretation of the integral: area under a torque-angle graph is rotational work.
Derivation 2: Connect work, energy, and power
Net rotational work changes rotational kinetic energy. Power is the time rate of doing that work.
Work-energy theorem
\[W_{\mathrm{net}}=\Delta K_{\mathrm{rot}}\]
Rotational kinetic energy
\[K_{\mathrm{rot}}=\frac12I\omega^2\]
Power
\[P=\frac{dW}{dt}=\tau\frac{d\theta}{dt}=\tau\omega\]
Rules
These are the compact results from the method above.
Rotational work
\[W=\int_{\theta_i}^{\theta_f}\tau\,d\theta\]
Constant torque
\[W=\tau\Delta\theta\]
Rotational energy
\[K_{\mathrm{rot}}=\frac12I\omega^2\]
Rotational power
\[P=\tau\omega\]
Examples
Question
A constant
\[6.0\,\mathrm{N\,m}\]
torque turns a wheel through \[5.0\,\mathrm{rad}\]
Find the work.Answer
Use
\[W=\tau\Delta\theta=6.0(5.0)=30\,\mathrm{J}\]
Checks
- Use radians in \(W=\\tau\\theta\).
- Positive torque over positive angular displacement does positive work.
- Power depends on instantaneous \(\omega\), not total angular displacement.
- A resisting torque has negative work for forward rotation.