Angular Momentum & its Conservation AP Physics Section 8-8 Angular Momentum & its Conservation

Angular Momentum τ∆t = L–L0 L L = I ω p = mv F∆t = p–p0 Recall: Linear momentum: p = mv Recall that angular velocity, ω, is measured in rad/s. The angular analog to inertial mass is the moment of inertia, I. Therefore, angular momentum, L, equals: L L = I ω p = mv Angular Impulse–Momentum Theorem ω F∆t = p–p0 τ∆t = L–L0

From the impulse–momentum theorem we can derive the more general version of Newton’s second law. It will take into account not only changing angular speed, but changing moment of inertia as well. Moment of inertia can often be changed by changing the shape of the object. p–p0 ∆p L–L0 ∆L Fnet = = τnet = = ∆t ∆t ∆t ∆t

Conservation of Angular Momentum In the same way that linear momentum was conserved when the net force on a system was zero, the angular momentum is conserved if the net torque is zero. L–L0 τnet = If τnet = 0, then L – L0 = 0. ∆t ∴ L = L0 The total angular momentum of a rotating object remains constant if the net torque acting on it is zero. L = L0 Iω = I0ω0 = constant

Gyroscopes A rotating object such as a heavy disk will not change its angular momentum if there is no net torque. A gyroscope is a devise featuring a massive rotating object for the purpose of maintaining a constant direction. They are used in gyroscopic compasses and in the guidance systems of missiles.

Precession If there is some net torque on the rotating object, the axis of rotation will change. Toy tops are given a torque and set to spinning, however any movement of the center of mass from vertical will create a new torque. This torque causes the axis to precess, or change its orientation over time.

Due to complex, changing forces on the spinning Earth, it undergoes several motions, including precession. The period of precession of the Earth is 26000 years. This causes the north pole to point at various stars over millennia. Our current pole star is Polaris. However, for the ancient Egyptians it was Thuban in Draco.