Rotational Motion: Angular Kinematics and Dynamics

Rotational Motion

Introduction to Rotational Motion

Rotational motion describes the movement of objects around a fixed axis. Many physical systems, such as spinning tops, wheels, and planets, exhibit rotational motion. Understanding the kinematics and dynamics of rotation is essential for analyzing these systems.

• Rigid Body: An object whose size and shape do not change as it moves. Examples include boomerangs and bicycle wheels.

• Rotation of a Rigid Body: All points in a rigid body rotate with the same angular velocity, regardless of their distance from the axis of rotation.

Angular Position, Velocity, and Acceleration

Angular quantities are used to describe rotational motion, analogous to linear quantities in straight-line motion.

• Angular Position (θ): The angle (in radians) that a point or line has rotated about a specified axis.

• Angular Velocity (ω): The rate of change of angular position, measured in radians per second (rad/s). $\omega = \frac{d\theta}{dt}$

• Angular Acceleration (α): The rate of change of angular velocity, measured in radians per second squared (rad/s2). $\alpha = \frac{d\omega}{dt}$

Relationship Between Linear and Angular Quantities

Linear and angular quantities are related through the radius of rotation.

• Arc Length (s): $s = r\theta$

• Linear Velocity (v): $v = \frac{ds}{dt} = r\frac{d\theta}{dt} = r\omega$

• Linear Acceleration (a): $a = \frac{dv}{dt}$

Angular Velocity as a Vector

Angular velocity is a vector quantity, with direction along the axis of rotation. The direction is determined by the right-hand rule: curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the angular velocity vector.

• Sign Convention: Counterclockwise rotation is taken as positive by convention.

Uniform Circular Motion

Uniform circular motion occurs when an object moves in a circle at constant speed. Several key quantities describe this motion:

• Angular Velocity: $\omega = \frac{2\pi}{T}$

• Linear Velocity: $v = 2\pi r / T = r\omega$

• Centripetal Acceleration: $a_c = \frac{v^2}{r} = r\omega^2$

• Centripetal Force: $F_c = m a_c = m \frac{v^2}{r} = m r \omega^2$

• Period (T): The time for one complete revolution. $T = \frac{2\pi r}{v}$

• Frequency (f): Number of revolutions per unit time. $f = \frac{1}{T}$

Angular Acceleration

Angular acceleration measures how quickly the angular velocity changes. It is analogous to linear acceleration in straight-line motion.

• $\alpha = \frac{d\omega}{dt}$

• Occurs when the rotational speed of an object increases or decreases.

Angular Velocity and Angular Acceleration: Direction and Sign

The direction and sign of angular velocity and angular acceleration determine whether an object is speeding up or slowing down.

• If $\omega$ and $\alpha$ point in the same direction, the rotation speeds up.

• If $\omega$ and $\alpha$ point in opposite directions, the rotation slows down.

• Counterclockwise (CCW) is positive; clockwise (CW) is negative.

Graphical Analysis

Graphs of angular velocity vs. time and angular acceleration vs. time are useful for visualizing rotational motion.

• The slope of the $\omega$ vs. $t$ graph represents angular acceleration $\alpha$.

• The area under the $\alpha$ vs. $t$ graph gives the change in angular velocity.

Linear vs. Angular Motion: Comparison Table

The following table summarizes the correspondence between linear and angular motion:

Linear Motion	Circular Motion
Position: $x$	Angular Position: $\theta$
Velocity: $v_x = \frac{dx}{dt}$	Angular Velocity: $\omega = \frac{d\theta}{dt}$
Acceleration: $a_x = \frac{dv_x}{dt}$	Angular Acceleration: $\alpha = \frac{d\omega}{dt}$

Kinematic Equations: Linear and Rotational

For constant acceleration (linear) and constant angular acceleration (rotational), the following kinematic equations apply:

• Linear:

  • $v_f = v_i + a \Delta t$

  • $x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

  • $v_f^2 = v_i^2 + 2a (\Delta x)$

  • $x_f = x_i + \frac{1}{2}(v_i + v_f) \Delta t$

• Rotational:

  • $\omega_f = \omega_i + \alpha \Delta t$

  • $\theta_f = \theta_i + \omega_i \Delta t + \frac{1}{2} \alpha (\Delta t)^2$

  • $\omega_f^2 = \omega_i^2 + 2\alpha (\Delta \theta)$

  • $\theta_f = \theta_i + \frac{1}{2}(\omega_i + \omega_f) \Delta t$

Centripetal and Tangential Acceleration

In circular motion, two types of acceleration are present:

• Tangential Acceleration ($a_t$): Changes the speed of the particle along the circular path. $a_t = r\alpha$

• Centripetal Acceleration ($a_c$): Changes the direction of the velocity, keeping the particle moving in a circle. $a_c = r\omega^2$

• Total Acceleration ($a$): $a = \sqrt{a_t^2 + a_c^2}$

Math Review: Radians and Arc Length

Radians are the standard unit for measuring angles in rotational motion. The arc length $s$ of a circle of radius $r$ and central angle $\theta$ (in radians) is:

• $s = r\theta$

• There are $2\pi$ radians in $360^\circ$.

• Sign convention: $\theta$ is positive in the counterclockwise direction.

Examples and Applications

• Example 1: Two coins on a turntable at different radii have the same angular velocity but different linear speeds. The coin farther from the axis moves faster linearly.

• Example 2: A fan blade slowing down has negative angular velocity and positive angular acceleration if rotating clockwise.

Summary Table: Key Rotational Equations

Quantity	Equation	Units
Angular Position	$\theta$	radians (rad)
Angular Velocity	$\omega = \frac{d\theta}{dt}$	rad/s
Angular Acceleration	$\alpha = \frac{d\omega}{dt}$	rad/s2
Linear Velocity	$v = r\omega$	m/s
Tangential Acceleration	$a_t = r\alpha$	m/s2
Centripetal Acceleration	$a_c = r\omega^2$	m/s2

Additional info: These notes expand on the provided lecture slides by including definitions, equations, and examples for clarity and completeness. The tables are reconstructed for comparison and summary purposes.