Lecture 20: Angular Momentum - Torques - Conservation of Angular Momentum - Spinning Neutron Stars - Stellar Collapse
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1. Angular Momentum of a Particle:
Angular momentum is defined relative to an origin whose position can be freely chosen. Angular momentum is therefore not an intrinsic property of a moving object, it depends on the position of the origin. The angular momentum of a projectile's motion is explored. The angular momentum changes along its trajectory. The angular momentum of the Earth's orbit if measured relative to the sun's position, is constant (it is NOT constant if it is measured relative to any other origin).
2. Rate of Change of Angular Momentum:
Torque equals the time derivative of the angular momentum. The torque acting on the Earth is zero if we choose the sun as our origin. It is NOT zero relative to any other origin. Thus the orbital angular momentum of the Earth (sun as origin) is constant.
3. Angular Momentum of Rigid Bodies:
The angular momentum of a disk rotating about its center of mass is proportional to its moment of inertia. The angular momentum associated with rotational motion of a rigid body about a stationary axis through the center of mass is called spin angular momentum. Spin angular momentum is an intrinsic property of a spinning object. It is independent of the point of origin chosen. The Earth spins about an axis through its center of mass. The total angular momentum of the Earth with the sun as the origin is the vectorial sum of the spin angular momentum and the orbital angular momentum.
4. Ice Skater's Delight:
A person stands on a turntable with weights in each hand; this person can change her moment of inertia by moving the weights near to and away from her body. A numerical example is worked out showing that the moment of inertia can easily change by a factor of three, and a live demo is given. Angular momentum is conserved, so the angular velocity increases when the weights are pulled in.
5. Angular Momentum of a System and Stellar Spin-up:
The angular momentum of a system of objects can only be changed by external torques. In stellar collapse (resulting in neutron stars and black holes) there is conversion of gravitational potential energy into kinetic energy (heat). As the star collapses, its moment of inertia decreases dramatically but its spin angular momentum is conserved. If a solar-size star with a 100 day spin period collapses into a neutron star, its spin period will become about 1 ms.
6. Supernova Explosions and Stellar Spin-up:
Slides and commentary cover: Jocelyn Bell's discovery of radio pulsars (at first called LGMs for "Little Green Men"), the mis-alignment of the magnetic dipole axis and the spin axis of a neutron star, the Crab Nebula supernova remnant, and other supernova observations.
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