Conservation of Angular Momentum and Rotational Dynamics

Conservation of Angular Momentum

Direction of Angular Velocity and Angular Acceleration

Angular velocity (\( \vec{\omega} \)) and angular acceleration (\( \vec{\alpha} \)) are vector quantities that describe rotational motion. Their directions are defined using the right-hand rule, which helps specify the orientation of rotation in three-dimensional space.

• Angular Velocity (\( \vec{\omega} \)): Direction is along the axis of rotation, determined by the right-hand rule (curl fingers in the direction of rotation; thumb points in the direction of \( \vec{\omega} \)).

• Angular Acceleration (\( \vec{\alpha} \)): Direction is that of the change in angular velocity (\( \Delta \vec{\omega} \)), not necessarily the same as \( \vec{\omega} \).

• Physical Meaning: The direction of these vectors is abstract; it does not correspond to the motion of any physical part of the object, but rather to the orientation of the axis about which the object rotates.

Vector Cross Product and Torque

The cross product is a mathematical operation used to define vector quantities such as torque and angular momentum.

• Cross Product Definition: For vectors \( \vec{A} \) and \( \vec{B} \): $|\vec{A} \times \vec{B}| = AB \sin \theta$ The direction is perpendicular to the plane formed by \( \vec{A} \) and \( \vec{B} \), determined by the right-hand rule.

• Unit Vector Cross Products:

  • $\hat{i} \times \hat{j} = \hat{k}$

  • $\hat{j} \times \hat{k} = \hat{i}$

  • $\hat{k} \times \hat{i} = \hat{j}$

Vector Torque

Torque is the rotational analogue of force and is defined as:

• $\vec{\tau} = \vec{r} \times \vec{F}$

• Torque depends on the choice of origin, as the position vector \( \vec{r} \) is measured from the origin to the point of force application.

Rotational Dynamics Equation

The rotational equivalent of Newton's second law relates torque, moment of inertia, and angular acceleration:

• $\vec{\tau} = I \vec{\alpha}$

• In terms of angular velocity: $\vec{\tau} = I \frac{d\vec{\omega}}{dt}$

Angular Momentum

Definition and Properties

Angular momentum (\( \vec{L} \)) is a vector quantity representing the rotational analogue of linear momentum.

• For a particle: $\vec{L} = \vec{r} \times \vec{p}$ where \( \vec{p} = m\vec{v} \) is the linear momentum.

• For a rigid body rotating about a fixed axis: $\vec{L} = I \vec{\omega}$

• The direction of \( \vec{L} \) is the same as the direction of \( \vec{\omega} \).

• Angular momentum is defined with respect to a chosen origin (often the axis of rotation).

Angular Momentum for a System of Particles

• Total angular momentum: $\vec{L}_{\text{tot}} = \sum_i \vec{r}_i \times m_i \vec{v}_i$

• For rotation about the z-axis: $\vec{L}_{\text{tot}} = I \vec{\omega}$

Conservation of Angular Momentum

Angular momentum is conserved in a system with no external torque.

• Conservation Law: $\vec{\tau}_{\text{ext}} = 0 \implies \vec{L}_{\text{tot}} = \text{constant}$

• Analogous to conservation of linear momentum when \( \vec{F}_{\text{ext}} = 0 \).

• For a rotating object: $I_i \omega_i = I_f \omega_f$

• If the moment of inertia decreases (e.g., by "pulling in arms"), angular velocity increases to keep \( L \) constant.

Proof of Conservation

• Time derivative of angular momentum: $\frac{d\vec{L}}{dt} = \vec{\tau}_{\text{net}}$

• If \( \vec{\tau}_{\text{net}} = 0 \), then \( \vec{L} \) is constant.

Examples and Applications

• Spinning Skater: When a skater pulls in their arms, their moment of inertia decreases, so their angular velocity increases to conserve angular momentum.

• Collapsing Star: As a star collapses (radius decreases), its rotation rate increases dramatically due to conservation of angular momentum. For example, a star's radius may shrink from about 1 million miles to 30 miles, causing its rotation period to decrease from days to milliseconds (as seen in neutron stars).

Quantitative Example: Collapsing Star

• Moment of inertia for a sphere: $I = \frac{2}{5} M R^2$

• Conservation equation: $I_i \omega_i = I_f \omega_f$

• Relating periods and radii: $\omega = \frac{2\pi}{T}$ $\frac{R_i^2}{T_i} = \frac{R_f^2}{T_f}$

• If \( R_i \gg R_f \), then \( T_i \gg T_f \): the rotation period decreases dramatically.

• Example: The Sun rotates once every 27 days; a neutron star (\( R \approx 30 \) miles) can rotate 100 times per second.

Translation and Rotation: Analogies

The following table summarizes the correspondence between translational and rotational motion:

Translational Quantity	Rotational Quantity
Displacement: x	Angular Displacement: \( \theta \)
Velocity: \( v = \frac{\Delta x}{\Delta t} \)	Angular Velocity: \( \omega = \frac{\Delta \theta}{\Delta t} \)
Acceleration: \( a = \frac{\Delta v}{\Delta t} \)	Angular Acceleration: \( \alpha = \frac{\Delta \omega}{\Delta t} \)
Force: F	Torque: \( \tau = r F_\perp \)
Mass: M	Moment of Inertia: \( I = \sum m r^2 \)
Newton's 2nd Law: \( F_{\text{net}} = M a \)	Rotational 2nd Law: \( \tau_{\text{net}} = I \alpha \)
Kinetic Energy: \( KE_{\text{trans}} = \frac{1}{2} M v^2 \)	Rotational Kinetic Energy: \( KE_{\text{rot}} = \frac{1}{2} I \omega^2 \)
Linear Momentum: \( p = m v \)	Angular Momentum: \( L = I \omega \)
Impulse-Momentum: \( F_{\text{net}} = \frac{\Delta p}{\Delta t} \)	Torque-Angular Momentum: \( \tau_{\text{net}} = \frac{\Delta L}{\Delta t} \)
If \( F_{\text{ext}} = 0 \), \( p_{\text{tot}} = \text{constant} \)	If \( \tau_{\text{ext}} = 0 \), \( L_{\text{tot}} = \text{constant} \)

Additional info: Only four physical quantities are strictly conserved in all interactions: energy, linear momentum, angular momentum, and electric charge.