Angular Momentum CONSERVATION OF ANGULAR MOMENTUM Objectives Calculate the angular momentum vector for a moving particle Calculate the angular momentum vector for a rotating rigid object where angular momentum is parallel to the angular velocity. 1
Quick Review of Linear Momentum Momentum p is a vector describing how difficult it is to stop a moving object. Total momentum is the sum of the individual momenta. A mass with velocity v has momentum p = mv (momentum and velocity vectors point in the same direction). Angular Momentum Angular Momentum L is a vector describing how difficult it is to stop a rotating object. Total angular momentum is the sum of individual angular momenta. A mass with velocity v moving at some position r about point G has angular momentum L Q. 2
Calculating Angular Momentum L Q = r P = r mv = r v m Use the right hand rule to determine the direction of the angular momentum. Write an equation for the magnitude of the momentum. What about an object moving in a circle in terms of angular velocity? L = mvr sin Ө = mr 2 ω Does this look familiar? 3
Spin Angular Momentum For an object rotating about its center of mass L = Iω This is known as an object s spin angular momentum Spin angular momentum is constant regardless of your reference point. (considering the COM) Intrinsic property of an object Object in Circular Orbit Find the angular momentum of a planet orbiting the sun. Assume a perfectly circular orbit. L = r p = r mv = r v m L = mvrsinө = mvr 4
Angular momentum of a point particle Find the angular momentum for a 5 kg point particle located at (2,2) with a velocity of 2 m/s east a) About point O at the origin b) About point p at (2,0) c) About point Q at (0,2) ANGULAR MOMENTUM 5
Objectives Recognize conditions under which angular momentum is conserved and relate this to systems such as satellite orbits. State the relation between net torque and angular momentum. Analyze problems in which the moment of inertia of an object is changed as it rotates freely about a fixed axis. Analyze a collision between a moving particle and a rigid object that can rotate. Angular Momentum and Net Torque L = r p What if we take the derivative? Rule for taking the derivative of a cross product: d da db A B = B + A dt dt dt So dl dt =? 6
dl dt = dr dp p + r dt dt dr dt = v So, Or, = v p + r F dl dt = r F dl dt = τ dp dt = F But what does that mean? A torque on an object is going to change its angular momentum. A force that is applied at some distance r from some axis of rotation, is going to cause a torque that will cause the object to begin to spin which will result in a change in angular momentum. 7
Conservation of Angular Momentum IN WORDS Spin angular momentum, the product of an object s moment of inertia and its angular velocity about the center of mass, is conserved in a closed system with no external net torque applied. IN MATHEMATICA L 0 = L f Iω 0 = Iω f Some examples ICE SKATER https://www.youtube.com/watc h?v=vmem0bnngr0 BALLET https://www.youtube.com/wat ch?v=bode0p7k0hm 8
Ice Skater Problem An ice skater spins with a specific angular velocity. She brings her arms and legs closer to her body reducing he moment of inertia to half its original value. What happens to her angular velocity? What happens to her rotational kinetic energy? Combining Spinning Discs A disc with moment of inertia 1 kgm 2 spins about an axel through its center of mass with angular velocity 10 rad/s. An identical disc which is not rotating is slid along the axle until it makes contact with the first disc. If the two discs stick together, what is their combined angular velocity? 9