11.1.2 Rotational Kinetic Energy

Deriving the Formula for Rotational Kinetic Energy

1. For each point mass in a rotating object:

$$E_k = \frac{1}{2}mv^2$$

Where $v$ is the linear speed of the point mass.

2. Since linear speed $v$ can be related to the angular speed $\omega$ $($where $v = \omega r )$, we can substitute $v = \omega r$ into the equation:

$$E_k = \frac{1}{2}m(\omega r)^2 = \frac{1}{2}m\omega^2 r^2$$

3. Total rotational kinetic energy for the object:

• Sum the kinetic energy of all point masses:

$$E_k = \frac{1}{2} \omega^2 \sum (mr^2)$$

4. Since $\sum mr^2 = I$ (where $I$ is the moment of inertia), the formula for rotational kinetic energy becomes:

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

Here:

• $I$ is the moment of inertia.
• $\omega$ is the angular speed of the object.

Factors Affecting Energy Storage in a Flywheel

1. Mass of the Flywheel: Increasing the flywheel's mass increases its moment of inertia, thereby increasing the energy it can store.
2. Angular Speed: Energy stored in a flywheel is proportional to the square of its angular speed, so higher speeds allow for more stored energy.
3. Friction: Friction can reduce the energy stored. Reducing friction with lubricated or vacuum-sealed bearings can help maintain stored energy.
4. Mass Distribution: A flywheel with mass concentrated further from the axis (like a spoked flywheel) has a higher moment of inertia and can store more energy.

Uses of Flywheels

• Regenerative Braking in Vehicles: Energy from braking is stored in the flywheel and later used to accelerate the vehicle.
• Wind Turbines: Flywheels store excess power during high wind and release it when there is no wind.
• Smoothing Torque and Angular Velocity: Flywheels help manage power fluctuations by storing bursts of energy and releasing it steadily.
• Production Processes: In industries like riveting, flywheels provide a consistent burst of energy without needing a high-power motor.