Rotation of Rigid Bodies

Section 9: Rotation of Rigid Bodies

• Key Concepts:

  • Angular velocity and angular acceleration

  • Rotational kinetic energy

  • Moment of inertia

  • Newton's second law for rotation

  • Rolling objects

Angular Velocity and Angular Acceleration

• Angular velocity ($ heta$) has units of rad/s; angular acceleration ($eta$) has units of rad/s².

• Linear tangential velocity ($v$) and centripetal acceleration ($a_c$) vary with distance from axis of rotation.

Comparison of Linear and Angular Motions

• Linear Motion Equations:

  • $x = x0 + v{0x}t + \frac{1}{2} a_xt^2$

  • $v^2 = v{0x}^2 + 2ax(x - x_0)$

• Rotational Motion Equations:

  • $ heta = heta0 + eta0t + \frac{1}{2}\alpha t^2$

  • $ orall angles: \;\omega^2 = \omega0^2 + 2\alpha(\theta - \theta0)$

Moment of Inertia

• Definition: Moment of inertia ($I$) is a measure of an object's resistance to rotational acceleration about an axis.

• For various shapes:

  • Uniform disk: $I = \frac{1}{2} MR^2$

  • Solid sphere: $\; I = \frac{2}{5} MR^2$

  • Hollow sphere: $\; I = \frac{2}{3} MR^2$

Newton’s Second Law for Rotation

• Relates torque ($\tau$) to angular acceleration: $ au = I\alpha$

• Torque is the rotational equivalent of force; it is affected by the distance from the axis of rotation.

Additional Key Concepts

• Parallel Axis Theorem: $I = I_{cm} + Md^2$

• Power in rotational systems is given by work over time: $P = \frac{\tau \omega}{t}$

• Rolling without slipping: Objects roll down inclines at different rates based on their moment of inertia.