Overview
This lecture covers rotational kinetic energy, extends the concept to systems of multiple point masses, introduces moment of inertia, and examines how rotational and translational kinetic energies combine in rolling objects.
Rotational Kinetic Energy for Point Masses
- A point mass moving in a circle has kinetic energy due to its tangential speed.
- Rotational kinetic energy for a point mass: ( KE_{rot} = \frac{1}{2} m (r \omega)^2 ).
- Angular velocity (( \omega )) is the same for all points in a rigid object, but tangential velocity depends on radius.
Systems of Point Masses and Moment of Inertia
- The total rotational kinetic energy of multiple masses: sum individual kinetic energies.
- For two point masses: ( KE_{total} = \frac{1}{2} m_1 r_1^2 \omega^2 + \frac{1}{2} m_2 r_2^2 \omega^2 ).
- This generalizes to ( KE_{rot, total} = \frac{1}{2} \sum_{i=1}^{N} m_i r_i^2 \omega^2 ).
- The term ( \sum m_i r_i^2 ) is defined as the moment of inertia (( I )).
Moment of Inertia and Mass Distributions
- Moment of inertia (( I )) depends on how mass is distributed relative to the axis of rotation.
- Units of moment of inertia: ( \text{kg} \cdot \text{m}^2 ).
- For many rigid objects, moments of inertia are tabulated for common shapes.
Common Moments of Inertia
- Solid cylinder: ( I = \frac{1}{2} M R^2 ), about central axis.
- Cylindrical ring or hoop: ( I = M R^2 ), about central axis.
- Solid sphere: ( I = \frac{2}{5} M R^2 ), about center.
- Spherical shell: ( I = \frac{2}{3} M R^2 ), about center.
Rotational and Translational Kinetic Energy in Rolling Objects
- Rolling objects have both rotational (( KE_{rot} = \frac{1}{2} I \omega^2 )) and translational (( KE_{trans} = \frac{1}{2} M v^2 )) kinetic energies.
- For rolling without slipping, ( v = R\omega ).
- Total kinetic energy: ( KE_{total} = KE_{rot} + KE_{trans} ).
Example Problem: Solid Sphere on a Flat Surface
- Given a solid sphere (( m = 2.50,\text{kg}, r = 0.350,\text{m}, \alpha = 3,\text{rad/s}^2 )), initially at rest.
- At ( t = 4,\text{s} ):
- Rotational KE: ( 14.7,\text{J} ).
- Translational KE: ( 22.05,\text{J} ).
- Total KE: ( 36.75,\text{J} ).
- Time when total KE reaches ( 75,\text{J} ): ( 5.71,\text{s} ).
Key Terms & Definitions
- Rotational Kinetic Energy (( KE_{rot} )) — Energy due to rotation, ( \frac{1}{2} I \omega^2 ).
- Moment of Inertia (( I )) — Quantifies mass distribution relative to rotation axis, ( I = \sum m_i r_i^2 ).
- Angular Velocity (( \omega )) — Rate of rotation (radians/second).
- Angular Acceleration (( \alpha )) — Rate of change of angular velocity.
- Translational Kinetic Energy (( KE_{trans} )) — Energy due to linear motion, ( \frac{1}{2} M v^2 ).
Action Items / Next Steps
- Review and memorize common moments of inertia for standard shapes.
- Practice problems combining rotational and translational kinetic energy.
- Look up moments of inertia for other shapes as needed for assignments.