A uniform spherical shell of mass M = 5.0 kg and radius R = 8.6 cm can rotate ab

Question

A uniform spherical shell of mass M = 5.0 kg and radius R = 8.6 cm can rotate about a vertical axis on frictionless bearings (see figure below). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 3.0 x 10^-3 kg. m^2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60 kg. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object after it has fallen 75 cm after being released from rest? Use energy considerations. m/s The thin uniform rod in the figure below has length 8.0 m and can pivot about a horizontal, frictionless pin through one end. It is released from rest at angle theta = 36 degrees above the horizontal. Use the principle of conservation of energy to determine the angular speed of the rod as it passes through the horizontal position. rad/sec

Explanation / Answer

Applying work energy theorem,

work done by gravity = change in ke

mgh = [ ( m v^2 / 2 ) + ( I (v/r)^2/ 2) + ( (2 M R^2 / 3) ( v /R)^2 / 2) ] - 0

rotational KE = I w^2 / 2   and w = v/r

0.60 x 9.8 x 0.75 = [ (0.60 v^2 / 2) + ( 3 x 10^-3 x (v / 0.05)^2 / 2) + ( 2 x 5 x v^2 / 6)]

4.41 = 2.57v^2

v = 1.31 m/s

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