Aim: How do we explain the conservation of angular momentum?

Conservation of Angular Momentum of a System The total angular momentum of a system remains constant if the net external torque acting on the system is zero. ∑τ=dL/dt = 0 Thus, if the net Torque is 0, then L (angular momentum) is constant

List the three quantities which are conserved in isolated systems (Limit to Mechanics) Conservation of Energy Conservation of Momentum Conservation of Angular Momentum

More on Conservation of Angular Momentum The angular momentum of an isolated system is conserved whether the system is a rigid body or not. Li=Lf

Examples of Conservation of Angular Momentum 1.Figure Skater 2. Spiral Football 3.Gyroscope

Figure Skater L=Iω Compare the moment of inertia of the skater in each case. Compare the angular velocity in each case. How do you know this?

More on the Skater

Football If the football didn’t spiral, there is no angular momentum to be conserved and forces from the air might cause the ball to tumble as it moves along its trajectory.

Gyroscope

Problem 1 A cylinder with a moment of inertia of I1 rotates about a vertical, frictionless axle with angular speed ωi . A second cylinder that has a moment of inertia of I2 and initially is not rotating drops onto the first cylinder. Because of friction between the surfaces, the two eventually reach the same angular speed ωf . Calculate ωf Show that the kinetic energy of the system decreases. ωf = ωiI1/(I1 + I2) I1/(I1 + I2)

Problem 1

Problem 2 A star undergoes a supernova explosion. The material left behind forms a sphere of radius 8 x 106 m just after the explosion with a rotation period of 15 hours. This remaining material collapses into a neutron star of radius 8000 m. What is the rotation period of the neutron star? .054 s

Problem 3 A playground merry go round of radius R = 2.00 m has a moment of inertia I = 250 kg m2 and is rotating about a frictionless vertical axle. Facing the axle, a 25 kg child hops onto the merry go round from the ground and manages to sit down on its edge. What is the new angular speed of the merry go round? 7.14 rev/min

Problem 4 A space station shaped like a giant wheel has a radius of 100 m and a moment of inertia of 5.00 x 108 kg m2 . A crew of 150 are living on the rim, and the station’s rotation causes the crew to experience an acceleration of 1 g. When 100 people move to the center of the station for a union meeting, the angular speed changes. What acceleration is experienced by the managers remaining at the rim? Assume that the average mass of each inhabitant is 65 kg. 12.3 m/s2

Problem 4