Work Done by Torque

When a force acts on an object and moves it through a displacement, work is done. In rotational motion, the equivalent concept involves a torque acting through an angular displacement. Just as linear work is defined as $W = F \cdot d$ , rotational work is defined as $W = \tau \cdot \theta$ , where $\tau$ is the torque and $\theta$ is the angular displacement in radians. This relationship is fundamental to understanding how rotational machines — from engines to wind turbines — transfer and transform energy. The concept of work done by torque is deeply connected to the work-energy theorem in rotation. When a net torque acts on a rigid body, it causes angular acceleration, and in doing so it transfers kinetic energy to or from the body. The rotational kinetic energy of an object is given by $KE_{rot} = \frac{1}{2}I\omega^2$ , where $I$ is the moment of inertia and $\omega$ is the angular velocity. The work-energy theorem for rotation states that the net work done by all torques equals the change in rotational kinetic energy: $W_{net} = \Delta KE_{rot}$ . Understanding work done by torque allows physicists and engineers to analyze the energy efficiency of rotating systems. For example, a motor applies a torque to a shaft over many rotations, and the total work delivered equals $\tau$ multiplied by the total angle swept. This principle is used to calculate power output, efficiency losses, and design specifications for mechanical systems encountered in everyday technology, from bicycle gears to jet engines.

Key takeaways

Start with linear work
Substitute into the work integral
Integrate to get total work
Connect to the rotational work-energy theorem
Rotational Power

Start with linear work

Linear work is defined as $W = F \cdot d$ , where $F$ is the component of force along the displacement $d$ . For a force acting tangentially on a rotating object at radius $r$ , the small linear displacement of the contact point is $ds = r\, d\theta$ , where $d\theta$ is the infinitesimal angular displacement in radians.

Substitute into the work integral

The infinitesimal work done by the tangential force is $dW = F_t \cdot ds = F_t \cdot r\, d\theta$ . Since torque is defined as $\tau = F_t \cdot r$ , we can immediately write $dW = \tau\, d\theta$ . This is the rotational analogue of $dW = F\, dx$ in linear motion.

Integrate to get total work

For the total work done as the object rotates from angle $\theta_1$ to $\theta_2$ , we integrate: W = \int_{\theta_1}^{\theta_2} \tau\, d\theta. If the torque is constant (a common case in problems), this simplifies to W = \tau (\theta_2 - \theta_1) = \tau \Delta\theta. This elegant result mirrors $W = F \Delta x$ exactly, confirming the deep symmetry between linear and rotational mechanics.

Connect to the rotational work-energy theorem

By Newton's second law for rotation, $\tau = I\alpha$ . Substituting into $dW = \tau\, d\theta$ and using the kinematic relation $\alpha\, d\theta = \omega\, d\omega$ (derived from $\alpha = \omega\, d\omega/d\theta$ ), we get $dW = I\omega\, d\omega$ . Integrating both sides: W = \int_{\omega_i}^{\omega_f} I\omega\, d\omega = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2 = \Delta KE_{rot}. This is the rotational work-energy theorem.

Rotational Power

Power is the rate of doing work. Differentiating $W = \tau\,\theta$ with respect to time gives the instantaneous rotational power: P = \frac{dW}{dt} = \tau \frac{d\theta}{dt} = \tau \omega. This means a motor delivering a constant torque $\tau$ while spinning at angular velocity $\omega$ outputs power $P = \tau\omega$ . This formula is used constantly in engineering to rate motors and engines.

The Key Equation

$$W = \tau \,\Delta\theta

- **$W$** — Work (Joule (J)): Energy transferred to or from the rotating body by the torque. - **$\tau$** — Torque (Newton-metre (N\cdot m)): The rotational equivalent of force; $\tau = r \times F$, the tendency to cause angular acceleration. - **$\Delta\theta$** — Angular Displacement (Radian (rad)): The angle through which the object rotates, measured in radians (dimensionless but physically meaningful). ## Worked Example **Problem.** An electric motor applies a constant torque of 25 N\cdot m to a flywheel. Starting from rest, the flywheel completes 8 full revolutions before the motor is switched off. (a) How much work does the motor do on the flywheel? (b) If the flywheel has a moment of inertia of $I = 4.0\,\text{kg}\,\cdot\,\text{m}^2$, what is its final angular velocity?

W = 400\pi \approx 1257\,\text{J},\quad \omega_f = \sqrt{200\pi} \approx 25.1\,\text{rad/s}

**Answer.** The motor does approximately 1257 J of work on the flywheel, and the flywheel reaches a final angular velocity of approximately 25.1 rad/s.

Figure

Diagram

Origins · 1 minJames Prescott Joule and how this was discovered

James Prescott Joule

1818–1889 · British

Joule established the mechanical equivalent of heat and the law of conservation of energy, showing rigorously that work of any kind — including rotational work — converts into a quantifiable amount of energy. The SI unit of energy, the joule (J), is named in his honour. His paddle-wheel experiments directly involved rotational work (torque through angular displacement) being converted into thermal energy in water, forming a cornerstone of thermodynamics. Joule was so dedicated to measurement that he allegedly brought a thermometer on his honeymoon to measure the temperature difference at the top and bottom of a waterfall, testing whether the water's kinetic energy converted to heat upon impact!

Torque and Rotational Force — Torque is the central quantity in the work formula $W = \tau\,\Delta\theta$ ; understanding its definition and calculation is a prerequisite for this concept.
Moment of Inertia — The rotational work-energy theorem $W = \Delta(\frac{1}{2}I\omega^2)$ directly involves the moment of inertia $I$ , the rotational analogue of mass.
Angular Kinematics — Angular displacement $\Delta\theta$ , angular velocity $\omega$ , and angular acceleration $\alpha$ are the kinematic quantities that appear throughout the derivation of rotational work.
Work-Energy Theorem (Linear) — The rotational work-energy theorem is the direct analogue of the linear theorem $W_{net} = \Delta KE$ ; comparing the two deepens understanding of both.
Rotational Power — Power in rotation is $P = \tau\omega$ , derived directly from the definition of work done by torque, and is essential for engineering applications.