Rotational Motion and Torque: Angular Kinematics, Forces, and Equilibrium

Rotational Motion and Angular Kinematics

Definitions and Fundamental Concepts

Rotational motion involves objects spinning about an axis, and is described using angular quantities analogous to linear motion. Understanding these concepts is essential for analyzing systems such as wheels, disks, and bars.

• Axis of Rotation: The point or line about which a system of particles rotates.

• Rigid Body: A solid object that maintains its shape and size during rotation (e.g., bar, disk, wheel).

• Rotational (Angular) Motion: Motion around the center of mass or an axis of rotation.

• Translational (Linear) Motion: Motion of the center of mass in a straight line.

Rule #1: Any point along a radius of a rotating rigid body has the same instantaneous angular velocity (\( \omega \)) and angular acceleration (\( \alpha \)), but different tangential velocity (\( v_t \)), tangential acceleration (\( a_t \)), and centripetal acceleration (\( a_c \)).

• Key Point: Points farther from the axis of rotation move faster tangentially to cover the same angle in the same time.

Angular Speed and Tangential Speed

• Angular Speed (\( \omega \)): The rate at which an object rotates, measured in radians per second (SI units).

• Conversion Example: For a disc spinning at 5.0 rpm (revolutions per minute): $\omega = 5.0 \ \text{rpm} \times \frac{2\pi \ \text{rad}}{1 \ \text{rev}} \times \frac{1 \ \text{min}}{60 \ \text{s}} = 0.52 \ \text{rad/s}$

• Tangential Speed (\( v_t \)): The linear speed of a point at a distance \( r \) from the axis: $v_t = r\omega$

• Example: At 3.0 cm from the center, $v_t = 0.030 \ \text{m} \times 0.52 \ \text{rad/s} = 1.6 \times 10^{-2} \ \text{m/s}$

• Angular Displacement: The angle (in radians) a point travels in a given time: $\theta = \omega t$

• Example: In 7.0 s, $\theta = 0.52 \ \text{rad/s} \times 7.0 \ \text{s} = 3.7 \ \text{rad}$

Period and Circular Motion

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

• Linear Distance in One Revolution: $2\pi r$

• Relationship: $v_t = \frac{2\pi r}{T}$

Torque: The Rotational Equivalent of Force

Definition and Calculation

Torque (\( \tau \)) is the measure of the tendency of a force to rotate an object about an axis. It depends on the magnitude of the force, the distance from the axis, and the angle at which the force is applied.

• Formula: $\vec{\tau} = \vec{r} \times \vec{F}$

• Magnitude: $\tau = r F \sin \phi$ Where: - $r$ = distance from axis to point of force application - $F$ = magnitude of force - $\phi$ = angle between $\vec{r}$ and $\vec{F}$

• Units: Newton-meter (N·m), not Joule.

• Only the perpendicular component of force (\( F_\perp \)) creates torque.

Factors Affecting Torque

• Magnitude of Force: Greater force produces greater torque.

• Distance from Axis: Applying force farther from the axis increases torque.

• Direction of Force: Only the tangential (perpendicular) component contributes to torque.

Right-Hand Curl Rule

• To determine the direction of torque, curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the torque vector.

Comparing Torque and Force

	Torque	Force
Effect	Changes rotation only	Changes rotation and/or translation
Direction	Acts tangentially to radius	Acts in any direction
Components	Categorized as CW or CCW	Broken into x, y, tangent, and centripetal

Equilibrium and Stability

Conditions for Equilibrium

A rigid body is in true equilibrium when both the net force and net torque acting on it are zero:

• Translational Equilibrium: $\sum \vec{F} = 0$ (no linear acceleration)

• Rotational Equilibrium: $\sum \tau_{ccw} + \sum \tau_{cw} = \tau_{net} = 0$ (no angular acceleration)

Only a net torque can produce a change in rotational motion.

Application: Forces on a Pinned Object

• Forces applied through the center of mass cause translation (linear acceleration).

• Forces applied away from the center of mass (or axis) cause rotation (torque).

• Forces acting at the axis of rotation (e.g., gravity, normal force at the pivot) produce zero torque because $r = 0$.

Gravitational Torque

• Gravity acts at the center of mass and can produce torque if the center of mass is not aligned with the axis of rotation.

• If the force of gravity acts through the axis, it produces no torque.

Ranking Examples

• Ranking Torque: For forces of equal magnitude applied at the same radius but different angles, torque is maximized when the force is perpendicular to the radius ($\sin \phi = 1$).

• Ranking Net Force and Net Torque: Systems can have different net forces and torques depending on the number, direction, and location of applied forces.

Independence of Net Force and Net Torque

• Net force and net torque are independent; a system can have zero net force but nonzero net torque, and vice versa.

Summary Table: Net Force vs. Net Torque

System	Net Force (Fnet)	Direction	Net Torque (\( \tau_{net} \))	Rotation Direction
A	4	Up	0	None
B	0	--	4	CCW
C	3	Up	1	CW
D	1	Up	3	CW

Key Equations

• Angular Speed: $\omega = \frac{\theta}{t}$

• Tangential Speed: $v_t = r\omega$

• Period: $T = \frac{2\pi}{\omega}$

• Torque: $\tau = r F \sin \phi$

• Equilibrium (Rotation): $\sum \tau_{ccw} + \sum \tau_{cw} = 0$

Example Problems

• Example 1: A disc of diameter 12.0 cm spins at 5.0 rpm. Find its angular speed in SI units. $\omega = 0.52 \ \text{rad/s}$

• Example 2: Find the tangential speed 3.0 cm from the center. $v_t = 1.6 \times 10^{-2} \ \text{m/s}$

• Example 3: How many radians does a spot 3.0 cm from the center travel in 7.0 s? $\theta = 3.7 \ \text{rad}$

Additional info: The notes above expand on the provided content with definitions, formulas, and context for clarity and completeness, suitable for college-level physics students studying rotational motion and torque.