Angular Momentum Study Notes
Angular Momentum
Definition: Angular momentum is defined as the product of moment of inertia and angular velocity.
Units: kg m^2/s
Linear Momentum: Defined as where:
mass
linear velocity
Just as net force is related to linear momentum by Newton's Second Law, net torque relates to angular momentum:
Relationship between Angular and Linear Quantities
Angular momentum is associated with linear momentum:
Where is the position vector from the reference point to the mass with momentum .
Magnitude of Angular Momentum:
Where is the angle between vectors and .
Examples
Calculating Angular Momentum for Circular Motion:
For mass moving in a circle of radius with speed :
(since )
Alternatively, using , where and also yields the same result.
Determining Direction: Use the Right-Hand Rule:
Position your right hand with fingers pointing along , then curl fingers towards ; the thumb indicates the direction of .
Angular momentum can be into or out of the page based on this orientation.
Example of Angular Momentum in Linear Motion
For a plane flying at constant altitude :
At different points and , calculate angular momentum using:
; both points yield the same result since altitude remains constant.
Direction determined via right-hand rule yields that both points have the same direction of angular momentum: into the page.
Conservation of Angular Momentum
Introduction to Angular Momentum
The concept of angular momentum is crucial in physics, especially when analyzing systems in motion.
Angular momentum (denoted as L) can be calculated for an object or a system experiencing motion, such as an airplane flying over a point on the ground.
The idea of conservation of angular momentum is linked closely to conservation of linear momentum, making it a fundamental law in mechanics.
Definition of Angular Momentum
Angular momentum (L) can be defined using two primary concepts:
For a point mass or linear motion: It is the product of the radius vector from the pivot point to the object and the linear momentum.
For rotational motion: It is the product of the moment of inertia and the angular velocity.
Conservation of Angular Momentum
The principle states that the total angular momentum of a system remains constant if no external torques act on it.
This relation holds true immediately before and after an event.
Key conditions for conservation:
No significant external torques must be acting on the system.
There must be no substantial external impulses affecting the angular momentum.
Events Affecting Angular Momentum
Angular momentum is conserved under specific types of events, including:
A change in the moment of inertia of an object or system.
Collisions that result in either rotation or linear motion.
Example 1: Ice Skater
Consider an ice skater spinning on ice:
When the skater pulls in their arms, they decrease their moment of inertia.
Since angular momentum remains constant, when the moment of inertia decreases, angular velocity must increase, resulting in the skater spinning faster.
Example 2: Collision with Rotation
Let
hanging vertically in a gravitational field and a mass colliding with it:The mass moves in a straight line and sticks to the bar, causing it to swing upwards.
The initial linear momentum of the mass about the pivot point must equal the combined angular momentum of the bar and mass after the collision.
This demonstrates the conservation of angular momentum since the linear motion has now resulted in rotational motion.
Example 3: Rotation Leading to Linear Motion
Conversely, analyze a rotating object:
If it swings down and collides with another object, causing that object to move in a straight line:
Again, the principle of conservation of angular momentum applies as the changes in motion stem from the interconnectedness of linear and rotational dynamics.
Consistency with Pivot Points
It is vital to maintain consistency when defining the pivot point for calculations, both before and after an event.