Rotation of Rigid Bodies: Angular Kinematics, Kinetic Energy, and Moment of Inertia

Rotation of Rigid Bodies

Introduction to Rotational Motion

Rotational motion is a fundamental aspect of physics, describing the movement of objects around a fixed axis. Examples include airplane propellers, revolving doors, ceiling fans, and Ferris wheels. In this chapter, we assume the rotating body is perfectly rigid, meaning it does not deform during rotation.

• Rigid body: An object with a fixed shape that does not stretch or twist during rotation.

• Axis of rotation: The straight line about which the body rotates.

Angular Kinematics

Angular Coordinate

The angular coordinate, denoted by fits, specifies the rotational position of a point or object relative to a reference axis. For example, the angle of a car's speedometer needle from the +x-axis describes its position.

• Angle (θ): Measured in radians, it is the ratio of the arc length (s) to the radius (r):

$\theta = \frac{s}{r}$

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

Units of Angles

• Radian: The angle subtended by an arc equal in length to the radius of the circle.

• Always use radians when relating linear and angular quantities.

Angular Velocity

Angular velocity describes how fast an object rotates and in which direction. It is defined as the rate of change of angular displacement with respect to time.

• Average angular velocity:

$\omega_{\text{avg},z} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous angular velocity:

$\omega_z = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$

• The subscript z indicates rotation about the z-axis.

• Counterclockwise rotation: $\omega_z > 0$; Clockwise rotation: $\omega_z < 0$.

Angular Velocity as a Vector

Angular velocity is a vector quantity. Its direction is determined by the right-hand rule: curl the fingers of your right hand in the direction of rotation, and your thumb points in the direction of the angular velocity vector.

• Positive z-direction: $\omega_z > 0$

• Negative z-direction: $\omega_z < 0$

Angular Acceleration

Angular acceleration measures how quickly the angular velocity changes with time.

• Average angular acceleration:

$\alpha_{\text{avg},z} = \frac{\omega_{z,2} - \omega_{z,1}}{t_2 - t_1} = \frac{\Delta \omega_z}{\Delta t}$

• Instantaneous angular acceleration:

$\alpha_z = \frac{d\omega_z}{dt}$

• If $\vec{\alpha}$ and $\vec{\omega}$ are in the same direction, rotation speeds up; if opposite, rotation slows down.

Rotation with Constant Angular Acceleration

The equations for rotational motion with constant angular acceleration are analogous to those for straight-line motion with constant linear acceleration.

Straight-Line Motion (Linear)	Rotational Motion (Angular)
$a_x = \text{constant}$	$\alpha_z = \text{constant}$
$v_x = v_{0x} + a_x t$	$\omega_z = \omega_{0z} + \alpha_z t$
$x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$	$\theta = \theta_0 + \omega_{0z} t + \frac{1}{2} \alpha_z t^2$
$v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$	$\omega_z^2 = \omega_{0z}^2 + 2 \alpha_z (\theta - \theta_0)$
$x - x_0 = \frac{1}{2} (v_x + v_{0x}) t$	$\theta - \theta_0 = \frac{1}{2} (\omega_z + \omega_{0z}) t$

Relating Linear and Angular Kinematics

For a point at a distance $r$ from the axis of rotation:

• Linear speed: $v = r \omega$

• Tangential acceleration: $a_{\text{tan}} = r \alpha$

• Centripetal (radial) acceleration: $a_{\text{rad}} = \frac{v^2}{r} = r \omega^2$

Important: Always use radians when relating linear and angular quantities. For example, $s = r \theta$ (with $\theta$ in radians).

Rotational Kinetic Energy and Moment of Inertia

Rotational Kinetic Energy

The rotational kinetic energy of a rigid body is given by:

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

• Moment of inertia (I): A measure of how mass is distributed relative to the axis of rotation. It determines how much torque is needed for a given angular acceleration.

• SI unit: kilogram-meter squared (kg·m2).

Moment of Inertia

The moment of inertia is calculated by summing (or integrating) the products of each mass element and the square of its distance from the axis of rotation:

$I = \sum m_i r_i^2$

or, for continuous mass distributions,

$I = \int r^2 d m$

• Moving mass closer to the axis decreases $I$ (easier to rotate).

• Moving mass farther from the axis increases $I$ (harder to rotate).

Examples: Moment of Inertia in Nature

• Hummingbirds have small wings (small $I$), allowing rapid flapping.

• Andean condors have large wings (large $I$), resulting in slower flapping.

Moments of Inertia for Common Bodies

Standard formulas for the moment of inertia of various shapes (about specified axes):

Body	Axis	Moment of Inertia ($I$)
Slender rod	Through center	$\frac{1}{12} M L^2$
Slender rod	Through end	$\frac{1}{3} M L^2$
Rectangular plate	Through center	$\frac{1}{12} M (a^2 + b^2)$
Thin rectangular plate	Along edge	$\frac{1}{3} M a^2$
Hollow cylinder	Central axis	$\frac{1}{2} M (R_1^2 + R_2^2)$
Solid cylinder	Central axis	$\frac{1}{2} M R^2$
Thin-walled hollow cylinder	Central axis	$M R^2$
Solid sphere	Diameter	$\frac{2}{5} M R^2$
Thin-walled hollow sphere	Diameter	$\frac{2}{3} M R^2$

Additional Concepts

Gravitational Potential Energy of an Extended Body

The gravitational potential energy of an extended body is calculated as if all its mass were concentrated at its center of mass:

$U_{\text{grav}} = M g y_{\text{cm}}$

• Application: High jumpers arch their bodies so their center of mass passes under the bar, minimizing the required energy.

The Parallel-Axis Theorem

The parallel-axis theorem relates the moment of inertia about any axis to that about a parallel axis through the center of mass:

$I_p = I_{\text{cm}} + M d^2$

• $I_p$ = moment of inertia about the parallel axis

• $I_{\text{cm}}$ = moment of inertia about the center of mass axis

• $M$ = total mass

• $d$ = distance between axes

Moment of Inertia Calculations

For arbitrary mass distributions, the moment of inertia is found by integrating over the volume:

$I = \int r^2 dV$

• Geophysicists use satellite data to measure Earth's moment of inertia, revealing that Earth's core is much denser than its outer layers.

Example: Rotational Motion in Bacteria

• Escherichia coli bacteria swim by rotating their flagella at angular speeds of 200–1000 revolutions per minute (about 20–100 rad/s), demonstrating the biological application of rotational kinematics.

Additional info: The notes provide a comprehensive overview of rotational kinematics, kinetic energy, and moment of inertia, including practical examples and standard formulas for common shapes. The parallel-axis theorem and the importance of using radians in calculations are emphasized for accuracy in physics problems.