Rotational Motion: Key Concepts and Formulas

Rotational Motion

Introduction to Rotational Motion

Rotational motion describes the movement of objects that rotate about a fixed axis. Each point on a rotating object follows a circular path, and the study of rotational motion is fundamental to understanding the dynamics of objects such as wheels, planets, and machinery.

• Rotational motion involves objects that rotate about an axis.

• Each point on the object traces a circular path.

Angular Position

The angular position of a point on a rotating object is a measure of its orientation relative to a reference direction.

• Angular position is represented by the symbol \( \theta \).

• Angles are measured in radians, not degrees.

• Counterclockwise rotation is considered positive.

Angular Displacement and Angular Velocity

Angular displacement and angular velocity describe how the angle changes over time during rotation.

• Angular displacement is the change in angle (\( \Delta \theta \)).

• Angular velocity is the change in angle per unit time:

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

• Units of angular velocity: radians per second (rad/s).

Period and Frequency

The period and frequency are measures of how often a rotation occurs.

• Period (T): Time for one complete rotation.

• Frequency (f): Number of rotations per second.

• Relationship between angular velocity, period, and frequency:

$\omega = 2\pi f = \frac{2\pi}{T}$

Linear and Angular Speed

Linear speed is related to angular speed, especially for points at different distances from the axis.

• Relationship: $v = \omega r$

• Points farther from the axis have greater linear speed.

• Example: The rim of a spinning wheel moves faster than points near the center.

Angular Acceleration

Angular acceleration measures how quickly angular velocity changes.

• Angular acceleration (\( \alpha \)) is defined as:

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

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

• When an object slows down, \( \omega \) and \( \alpha \) have opposite signs.

Tangential Acceleration

Tangential acceleration is the rate at which the linear speed of a point on a rotating object changes.

• Measures the rate of change of velocity for a point at radius r.

• Relationship: $a_t = \alpha r$

Torque

Torque is the rotational equivalent of force and causes objects to rotate.

• Torque (\( \tau \)) is defined as:

$\tau = r F$ $\tau = r F \sin \theta$

• Only the perpendicular component of force produces torque.

• Counterclockwise torque is positive (+); clockwise torque is negative (-).

Center of Gravity

The center of gravity is the point where the object's weight can be considered to act.

• If the center of gravity is directly below the pivot, the gravitational torque is zero.

Moment of Inertia

Moment of inertia quantifies how mass is distributed relative to the axis and affects rotational acceleration.

• Rotational equivalent of mass in linear motion.

• Formula: $I = \sum m_i r_i^2$

• Depends on mass and distance from the axis.

Newton’s Second Law for Rotation

Newton's second law applies to rotational motion, relating torque, moment of inertia, and angular acceleration.

• Rotational form: $\tau = I \alpha$

• A larger moment of inertia results in smaller angular acceleration for the same torque.

Rolling Motion

Rolling without slipping occurs when the point of contact with the surface is momentarily at rest.

• Condition: $v = \omega r$

• The bottom point of a rolling wheel is momentarily at rest (zero velocity).

Summary Table: Rotational Motion Quantities

Quantity	Symbol	Formula	Units
Angular Position	\( \theta \)	—	radians
Angular Velocity	\( \omega \)	$\omega = \frac{\Delta \theta}{\Delta t}$	rad/s
Angular Acceleration	\( \alpha \)	$\alpha = \frac{\Delta \omega}{\Delta t}$	rad/s2
Torque	\( \tau \)	$\tau = r F \sin \theta$	N·m
Moment of Inertia	\( I \)	$I = \sum m_i r_i^2$	kg·m2
Linear Speed	\( v \)	$v = \omega r$	m/s
Tangential Acceleration	\( a_t \)	$a_t = \alpha r$	m/s2