Is Rotational Kinetic Energy Conserved?

Title: Is Rotational Kinetic Energy Conserved? Category: physics

Short answer: Rotational kinetic energy is not conserved in general. A net torque can do work and change it, while energy can also transfer between rotation, translation, potential energy, and internal energy. For a rigid body rotating about a fixed axis, its rotational kinetic energy remains constant if the net torque’s power is zero throughout the motion; zero torque is sufficient, but not necessary.

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Rotational kinetic energy is the energy associated with a body’s rotation. It can change as the body speeds up or slows down, or as energy is transferred into or out of rotation. Whether it stays constant depends on the torque and on what other forms of energy you include in the system.

What rotational kinetic energy means

For a rigid body rotating about a fixed axis with moment of inertia $I$ and angular speed $\omega$, the rotational kinetic energy is

$$K_{\mathrm{rot}}=\frac{1}{2}I\omega^2.$$

For general three-dimensional rotation, the corresponding expression is $K_{\mathrm{rot}}=\tfrac{1}{2}\boldsymbol{\omega}\cdot\mathbf{L}_{\mathrm{CM}}$, where $\boldsymbol{\omega}$ is angular velocity and $\mathbf{L}_{\mathrm{CM}}$ is angular momentum about the center of mass. A moving rigid body may also have translational kinetic energy of its center of mass. Rotational and translational kinetic energy are separate parts of its total kinetic energy.

How torque changes rotational kinetic energy

For a rigid body about its center of mass, the rate at which the net external torque about the center of mass transfers energy into rotation is the rotational power:

$$\frac{dK_{\mathrm{rot}}}{dt}=\boldsymbol{\tau}_{\mathrm{ext,CM}}\cdot\boldsymbol{\omega}.$$

For rotation about one fixed axis, this reduces to $dK_{\mathrm{rot}}/dt=\tau\omega$. A torque can therefore change rotational kinetic energy by doing work. But torque alone does not tell you whether the energy changes: the torque’s direction relative to the angular velocity matters. If they are perpendicular, their dot product is zero, so the torque does no rotational work at that instant.

Zero net external torque about the center of mass is a sufficient condition for rotational kinetic energy to remain constant for a rigid body. It is not necessary: a nonzero torque perpendicular to the angular velocity has zero rotational power at that instant. If it remains perpendicular throughout an interval, rotational kinetic energy stays constant over that interval, and it may redirect the angular velocity. For a nonspherical rigid body, however, this does not by itself imply that the angular speed stays constant. In a fixed-axis problem, a nonzero torque generally changes the angular speed and hence the energy.

A worked example: a torque speeds up a wheel

Suppose a wheel has moment of inertia $I=2\,\mathrm{kg\,m^2}$ and rotates at $\omega=3\,\mathrm{rad/s}$. Its rotational kinetic energy is

$$K_{\mathrm{rot}}=\frac{1}{2}(2)(3^2)=9\,\mathrm{J}.$$

A constant torque of $4\,\mathrm{N\,m}$ acts in the direction of rotation through an angle of $2\,\mathrm{rad}$. The work done is $W=\tau\Delta\theta=8\,\mathrm{J}$. If the wheel’s moment of inertia stays fixed and no other energy transfer is involved, its final rotational kinetic energy is $9+8=17\,\mathrm{J}$. From $K_{\mathrm{rot}}=\tfrac{1}{2}I\omega^2$, its final angular speed is $\sqrt{17}\,\mathrm{rad/s}$, about $4.12\,\mathrm{rad/s}$.

The rotational kinetic energy has increased, so it was not conserved. The increase is accounted for by the work done by the torque.

Changing shape can change rotational energy

A figure skater pulling in their arms is a useful example. If external torque is negligible, angular momentum is approximately conserved. But pulling the arms in changes the skater’s moment of inertia, so angular speed and rotational kinetic energy can change even though angular momentum does not.

For instance, suppose $I$ decreases from $4$ to $2\,\mathrm{kg\,m^2}$. If angular momentum stays at $8\,\mathrm{kg\,m^2/s}$, the angular speed rises from $2$ to $4\,\mathrm{rad/s}$. The rotational kinetic energy changes from

$$\frac{1}{2}(4)(2^2)=8\,\mathrm{J} \quad\text{to}\quad \frac{1}{2}(2)(4^2)=16\,\mathrm{J}.$$

The added rotational energy comes from work done by the skater’s muscles. This illustrates why conservation of angular momentum does not imply conservation of rotational kinetic energy.

Rotational energy versus total energy

Rotational kinetic energy is not a universal conservation law, but total energy is conserved when you account for all relevant forms and the system is isolated. For example, as a ball rolls down a hill without slipping, gravitational potential energy can become both translational and rotational kinetic energy. The rotational part increases, even if the total mechanical energy stays constant.

In a real system, energy may also become heat, sound, or deformation. Whether mechanical energy is conserved depends on whether those transfers matter in the model. The energy has not disappeared; it has moved to a form that may not be included in the mechanical-energy calculation.

Common mistakes

• Confusing angular momentum with energy: A system can conserve angular momentum while its rotational kinetic energy changes, as in the skater example.
• Assuming any torque changes energy: A torque perpendicular to angular velocity has zero rotational power at that instant.
• Considering rotation alone: A change in rotational kinetic energy may be balanced by a change in translation, potential energy, or internal energy.
• Using $\tfrac{1}{2}I\omega^2$ without checking the model: This is the standard expression for a rigid body rotating about a fixed axis. If the body changes shape, its moment of inertia changes too; for general three-dimensional rotation, use the vector form.