A child of mass m = 40 kg is initially placed on the edge of a uniform disk, of mass M = 100 kg and radius R = 2.0 m, which is rotating with an angular speed w_i about a vertical axis passing through its center. The child moves on a radius and stops at the distance a = 1.5 m from the edge of the disk as indicated. If the final angular speed of the disk-child system is w_f= 4.5 rad/s, determine the initial angular speed w_i. Wf a m

A playground merry-go-round with an axis at the center (radius R = 1.5 m and rotational inertia | = 1.33 x 103 kgm²) is initially rotating at angular velocity wo = 1.1 rad/s (counter-clockwise). A girl of mass m = 39 kg is running at speed vo = 3.3 m/s in a direction tangent to the disk of the merry-go-round, intending to jump on to the point labeled with the green "x" (she is indicated by the purple circle in the figure). What is the angular momentum of the girl relative to the rotation axis of the merry-go-round? Your answer should be in kgm²/s, but enter only the numerical part in the box. R Wo

A hollow sphere (I=\frac{2}{3}mr^2I=32​mr2) of radius R rotates about a diameter with an angular speed \omegaω. The sphere then collapses (magically) under the action of internal forces to a final radius of R/2 with no change in its mass. What is the final angular speed of the sphere?

A thin stick of mass 0.2 kg and length L = 0.5 m is attached to the rim of a metal disk of mass M = 2.0 kg and radius R = 0.3 m . The stick is free to rotate around a horizontal axis through its other end (see the following figure). (a) If the combination is released with the stick horizontal, what is the speed of the center of the disk when the stick is vertical? (b) What is the acceleration of the center of the disk at the instant the stick is released? (c) At the instant the stick passes through the vertical?

A girl of mass 20.3 kg stands on the rim of a frictionless merry-go-round of radius 1.29 m and rotational inertia 455 kg-m2 that is not moving. She throws a rock of mass 584 g horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is 5.22 m/s. Afterward, what are (a) the angular speed of the merry-go-round and (b) the linear speed of the girl? (a) Number i Units (b) Number i Units

A uniform thin spherical shell of mass M and radius R rotates about a vertical axis on frictionless bearings.  A massless cord passes around the equator of the shell, over a pulley of rotational inertia I and radius r, and is attached to a small object of mass m.  There is no friction on the pulley’s axle; the cord does not slip on the pulley.  Determine the expression for the speed of the object after it falls a distance h from rest.

A flat uniform circular disk (radius = 6.22 m, mass=320 kg) is initially stationary. The disk is free to rotate in the horizontal plane about a frictionless axis perpendicular to the center of the disk. A 51.0-kg person, standing 2.32 m from the axis, begins to run on the disk in a circular path and has a tangential speed of 3.00 m/s relative to the ground. Find the resulting angular speed (in rad/s) of the disk.

A ball of mass M = 1.30 kg and radius r = 4.50 cm is attached to one end of a thin, cylindrical rod of length L = 14.0 cm and mass m = 0.880 kg. The ball and rod, initially at rest in a vertical position and free to rotate around the axis shown in the figure below, are nudged into motion. L m M (a) What is the rotational kinetic energy of the system when the ball and rod reach a horizontal position? J (b) What is the angular speed of the ball and rod when they reach a horizontal position? rad/s V(c) Vfreefall (c) What is the linear speed of the center of mass of the ball when the ball and rod reach a horizontal position? m/s (d) What is the ratio of the speed found in part (c) to the speed of a ball that falls freely through the same distance?

Assume mars is a uniform solid sphere of mass 6.42x10^23 kg and a radius 3390 km.The length of a day on mars is 24 hours and 37 min. What is the rotational kinetic energy of the planet about its axis.

A 20-kg piece of steel pipe that is 20 cm in diameter, 80 cm long, and 1 cm thick is rolling toward you at a speed of 3.0 m/s.   (a)What is the magnitude and direction of the pipe’s angular velocity vector w?   (b)What is the pipe’s rotational energy?  (c) What is its total energy K + Urot?