Summer 2012 PHYS 172: Modern Mechanics Lecture 19 – Angular momentum Read 11.4 – 11.7

The angular momentum principle The angular momentum principle for a point particle A Note: The angular momentum principle is derived from the momentum principle

Clicker 1. Which case corresponds to largest torque? ABCABC 2. The direction of Torque in case B is: A) into page B) out of page C) it has no direction since it is zero Torque at angle

Example: momentum and angular momentum principles Use the momentum principle: Use the angular momentum principle: Falling object (nonrelativistic)

Conservation of angular momentum Example: Important: both L’s must be about the same point (axis)

A comet CLICKER: What is the direction of the torque on the comet in point B about the star due to gravitational pull? A)Into the page B)Out of the page C)It is zero B (nonrelativistic)

Example: Kepler and elliptical orbits Kepler, 1609: “a radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time” Can be easily proven using conservation of angular momentum See book p. 430 (11.4)

Clicker A ball falls straight down in the xy plane. Its momentum is shown by the blue arrow. What is the direction of the ball's angular momentum about the origin? x y A) +y B) –y C) +z (out of the page) D) –z (into the page) E) zero magnitude

Clicker A planet orbits a star, in a circular orbit in the xy plane. Its momentum is shown by the red arrow. What is the direction of the angular momentum of the planet in respect to the star? A) same direction as B) opposite to C) into the page D) out of the page E) zero magnitude

Multiparticle system Net torque caused by internal forces cancels out! m1m1 m2m2 m3m3 A

The angular momentum principle for a multiparticle system The angular momentum principle relative to the center of mass:

The three principles of mechanics Momentum Angular momentumEnergy External force: momentum changes External torque: angular momentum changes Energy input: energy of the system changes No external force: momentum is constant No external torque: angular momentum is constant No energy input: energy of the system is constant Location of object does not matter Location of object relative to A does matter Location of object does not matter (fundamental) (derived)

Angular momentum: a system with no torque Dorothy Hamill, 1985

Angular momentum: a system with no torque Cat always lands on its feet

Angular momentum: application A free-falling cat cannot alter its total angular momentum. Nonetheless, by swinging its tail and twisting its body to alter its moment of inertia, the cat can manage to alter its orientation See also book example: High dive page 437

Angular momentum: application A meteor rips through a satellite with solar panels. Calculate: v x,v y of center of mass f – angular velocity Momentum principle: x y ω

Angular momentum: application A meteor rips through a satellite with solar panels. Calculate: v x,v y of center of mass f – angular velocity Angular momentum principle: For sphere: Direction? ω

Static equilibrium: seesaw A