Rotation of Rigid Bodies – Study Notes (University Physics, Chapter 9)

Rotation of Rigid Bodies

Introduction

Rotational motion is a fundamental concept in physics, describing how objects spin about an axis. Examples include airplane propellers, revolving doors, ceiling fans, and Ferris wheels. In this chapter, we focus on the ideal case of rigid bodies, which do not deform during rotation.

• Rigid body: An object whose shape does not change during motion.

• Real-world rotations may involve stretching and twisting, but these are neglected for introductory analysis.

• Rotational motion is characterized by angular position, angular velocity, and angular acceleration.

Angular Coordinate

The angular coordinate specifies the rotational position of a point or object about a fixed axis.

• Definition: The angle $\theta$ from a reference axis (often the +x-axis) to the position vector of the rotating point.

• Measured in radians or degrees, but radians are preferred in physics for calculations.

• Example: A car's speedometer needle rotates about a fixed axis, and its position is described by $\theta$.

Units of Angles

Angles are measured in radians, which relate arc length to radius.

• Radian: The angle subtended at the center of a circle by an arc whose length equals the radius.

• One complete revolution: $360^\circ = 2\pi$ radians.

• Relationship: $\theta = \frac{s}{r}$, where $s$ is arc length and $r$ is radius.

Formula:

• $\theta = \frac{s}{r}$

Example: If the arc length $s$ is equal to the radius $r$, then $\theta = 1$ radian.

Angular Velocity

Angular velocity describes how fast an object rotates and in which direction.

• Average angular velocity: $\omega_{z,avg} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous angular velocity: $\omega_z = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$

• Direction is given by the right-hand rule: curl fingers in direction of rotation, thumb points along axis.

• Positive $\omega$ for counterclockwise rotation, negative $\omega$ for clockwise rotation.

Example: The angular position of a flywheel is given by $\theta = (2.0\, \text{rad/s}) t$. At $t = 2.0$ s, $\theta = 4.0$ rad.

Angular Acceleration

Angular acceleration measures how quickly the angular velocity changes.

• Average angular acceleration: $\alpha_{z,avg} = \frac{\Delta \omega_z}{\Delta t}$

• Instantaneous angular acceleration: $\alpha_z = \frac{d\omega_z}{dt}$

• Positive $\alpha$ means speeding up in the direction of rotation; negative $\alpha$ means slowing down.

Rotation with Constant Angular Acceleration

When angular acceleration is constant, rotational kinematic equations mirror those for linear motion.

Linear Motion	Rotational Motion
$v = v_0 + a t$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Relating Linear and Angular Kinematics

Points on a rotating body have both linear and angular properties.

• Linear speed: $v = r \omega$

• Tangential acceleration: $a_t = r \alpha$

• Centripetal (radial) acceleration: $a_c = r \omega^2$

• Always use radians when relating linear and angular quantities.

Example: An athlete whirls a discus of radius $0.80$ m at $\omega = 10.0$ rad/s, with $\alpha = 50.0$ rad/s$^2$. The tangential acceleration is $a_t = r \alpha = 40.0$ m/s$^2$.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to the axis of rotation, affecting how difficult it is to change the rotational motion.

• Definition: $I = \sum m_i r_i^2$ (for discrete masses), or $I = \int r^2 dm$ (for continuous bodies)

• SI unit: kg·m$^2$

• Mass closer to the axis: smaller $I$, easier to rotate.

• Mass farther from the axis: larger $I$, harder to rotate.

Example: A hummingbird's wings have a small moment of inertia, allowing rapid movement; a condor's large wings have a large moment of inertia, resulting in slower flapping.

Moments of Inertia for Common Bodies

Standard formulas exist for simple shapes:

Body	Axis	Moment of Inertia ($I$)
Slender rod	Center	$I = \frac{1}{12} M L^2$
Slender rod	End	$I = \frac{1}{3} M L^2$
Rectangular plate	Center	$I = \frac{1}{12} M (a^2 + b^2)$
Rectangular plate	Edge	$I = \frac{1}{3} M a^2$
Solid cylinder	Axis	$I = \frac{1}{2} M R^2$
Hollow cylinder	Axis	$I = M R^2$
Solid sphere	Center	$I = \frac{2}{5} M R^2$
Thin-walled hollow sphere	Center	$I = \frac{2}{3} M R^2$

Rotational Kinetic Energy

A rotating rigid body possesses kinetic energy due to its rotation.

• Formula: $K_{rot} = \frac{1}{2} I \omega^2$

• Depends on both the moment of inertia and the square of the angular speed.

Example: If a solid sphere's radius is doubled while keeping kinetic energy constant, its angular velocity must decrease to maintain the same energy.

Parallel-Axis Theorem

The parallel-axis theorem allows calculation of the moment of inertia about any axis parallel to one through the center of mass.

• Formula: $I = I_{cm} + M d^2$

• $I_{cm}$: moment of inertia about center of mass axis

• $d$: distance between axes

Example: Calculating the moment of inertia of a baseball bat about an axis through the handle (not the center of mass).

Gravitational Potential Energy of Extended Bodies

The gravitational potential energy of an extended body is treated as if all mass is concentrated at its center of mass.

• Formula: $U_{grav} = M g h_{cm}$

• Applications include analyzing the motion of athletes or objects in gravitational fields.

Additional info:

• Rotational motion concepts are foundational for understanding dynamics of machinery, planetary motion, and biological systems (e.g., bacterial flagella).

• All points on a rigid body rotating about a fixed axis share the same angular velocity and angular acceleration, but their linear speeds and accelerations depend on their distance from the axis.