Rotational Motion: Dynamics, Energy, and Applications

Rotational Motion

Angular Position, Velocity, and Acceleration

Rotational motion describes the movement of objects around a fixed axis. The key quantities are angular position (θ), angular velocity (ω), and angular acceleration (α).

• Angular Position (θ): The angle an object has rotated, measured in radians (rad).

• Angular Displacement (Δθ): The change in angular position.

• Angular Velocity (ω): The rate of change of angular position with respect to time. $\omega = \frac{\Delta \theta}{\Delta t}$ (unit: rad/s)

• Angular Acceleration (α): The rate of change of angular velocity with respect to time. $\alpha = \frac{\Delta \omega}{\Delta t}$ (unit: rad/s2)

For constant angular acceleration, the equations of motion are analogous to those for linear motion:

• $\omega = \omega_0 + \alpha t$

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

• $\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t$

Linear vs. Angular Quantities

There is a direct relationship between linear and angular quantities for a point at a distance r from the axis of rotation:

• Linear velocity: $v = r\omega$

• Linear acceleration: $a = r\alpha$

Moment of Inertia

The moment of inertia (I) quantifies how mass is distributed relative to the axis of rotation and determines the resistance to angular acceleration. It is the rotational analog of mass in linear motion.

• For a point mass: $I = mr^2$

• For continuous bodies, $I = \int r^2 dm$

• The moment of inertia depends on the axis of rotation and the mass distribution.

Kinetic Energy of Rotation

A rotating rigid body possesses rotational kinetic energy:

• $K = \frac{1}{2} I \omega^2$

For objects undergoing both translation and rotation (e.g., rolling), the total kinetic energy is the sum of translational and rotational parts:

• $K_{total} = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I_{CM} \omega^2$

Work-Energy Theorem for Rotation

The work done by torques on a rotating object changes its rotational kinetic energy, analogous to the linear work-energy theorem.

• $W = \Delta K_{rot} = \frac{1}{2} I \omega^2 - \frac{1}{2} I \omega_0^2$

Example: A cable unwinds from a winch drum (solid cylinder, mass 50 kg, diameter 0.12 m) as a 9.0 N force pulls it for 2.0 m. Find the final angular velocity and speed of the cable.

Rotation About a Moving Axis

When the axis of rotation itself moves (e.g., rolling objects), the motion can be decomposed into:

• Translational motion of the center of mass

• Rotation about the center of mass

The total kinetic energy is:

• $K_{total} = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I_{CM} \omega^2$

Rolling Motion Without Slipping

For an object rolling without slipping, the point of contact with the surface is instantaneously at rest. The condition for rolling without slipping is:

• $v_{CM} = \omega R$

For one complete revolution, the distance traveled is the circumference:

• $\Delta x = 2\pi R$

Rotational Dynamics of Rolling Objects

When objects roll down an incline, both their mass and how that mass is distributed (moment of inertia) affect their acceleration and final speed.

• Objects with smaller moments of inertia (relative to their mass and radius) reach the bottom faster.

• For a solid cylinder: $I = \frac{1}{2}MR^2$

• For a hollow cylinder: $I = MR^2$

Example: Yo-Yo and Rolling Objects

Yo-Yo Problem: A yo-yo (solid disk, mass M, radius R) is released from rest and unwinds as it falls. The speed of the center of mass after dropping a height h can be found using energy conservation:

• Potential energy lost = total kinetic energy gained

• $Mgh = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I\omega^2$

• For a solid disk, $I = \frac{1}{2}MR^2$ and $v_{CM} = \omega R$

Comparing Rolling Objects on an Incline

When different objects (solid cylinder, hollow cylinder, sphere, etc.) roll down the same incline, their final speeds depend on their moments of inertia. The object with the smallest moment of inertia relative to its mass and radius will reach the bottom first.

Object	Moment of Inertia (I)
Solid Cylinder	$\frac{1}{2}MR^2$
Hollow Cylinder	$MR^2$
Solid Sphere	$\frac{2}{5}MR^2$
Thin Spherical Shell	$\frac{2}{3}MR^2$

Quick Check: Solid vs. Hollow Cylinder

Given two cylinders of equal mass and diameter, one solid and one hollow, released from rest at the top of an incline, the solid cylinder reaches the bottom first because it has a smaller moment of inertia and thus accelerates faster.

Summary Table: Linear vs. Angular Motion

Straight-line motion with constant linear acceleration	Fixed-axis rotation with constant angular acceleration
$a = \text{constant}$	$\alpha = \text{constant}$
$v = v_0 + at$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$
$x - x_0 = \frac{1}{2}(v_0 + v)t$	$\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t$

Additional info: The above notes cover the main concepts of rotational motion, including kinematics, dynamics, energy, and applications to rolling motion and compound motion. The included images and tables reinforce the mathematical relationships and physical intuition for these topics.