Section 5.7: Rotational Energy
Rotational energy is the kinetic energy due to rotation of a body about an axis. It depends on the body's moment of inertia and angular velocity.
\[
KE_{rot} = \frac{1}{2} I \omega^2
\]
where:
\( KE_{rot} \) = rotational kinetic energy (J)
\( I \) = moment of inertia (kg·m²)
\( \omega \) = angular velocity (rad/s)
\( KE_{rot} \) = rotational kinetic energy (J)
\( I \) = moment of inertia (kg·m²)
\( \omega \) = angular velocity (rad/s)
The total kinetic energy of a rolling object is the sum of translational and rotational energies:
\[
KE_{total} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2
\]
where \( v \) is the linear velocity of the center of mass.
Example 1: Rotational Energy of a Disc
A solid disc of mass 2 kg and radius 0.5 m rotates at 10 rad/s. Find its rotational kinetic energy.
Moment of inertia: \( I = \frac{1}{2} m r^2 = 0.5*2*0.5^2 = 0.25 \, \text{kg·m²} \)
Rotational KE: \( KE_{rot} = 0.5*0.25*10^2 = 12.5 \, \text{J} \)
Practice Problems
- A solid cylinder of mass 3 kg and radius 0.2 m spins at 15 rad/s. Find its rotational energy.
- A hollow sphere of mass 5 kg and radius 0.3 m rotates at 8 rad/s. Calculate its rotational kinetic energy.
- A rolling wheel of mass 10 kg and radius 0.4 m moves with linear speed 5 m/s. Compute its total kinetic energy.
- A disc of 4 kg and radius 0.5 m rotates at 12 rad/s. Find rotational KE and compare with a cylinder of same mass and radius at same angular velocity.
- A solid sphere of mass 2 kg and radius 0.3 m rolls without slipping at linear speed 6 m/s. Determine translational and rotational KE, then total KE.