Rotational work is the energy transferred to or from a rotating system by a torque acting through an angular displacement. It is the rotational analog of W = Fd from Unit 3 — torque replaces force, angular displacement replaces linear displacement.
The key requirement: the system must actually rotate while the torque acts. A torque applied to a system that does not move (Δθ = 0) does zero work, regardless of how large the torque is. This is exactly analogous to pushing on a wall — huge force, zero displacement, zero work.
For a constant torque acting through an angular displacement Δθ:
Δθ must be in radians. The result is in Joules. This is the rotational counterpart of W = Fd, where force acts over a linear distance. The same dimensional check applies: [N·m] × [rad] = [J], since radians are dimensionless.
Set torque and angular displacement to calculate rotational work. See how that work changes the spinning speed via the work-energy theorem.
Set tau = 0 — no work done regardless of Dtheta. Set Dtheta = 0 — no work done regardless of tau. Both must be nonzero for work to occur.
A motor applies a constant torque of 15 N·m to a flywheel through 3 complete revolutions. How much work does the motor do?
For a variable torque, the work done equals the area under the torque vs. angular position graph. This is the direct rotational analog of finding work from the area under a F vs. x graph from Unit 3.
A tau vs. theta graph shows a triangular profile: torque rises linearly from 0 at theta = 0 to 12 N·m at theta = 4 rad, then drops back to 0 at theta = 8 rad. Find the total work done.
The net rotational work done on a system equals its change in rotational kinetic energy:
This is the rotational version of the work-energy theorem from Unit 3, applied to spinning systems. Positive net work increases K_rot (system spins faster). Negative net work decreases K_rot (system slows down). Zero net work means constant rotational KE — rotational equilibrium from 5.5.
A constant torque of 6 N·m is applied to a wheel (I = 0.8 kg·m²) starting from rest. The torque acts through 5 radians. Find the final angular velocity.
Power is the rate of doing work. For a rotating system:
This mirrors P = Fv from Unit 3. A motor delivering torque τ at angular velocity ω produces power P = τω in Watts. At higher angular velocity, the same torque delivers more power — this is why engine power ratings are usually specified at a particular RPM.
An electric motor delivers 25 N·m of torque while the output shaft spins at 120 rad/s. What is the motor's power output?