Chapter 9: Rotational Motion – Study Notes

Rotational Motion

Introduction to Rotational Motion

Rotational motion refers to the movement of objects around a fixed axis. Many physical systems, from spinning ice skaters to rotating wind turbines, exhibit rotational motion. Understanding the principles of rotational motion is essential for analyzing the dynamics of rigid bodies in physics.

• Examples: Ferris wheels, wind turbines, car wheels, and human joints.

• Key Concepts: Angular position, angular velocity, angular acceleration, torque, moment of inertia, rotational kinetic energy, angular momentum, and collisions.

Angular Position and Rotation Angle

The angular position of an object describes its orientation relative to a reference direction, typically measured in radians. The rotation angle ($\theta$) is defined as the ratio of the arc length ($s$) to the radius ($r$) of the circular path:

• $\theta = \frac{s}{r}$

• One complete revolution: $s = 2\pi r$, so $\theta = 2\pi$ radians

• Conversion: $2\pi$ radians = $360^\circ$; $1$radian$\approx 57.3^\circ$

Angular Displacement

Angular displacement ($\Delta \theta$) is the change in angular position over time. It is a vector quantity, with direction determined by the sense of rotation (clockwise or counterclockwise).

• $\Delta \theta = \theta_f - \theta_i$

• Measured in radians ($\text{rad}$)

Angular Velocity

Angular velocity ($\omega$) describes how fast an object rotates and is defined as the rate of change of angular displacement:

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

• Units: radians per second ($\text{rad/s}$)

• Sign convention: Counterclockwise is positive, clockwise is negative

• Example: A Ferris wheel rotating at 6 revolutions per minute (rpm) has $\omega = 6 \times \frac{2\pi}{60} = 0.628\ \text{rad/s}$

Angular and Linear Speed

The linear speed ($v$) of a point on a rotating object is related to its angular speed:

• $v = r\omega$

• Points farther from the axis move faster

• Direction of $v$ is tangent to the circular path

Centripetal Acceleration and Force

Objects in uniform circular motion experience centripetal acceleration ($a_c$) directed toward the center of the circle:

• $a_c = \frac{v^2}{r} = r\omega^2$

• Centripetal force ($F_c$) is the net force causing this acceleration:

• $F_c = m a_c = m r \omega^2$

• Applications: Cars turning on curves, centrifuges, amusement park rides

Angular Acceleration

Angular acceleration ($\alpha$) is the rate of change of angular velocity:

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

• Units: $\text{rad/s}^2$

• Direction: Positive if speeding up in the positive angular direction, negative if slowing down

Tangential Acceleration

Tangential acceleration ($a_t$) changes the magnitude of the linear velocity:

• $a_t = r\alpha$

• Acts tangent to the circular path

• Perpendicular to centripetal acceleration

Equations of Rotational Kinematics (Constant Angular Acceleration)

Analogous to linear kinematics, the following equations apply:

• $\omega_f = \omega_i + \alpha t$

• $\theta = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$

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

Torque

Torque ($\tau$) is the rotational equivalent of force, causing objects to rotate about an axis:

• $\tau = r F \sin \phi$

• $r$ = distance from axis to point of force application (lever arm)

• $F$ = applied force, $\phi$ = angle between $r$ and $F$

• Maximum torque when force is perpendicular to lever arm

Moment of Inertia

Moment of inertia ($I$) quantifies an object's resistance to changes in rotational motion:

• $I = \sum m_i r_i^2$ (for discrete masses)

• Depends on mass distribution and axis of rotation

• Common moments of inertia:

Object	Axis	Moment of Inertia ($I$)
Thin rod	Center	$\frac{1}{12} M L^2$
Solid cylinder	Center	$\frac{1}{2} M R^2$
Hollow cylinder	Center	$M R^2$
Sphere	Center	$\frac{2}{5} M R^2$

Newton's Second Law for Rotation

The rotational analog of Newton's second law relates net torque to angular acceleration:

• $\sum \tau = I \alpha$

Rolling Motion

When an object rolls without slipping, its rotational and translational motions are related:

• $v = r \omega$

• Energy is shared between translation and rotation

• Example: Cylinder rolling down an incline

Summary Table: Key Rotational Quantities

Quantity	Symbol	Equation	Units
Angular displacement	$\Delta \theta$	$\theta_f - \theta_i$	rad
Angular velocity	$\omega$	$\frac{\Delta \theta}{\Delta t}$	rad/s
Angular acceleration	$\alpha$	$\frac{\Delta \omega}{\Delta t}$	rad/s2
Torque	$\tau$	$r F \sin \phi$	N·m
Moment of inertia	$I$	$\sum m_i r_i^2$	kg·m2
Rotational kinetic energy	$K_{rot}$	$\frac{1}{2} I \omega^2$	J
Angular momentum	$L$	$I \omega$	kg·m2/s

Additional info: These notes expand on the provided slides and handwritten content, filling in definitions, equations, and examples for completeness and clarity.