Topic 6.1 Rotational Kinetic Energy: define the kinetic energy of a rotating rigid body and relate it to rotational inertia and angular velocity.

What this topic is asking

The College Board (Topic 6.1) wants you to define the rotational kinetic energy of a rotating rigid body and relate it to its rotational inertia $I$ and angular velocity $\omega$. Rotational kinetic energy is the rotational analogue of translational kinetic energy: where translation uses $\tfrac{1}{2}mv^2$, rotation uses $\tfrac{1}{2}I\omega^2$, with rotational inertia playing the role of mass and angular velocity the role of linear speed.

What rotational kinetic energy is

Every point in a rotating body is moving, so the body carries kinetic energy even if its center of mass is at rest. Summing $\tfrac{1}{2}mv^2$ over all the particles, using $v = r\omega$ for each, collapses neatly to $\tfrac{1}{2}I\omega^2$, because $\sum mr^2 = I$. This is why rotational inertia appears here in exactly the role mass plays in translational kinetic energy.

How it depends on $I$ and $\omega$

The squared dependence on $\omega$ is the feature most exam questions hinge on, mirroring the $v^2$ dependence of translational kinetic energy from Topic 3.1. Because $K_{rot}$ is just another form of kinetic energy, it slots straight into the work-energy theorem and conservation of energy: a torque doing work changes $K_{rot}$, and an object rolling down a ramp converts gravitational potential energy into both translational and rotational kinetic energy.

Total kinetic energy of a rolling object

An object that rolls is both translating (its center of mass moves) and rotating (it spins about that center). Its total kinetic energy is the sum:

For rolling without slipping, $v_{cm} = R\omega$ ties the two together, so the split between translational and rotational energy is fixed by the object's shape (through $I$). A hoop, with all its mass at the rim, puts more of its energy into rotation than a solid sphere of the same mass and radius does. This is the key to the classic "which object wins the race down a ramp" problem (Topic 6.5): for a given drop, the object with the smaller share of energy locked into rotation reaches the bottom faster. Understanding rotational kinetic energy as just another term in the energy ledger, on equal footing with $\tfrac{1}{2}mv^2$ and potential energy, is the conceptual bridge from Unit 3's energy methods into the dynamics of spinning and rolling systems.

Try this

Q1. A wheel has rotational inertia $2.0$ kg$\cdot$m squared and spins at $5.0$ rad/s. Calculate its rotational kinetic energy. [2 points]

• Cue. $K = \tfrac{1}{2}(2.0)(5.0)^2 = \tfrac{1}{2}(2.0)(25) = 25$ J.

Q2. A spinning object's angular velocity triples while its rotational inertia stays the same. By what factor does its rotational kinetic energy change? [1 point]

• Cue. $K \propto \omega^2$, so tripling $\omega$ gives a factor of $9$.