Physics 170 - Mechanics Lecture 27 Angular Momentum Angular Momentum For a particle of mass m moving in a circle of radius r: More generally: L = r x p Angular Momentum For more general motion, Sign of Angular Momentum L>0 L>0 The angular momentum L is taken to be positive if the angular position is increasing with time, i.e., if the motion associated with L is a counterclockwise rotation. Rotation as a Vector We can treat rotation and other angular motion quantities as vectors by using the right-hand rule: if the fingers of your right hand follow the rotation direction, then your thumb points along the rotation axis in the vector direction of the angular velocity ω. An alternative definition is that if a right-hand threaded screw is rotated, then ω points in the direction in which the screw advances. The Vector Nature of Rotational Motion Similar application of the right-hand rule gives the vector direction of the torque τ. The Vector Nature of Rotational Motion The vector directions of the angular velocity vector ω and the angular momentum vector L are along the axis of rotation. Applying the right-hand rule gives the direction. Angular Momentum and Angular Velocity as Vectors Example: Angular Momentum About the Origin Find the angular momentum about the origin for the following situations: (a) A car of mass 1200 kg moves in a counterclockwise circle in the xy plane of radius 20 m with a speed of 15 m/s; (b) A uniform disk in the xy plane of radius 20 m and mass 1200 kg rotates at 0.75 rad/s along its axis, which is the z axis. Example: Angular Momentum (a) What is the angular momentum of a 0.13 kg Frisbee, considered to be a uniform disk of radius 7.5 cm, spinning with ω = 11.5 rad/s? (b) What is the angular momentum of a 95 kg person running with a speed of 5.1 m/s around a circular track of radius 25 m? Example: The Spin Angular Momentum of the Earth What is the angular momentum of the Earth as it rotates on its axis? (Assume a uniform sphere.) Example: The Orbital Angular Momentum of the Earth What is the angular momentum of the Earth as it orbits the Sun? Changing Angular Momentum Looking at the rate at which angular momentum changes, Therefore, if τ = 0, then L is constant with time. If the net external torque on a system is zero, the angular momentum is conserved. Example: A Windmill In a light wind, a windmill experiences a constant torque of 255 N m. If the windmill is initially at rest, what is its angular momentum after 2.00 s? Notice that you do not need to know the moment of inertia of the windmill to do this calculation. Example: Jumping On Running with a speed of 4.10 m/s, a 21.2 kg child heads toward the rim of a merry-go-round of radius 2.00 m, as shown. What is the child’s angular momentum L with respect to the center of the merry-go-round? (L = r x p) Conservation of Angular Momentum If the net external torque on a system is zero, the angular momentum is conserved. The most interesting consequences occur in systems that are able to change shape: 3.74 rad/s 5.33 kg m2 1.60 kg m2 Example: A Stellar Performance 8 A star of radius Ri = 2.3 x 10 m rotates initially with -6 an angular speed of ωi = 2.4 x 10 rad/s. If the star collapses to a neutron star of radius Rf = 20.0 km, what will be its final angular speed ωf ?.