Rotational Motion: Concepts, Equations, and Applications

Rotational Motion of Rigid Bodies

Introduction to Rotational Motion

Rotational motion describes the movement of objects around a fixed axis. Unlike point particles, rigid bodies have a definite size and shape, and every point in the body moves in a circle about the axis of rotation. Examples include spinning wheels, rotating discs, and wind turbines.

• Rigid Body: An object with fixed size and shape that cannot be stretched, twisted, or squeezed.

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

• Examples: Wind turbines, gyroscopes, and playground merry-go-rounds.

Measuring Angles and Angular Position

Angle and Radian Measure

Angles in rotational motion are measured in radians, which relate the arc length to the radius of the circle. This unit is essential for describing angular displacement and velocity.

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

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

• Conversion: $360^\circ = 2\pi$ radians; $1 \text{ rad} = \frac{360^\circ}{2\pi} \approx 57.3^\circ$

• Direction: Angles measured counterclockwise from the positive x-axis are positive.

Angular Displacement

Definition and Calculation

Angular displacement is the change in the angular position of a rotating body. It is measured in radians and can be positive or negative depending on the direction of rotation.

• Formula: $\Delta \theta = \theta_2 - \theta_1$

• Sign Convention: Counterclockwise is positive, clockwise is negative.

Angular Velocity

Average and Instantaneous Angular Velocity

Angular velocity describes how fast an object rotates or revolves relative to another point, typically the axis of rotation. It is the rate of change of angular displacement with respect to time.

• Average Angular Velocity: $\omega_{av} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous Angular Velocity: $\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$

• Units: radians per second (rad/s)

• Other Units: 1 revolution/second = $2\pi$ rad/s; 1 rpm = $\frac{2\pi}{60}$ rad/s

• Sign Convention: Counterclockwise is positive, clockwise is negative.

Angular Acceleration

Definition and Calculation

Angular acceleration is the rate at which angular velocity changes with time. It is a vector quantity and can be positive or negative depending on whether the rotation is speeding up or slowing down.

• Average Angular Acceleration: $\alpha_{av} = \frac{\omega_2 - \omega_1}{t_2 - t_1} = \frac{\Delta \omega}{\Delta t}$

• Instantaneous Angular Acceleration: $\alpha = \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t}$

• Units: radians per second squared (rad/s2)

Rotation with Constant Angular Acceleration

Kinematic Equations for Rotational Motion

When angular acceleration is constant, the equations of rotational motion closely resemble those for linear motion with constant acceleration. These equations are essential for solving problems involving rotating objects.

• $\omega = \omega_0 + \alpha t$

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

• $\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t$

Straight-line motion with constant linear acceleration	Fixed-axis rotation with constant angular acceleration
$a = \text{constant}$	$\alpha = \text{constant}$
$v = v_0 + at$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$
$x - x_0 = \frac{1}{2}(v_0 + v)t$	$\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t$

Worked Example: Rotation of a Compact Disc

Application of Rotational Kinematics

Consider a CD with radius 6.0 cm spinning at 7200 rpm. We can calculate its angular velocity, the time to rotate through a certain angle, and its average angular acceleration if it starts from rest.

• Angular velocity: $\omega = 7200 \text{ rpm} \times \frac{2\pi}{60} = 754 \text{ rad/s}$

• Time to rotate through $90^\circ$: $\Delta \theta = \frac{\pi}{2}$ rad, $t = \frac{\Delta \theta}{\omega}$

• Average angular acceleration: $\alpha_{av} = \frac{\omega - 0}{4.0}$

Conceptual Questions and Quick Checks

Understanding Angular Velocity and Acceleration

Conceptual questions help reinforce the understanding of rotational motion. For example, all points on a rigid body rotating about a fixed axis have the same angular velocity, regardless of their distance from the axis.

• Example: On a merry-go-round, both a child at the edge and one near the center have the same angular velocity, but different linear velocities.

Sign of Angular Velocity and Acceleration

Direction and Slowing Down

The sign of angular velocity ($\omega$) and angular acceleration ($\alpha$) depends on the direction of rotation and whether the object is speeding up or slowing down. If an object is slowing down, $\omega$ and $\alpha$ have opposite signs.

• Example: If a fan blade is rotating counterclockwise (positive $\omega$) and slowing down, $\alpha$ is negative.

Summary Table: Linear vs. Angular Motion

Comparison of Kinematic Equations

The table below summarizes the analogy between linear and angular kinematics for constant acceleration.

Straight-line motion with constant linear acceleration	Fixed-axis rotation with constant angular acceleration
$a = \text{constant}$	$\alpha = \text{constant}$
$v = v_0 + at$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$
$x - x_0 = \frac{1}{2}(v_0 + v)t$	$\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t$

Practice Problems and Solutions

Sample Questions on Rotational Motion

Practice problems reinforce the application of concepts and equations in rotational motion.

• Example 1: Starting from rest, a wheel with constant angular acceleration turns through 25 rad in time $t$. After time $2t$, it will have turned through 100 rad ($\Delta \theta \propto (\Delta t)^2$).

• Example 2: Starting from rest, a wheel spins up to 25 rpm in time $t$. After time $2t$, its angular velocity will be 50 rpm ($\Delta \omega \propto \Delta t$).

Additional info: The notes above expand on the provided slides and images, adding definitions, formulas, and academic context for clarity and completeness.