Conservation of angular momentum

Problem:

A child of mass 25 kg stands at the edge of a rotating platform of mass 150 kg and radius 4.0 m.  The platform with the child on it rotates with an angular speed of 6.2 rad/s.  The child jumps off in a radial direction (i.e. has no tangential velocity in the inertial frame of an outside observer).  What happens to the angular speed of the platform?  Treat the platform as a uniform disk.  Give a quantitative explanation.

Solution:

Problem:

A 60 kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 kgm2, and a radius of 2 m.  The turntable is initially at rest and is free to rotate about a frictionless vertical axis through its center.  The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.5 m/s relative to the Earth.
(a)  In what direction and with what angular speed does the turntable rotate?
(b)  How much work does the woman do to set herself and the turntable in motion?

Solution:

Problem:

A mass M is attached to the end of a string which passes through a hole in a frictionless horizontal plane.  Initially the mass moves on a circle of radius R with kinetic energy T0.  The string is then slowly pulled so that the mass finally rotates on a circle of radius R/3.  How much work was done?

Solution:

Problem:

A small ball swings in a horizontal circle at the end of a cord of length L1 which forms an angle θ1 with the vertical.  Gravity is acting downward.  The cord is slowly shortened by pulling it through a hole in its support until the free length is L2 and the ball is moving at an angle θ2 from the vertical. 
(a)  Derive a relation between L1, L2, θ1, and θ2
(b)  If L1 = 50 cm, θ1 = 5o, and L2 = 30 cm, find θ2.

Solution:

Problem:

A point mass m attached to the end of a string revolves in a circle of radius R on a frictionless table at constant speed with initial kinetic energy E0.  The string passes through a hole in the center of the table and the string is pulled down until the radius of the circle is ½ of its initial value. Assuming no external torque acts on the system, how much work is done?

Solution:

Problem:

A solid sphere toy globe of mass M and radius R rotates freely without friction with an initial angular velocity ω0.  A bug of mass m starts at one pole N and travels with constant speed v to the other pole S along a meridian in time T.  The axis of rotation of the globe is held fixed.  Show that during the time the bug is traveling the globe rotates through an angle
Δθ = (πω0R/v) (2M/(2M + 5m))1/2.

Useful integral: ∫0dx/(a + b cosx) = 2π/(a2 - b2)1/2,  (a2 > b2).

Solution:

Problem:

A uniform spherical planet of radius a revolves around the sun in a circular orbit of radius r0 and angular velocity ω0.  It rotates about its axis with angular velocity Ω0 (period T0) normal to the plane of the orbit.  Due to tides raised on the planet by the sun, its angular velocity of rotation is decreasing.  Find an expression which gives the orbital radius r as a function of the angular velocity Ω of rotation and the parameters r0 and T0 at any later or earlier time.