Rotation of a Rigid Body and Angular Momentum: Study Notes and Worked Examples

Rotation of a Rigid Body

Angular Velocity and Acceleration

Rotational motion involves objects spinning about an axis. The angular velocity (ω) describes how fast an object rotates, while angular acceleration (α) describes how quickly the angular velocity changes.

• Angular velocity (ω): The rate of change of angular displacement, measured in radians per second (rad/s).

• Angular acceleration (α): The rate of change of angular velocity, measured in rad/s2.

• Relationship: $\alpha = \frac{d\omega}{dt}$

• Example: If a disk starts from rest and reaches 900 rpm in 60 seconds, its angular acceleration is $\alpha = \frac{\omega_f - \omega_i}{t}$, where $\omega_f$ is the final angular velocity.

Moment of Inertia

The moment of inertia (I) quantifies an object's resistance to changes in rotational motion. It depends on the mass distribution relative to the axis of rotation.

• Definition: $I = \sum m_i r_i^2$ for discrete masses, or $I = \int r^2 dm$ for continuous bodies.

• Common moments of inertia:

  • Solid disk about center: $I = \frac{1}{2}MR^2$

  • Solid sphere about center: $I = \frac{2}{5}MR^2$

  • Rod about center: $I = \frac{1}{12}ML^2$

• Example: For a disk of mass 2 kg and radius 0.05 m, $I = \frac{1}{2} \times 2 \times (0.05)^2 = 0.0025$ kg·m2.

Rotational Kinetic Energy

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

• Formula: $K_{rot} = \frac{1}{2}I\omega^2$

• Example: For $I = 5.0 \times 10^{-2}$ kg·m2 and $\omega = 69.3$ rad/s, $K_{rot} = \frac{1}{2} \times 5.0 \times 10^{-2} \times (69.3)^2 = 12$ joules.

Angular Momentum and Conservation

Angular Momentum

Angular momentum (L) is a measure of the rotational motion of an object and is conserved in the absence of external torques.

• Definition: $L = I\omega$

• Conservation: If no external torque acts, $L_{initial} = L_{final}$

• Example: If a rotating disk's moment of inertia changes, its angular velocity adjusts to conserve $L$.

Torque and Rotational Dynamics

Torque (τ) is the rotational equivalent of force and causes changes in angular momentum.

• Definition: $\tau = I\alpha$

• Relation to angular momentum: $\tau = \frac{dL}{dt}$

• Example: A pulley system with masses and a rotating cylinder requires analysis of torques to determine acceleration.

Applications and Problem Solving

Rotational Motion in Systems

Many problems involve pulleys, disks, and masses connected by strings. Analyzing forces, torques, and energy is essential.

• Free-body diagrams: Used to identify forces and torques acting on each component.

• Equations of motion: Combine Newton's laws for translation and rotation.

• Example: For a pulley of mass $m_p$ and radius $R$, with masses $m_1$ and $m_2$ attached, the acceleration is: $a = \frac{m_1g}{m_1 + m_2 + \frac{1}{2}m_p}$

Multiple-Choice Conceptual Questions

Conceptual understanding is tested with statements about angular momentum, torque, and rotational motion.

• Key concepts:

  • A particle moving in a straight line cannot have angular momentum about its own center.

  • Angular momentum is conserved in the absence of external torque.

  • Rotational inertia depends on mass distribution.

Summary Table: Rotational Quantities

Quantity	Symbol	Formula	Units
Angular velocity	ω	$\omega = \frac{\Delta \theta}{\Delta t}$	rad/s
Angular acceleration	α	$\alpha = \frac{d\omega}{dt}$	rad/s2
Moment of inertia	I	$I = \sum m_i r_i^2$	kg·m2
Rotational kinetic energy	$K_{rot}$	$K_{rot} = \frac{1}{2}I\omega^2$	J (joules)
Angular momentum	L	$L = I\omega$	kg·m2/s
Torque	τ	$\tau = I\alpha$	N·m

Additional info:

• Some problems involve the conservation of angular momentum when the moment of inertia changes (e.g., a figure skater pulling in arms).

• Rotational analogs of Newton's laws are essential for analyzing systems with pulleys and rotating masses.

• Understanding the direction of angular momentum and torque vectors is crucial for solving 3D problems.