10.4 Rotational Kinetic Energy: Work and Energy Revisited

Rotational Work and Energy

Rotational work and energy extend the familiar ideas of work and kinetic energy to objects that spin. Instead of forces and displacements, you'll use torques and angular displacements. The payoff is that conservation of energy still works the same way, which makes solving problems with rolling, spinning, or orbiting objects much more manageable.

Equation for Rotational Work

Rotational work ($W_r$) is the work done by a torque ($\tau$) acting through an angular displacement ($\theta$):

$W_r = \tau \theta$

This mirrors the linear equation $W = Fd$, with torque replacing force and angular displacement replacing distance.

Torque is the rotational equivalent of force. It equals the applied force times the perpendicular distance from the axis of rotation to the force's line of action:

$\tau = F r_\perp$

Think of pushing a door open: the farther from the hinge you push, the larger $r_\perp$ is, and the easier the door swings. That's why door handles are placed at the edge, not near the hinge.

Angular displacement ($\theta$) is the angle through which the object rotates, measured in radians. You can convert from degrees or revolutions, but radians are required in these equations.

Substituting the torque expression into the work equation gives:

$W_r = F r_\perp \theta$

So rotational work depends on three things: how hard you push, how far from the axis you push, and how far the object rotates. Pedaling a bicycle is a good example: your foot applies a force at some distance from the crank axle, and the crank rotates through some angle.

The work-energy theorem applies here too: the net rotational work done on an object equals its change in rotational kinetic energy.

Calculation of Rotational Kinetic Energy

Rotational kinetic energy ($K_r$) is the kinetic energy an object has because it's spinning:

$K_r = \frac{1}{2} I \omega^2$

This mirrors $K_t = \frac{1}{2}mv^2$, with moment of inertia ($I$) replacing mass and angular velocity ($\omega$) replacing linear velocity.

Moment of inertia ($I$) measures how much an object resists changes in its rotation. It depends on both the object's mass and how that mass is distributed relative to the axis of rotation.

• For a single point mass: $I = mr^2$, where $m$ is the mass and $r$ is the distance from the axis.
• For extended objects (disks, spheres, rods), each shape has its own formula, but the principle is the same: mass farther from the axis contributes more to $I$.

A figure skater illustrates this well. With arms extended, mass is spread far from the spin axis, so $I$ is large. Pulling the arms in reduces $I$, and the skater spins faster to conserve angular momentum.

Angular velocity ($\omega$) is how fast the object rotates, measured in radians per second:

$\omega = \frac{d\theta}{dt}$

A spinning wheel might have $\omega = 10$ rad/s, meaning it sweeps through 10 radians every second.

The parallel axis theorem lets you calculate the moment of inertia about any axis that's parallel to one through the center of mass: $I = I_{cm} + Md^2$, where $d$ is the distance between the two axes. You won't always need it, but it shows up when objects rotate about an axis that doesn't pass through their center.

Conservation of Energy in Linear vs. Rotational Motion

The law of conservation of energy still holds: in a closed system, total energy cannot be created or destroyed, only converted between forms.

For objects that are both moving through space and spinning (like a ball rolling down a ramp), the total kinetic energy is the sum of translational and rotational parts:

$K_{total} = K_t + K_r = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• Translational KE ($\frac{1}{2}mv^2$) comes from the object's center of mass moving through space.
• Rotational KE ($\frac{1}{2}I\omega^2$) comes from the object spinning about its center of mass.

A sliding block has only translational KE. A spinning top fixed in place has only rotational KE. A ball rolling down a ramp has both.

When no external work is done on the system, the total mechanical energy stays constant:

$\Delta K_{total} + \Delta U = 0$

Here's how energy converts for a ball rolling down a ramp:

1. At the top: The ball is at rest, so all energy is gravitational potential energy ($U = mgh$).
2. While rolling down: Potential energy converts into both translational and rotational kinetic energy.
3. At the bottom: Potential energy is at a minimum, and kinetic energy is at a maximum, split between $\frac{1}{2}mv^2$ and $\frac{1}{2}I\omega^2$.

This split matters. A rolling ball reaches the bottom of a ramp slower than a sliding block released from the same height, because some of the energy goes into spinning rather than into forward speed.

Angular Momentum and Related Concepts

• Angular momentum ($L = I\omega$) is the rotational equivalent of linear momentum. A rotating object tends to keep rotating unless acted on by an external torque, just as a moving object keeps moving unless acted on by an external force.
• Angular acceleration ($\alpha$) is the rate at which angular velocity changes over time: $\alpha = \frac{d\omega}{dt}$. A net torque causes angular acceleration, just as a net force causes linear acceleration.
• Centripetal force is the inward-directed force that keeps an object on a circular path. It doesn't do work on the object (since it's always perpendicular to the velocity), but without it, the object would fly off in a straight line.