Rotational Motion and Torque: Study Guide for College Physics

Rotational Motion and Angular Quantities

Introduction to Rotational Motion

Rotational motion describes the movement of objects around a fixed axis or point, following a circular path. Many physical systems, from wheels to planets, exhibit rotational motion, and understanding its principles is essential in physics.

• Rotational Motion: Motion around a fixed point or axis, following a circular trajectory.

• Rotational/Angular Position (θ): The rotational equivalent of linear position (x).

Linear vs. Rotational Position

Linear Position	Rotational Position
Measured in meters (m) Origin is where x = 0 Direction (+) is to the right	Measured in radians (rad) or degrees (°) Origin is where θ = 0 Direction (+) is counterclockwise (CCW), (-) is clockwise (CW)

Linear vs. Rotational Displacement

Displacement measures the change in position. In rotational motion, the equivalent is angular displacement (Δθ).

• Linear Displacement (Δx): Change in position, measured in meters.

• Rotational Displacement (Δθ): Change in angular position, measured in radians or degrees.

• Relationship: $\Delta x = r \Delta \theta$ (where r is the radius of the circular path)

• To convert between radians and degrees: $1 \text{ rad} \approx 57.3^\circ$

Examples and Applications

• Example: An object moves along a circle of radius 10 m from 30° to 120°. Find angular displacement and linear displacement.

• Practice: If a tire of radius 0.40 m rotates 10,000 times, how far did you drive?

Revolutions and Angular Displacement

Full Revolutions

• One full revolution: $\Delta \theta = 2\pi$ radians or $360^\circ$

• Linear distance for one revolution: $\Delta x = 2\pi r$

• N revolutions: $\Delta \theta = 2\pi N$

• To find final position after many revolutions: subtract multiples of $360^\circ$ or $2\pi$ until $0 < \theta < 2\pi$

Rotational Velocity and Acceleration

Angular Velocity and Angular Acceleration

Rotational motion has equivalents to linear velocity and acceleration.

• Angular Velocity (ω): Rate of change of angular position, measured in rad/s.

• Angular Acceleration (α): Rate of change of angular velocity, measured in rad/s².

• Formulas:

  • $\omega = \frac{\Delta \theta}{\Delta t}$

  • $\alpha = \frac{\Delta \omega}{\Delta t}$

Units and Conversions

Quantity	Unit	Conversion
Angular Velocity (ω)	rad/s	1 RPM = $\frac{2\pi}{60}$ rad/s; 1 Hz = $2\pi$ rad/s

Examples

• Example: A 30-kg disc of radius 2 m rotates at 120 RPM. Find its period and angular speed.

• Practice: Calculate the rotational velocity of a clock's minute hand in rad/s.

Rotational Kinematics: Equations of Motion

Rotational Motion Equations

Rotational motion uses equations analogous to linear kinematics, with angular variables.

Linear Equations	Rotational Equations
$v_f = v_i + a t$ $v_f^2 = v_i^2 + 2 a \Delta x$ $\Delta x = v_i t + \frac{1}{2} a t^2$ $x_f = x_i + v_{avg} t$	$\omega_f = \omega_i + \alpha t$ $\omega_f^2 = \omega_i^2 + 2 \alpha \Delta \theta$ $\Delta \theta = \omega_i t + \frac{1}{2} \alpha t^2$ $\theta_f = \theta_i + \omega_{avg} t$

Examples

• Example: A wheel accelerates from rest at 4 rad/s² until it reaches 80 rad/s. Find the time and angular displacement.

• Practice: A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration.

Linking Linear and Rotational Quantities

Relationships Between Linear and Rotational Motion

Linear and rotational quantities are connected through the radius of the circular path.

Linear	Rotational	Link
x	$\theta$	$x = r \theta$
$\Delta x / \Delta t$	$\Delta \theta / \Delta t$	$v = r \omega$
$a = \Delta v / \Delta t$	$\alpha = \Delta \omega / \Delta t$	$a_t = r \alpha$

• Linear Speed (v): $v = r \omega$

• Tangential Acceleration (a_t): $a_t = r \alpha$

Types of Acceleration in Rotational Motion

• Tangential Acceleration (a_t): Due to change in speed along the circular path.

• Radial (Centripetal) Acceleration (a_c): Due to change in direction, always points toward the center: $a_c = \frac{v^2}{r} = r \omega^2$

• Total Acceleration: $a_{total} = \sqrt{a_t^2 + a_c^2}$

• Angular Acceleration (α): Change in angular velocity.

Torque and Rotational Dynamics

Definition and Calculation of Torque

Torque is a measure of the tendency of a force to rotate an object about an axis. It is the rotational analogue of force in linear motion.

• Formula: $\tau = r F \sin \theta$

• Where r is the distance from axis to point of force application, F is the force, and θ is the angle between r and F.

• Torque is maximum when the force is perpendicular to r ($\theta = 90^\circ$).

• Direction: CW (clockwise) is negative, CCW (counterclockwise) is positive.

Net Torque

• If multiple torques act, net torque is the sum: $\tau_{net} = \sum \tau_i$

• Use signs (+/-) to indicate direction.

Examples and Applications

• Example: Calculate the torque produced by a force at different points on a door.

• Practice: Calculate the torque produced by pulling on a wrench at a given angle.

• Example: Calculate the net torque on a bar with forces applied at different points and angles.

• Example: Calculate the torque produced on a cylinder by attached masses.

Summary Table: Key Rotational Quantities

Quantity	Symbol	Unit	Formula
Angular Position	$\theta$	radian (rad)	-
Angular Displacement	$\Delta \theta$	radian (rad)	-
Angular Velocity	$\omega$	rad/s	$\omega = \frac{\Delta \theta}{\Delta t}$
Angular Acceleration	$\alpha$	rad/s²	$\alpha = \frac{\Delta \omega}{\Delta t}$
Torque	$\tau$	Newton-meter (Nm)	$\tau = r F \sin \theta$

Additional info:

• Practice problems throughout the notes reinforce concepts and provide opportunities for application.

• These notes cover the foundational aspects of rotational kinematics and dynamics, suitable for introductory college physics.