Chapter 9: Rotational Motion – Study Notes

Rotational Motion

Introduction to Rotational Motion

Rotational motion is a fundamental concept in physics, describing the movement of objects around a fixed axis. Unlike linear motion, rotational motion involves angular quantities and is essential for understanding the behavior of rigid bodies such as wheels, disks, and planets.

• Rigid Body vs. Deformable Body: A rigid body is an object whose shape does not change during motion, while a deformable body can change shape.

• Coordinate System: When analyzing rotational motion, it is important to define a coordinate system and choose a direction as positive. Conventionally, counterclockwise (CCW) rotation is considered positive.

Comparing Points on a Rotating Object

When an object rotates, different points on the object move at different linear speeds, even though they complete the same number of revolutions per unit time.

• Key Point: Points farther from the axis of rotation travel a greater distance in the same time, so their linear speed is higher.

• Example: On a spinning record, a point on the outer edge moves faster than a point near the center, although both complete one revolution in the same time.

Angular Quantities

Measuring Angles: The Radian

Angles in rotational motion are measured in radians, which relate the arc length to the radius of a circle.

• Definition of Radian: One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

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

• Example: If the arc length $s$ equals the radius $r$, then $\theta = 1$ radian.

Angular Position, Velocity, and Acceleration

Rotational motion is described using angular position, angular velocity, and angular acceleration, analogous to their linear counterparts.

• Angular Position ($\theta$): The orientation of a line with the body, measured from a reference axis.

• Angular Velocity ($\omega$): The rate of change of angular position with respect to time. $\omega = \frac{d\theta}{dt}$

• Angular Acceleration ($\alpha$): The rate of change of angular velocity with respect to time. $\alpha = \frac{d\omega}{dt}$

• Average and Instantaneous Values:

  • Average angular velocity: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$

  • Instantaneous angular velocity: $\omega = \frac{d\theta}{dt}$

  • Average angular acceleration: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$

  • Instantaneous angular acceleration: $\alpha = \frac{d\omega}{dt}$

Linear and Angular Motion Comparison

Equations of Motion

Rotational motion equations closely parallel those of linear motion, especially under constant acceleration.

Linear Motion	Rotational Motion
$v = v_0 + at$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Additional info: These equations are valid for motion with constant acceleration.

Linear and Rotational Quantities

Relationship Between Linear and Angular Quantities

Linear velocity and acceleration can be related to their angular counterparts for points at a distance $r$ from the axis of rotation.

• Tangential Velocity: $v_{tan} = r \omega$

• Tangential Acceleration: $a_{tan} = r \alpha$

• Radial (Centripetal) Acceleration: $a_{rad} = \frac{v^2}{r} = r \omega^2$

• Total Acceleration: $a = \sqrt{a_{tan}^2 + a_{rad}^2}$

Kinetic Energy in Rotational Motion

Rotational Kinetic Energy

Rotating objects possess kinetic energy due to their motion about an axis.

• Formula: $K_{rot} = \frac{1}{2} I \omega^2$ where $I$ is the moment of inertia and $\omega$ is the angular velocity.

• Moment of Inertia ($I$): A measure of how mass is distributed relative to the axis of rotation. For a point mass: $I = m r^2$

• Example: A disk rotating about its center has a different $I$ than a ring or a sphere.

Translational and Rotational Energy

Objects can have both translational and rotational kinetic energy.

• Translational Kinetic Energy: $K_{trans} = \frac{1}{2} M v^2$

• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$

• Total Kinetic Energy: $K_{total} = K_{trans} + K_{rot}$

Rolling Motion

Rolling Without Slipping

When an object rolls without slipping, there is a fixed relationship between its translational and rotational motion.

• Condition: $v_{cm} = R \omega$ where $v_{cm}$ is the velocity of the center of mass, $R$ is the radius, and $\omega$ is the angular velocity.

• Application: This condition is used to analyze rolling objects such as wheels and cylinders.

Potential Energy in Rotational Systems

Extended objects in a gravitational field possess gravitational potential energy.

• Formula: $U_{grav} = M g y_{cm}$ where $y_{cm}$ is the height of the center of mass.

Rotational Motion with Moving Axis

Center-of-Mass Speed and Angular Velocity

When the axis of rotation moves, such as in a yo-yo descending, both the center-of-mass speed and angular velocity must be considered.

• Example: For a yo-yo of mass $m = 0.10$ kg descending a height $h = 0.40$ m, the center-of-mass speed and angular velocity can be calculated using energy conservation and rotational kinematics.

• Typical Results: $v_{cm} = 2.9$ m/s, $\omega = 22.9$ rad/s

Additional info: Some equations and context have been expanded for clarity and completeness.