A uniform spherical shell of mass M = 8.50 kg and radius R = 0.670 m can rotate

Question

A uniform spherical shell of mass M = 8.50 kg and radius R = 0.670 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.190 kg·m2 and radius r = 0.140 m, and is attached to a small object of mass m = 4.10 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 when it has fallen a distance 0.924 m after being released from rest? Use energy considerations.

Explanation / Answer

Here ,

let the speed of the object is v

as the rope is not slipping ,

the angular speed of shell is w = v/R

angular speed of pulley = v/r

NOw , using conservation of energy

0.5 * m * v^2 + 0.5 * I * w^2(shell) + 0.5 * I* w^2 (pulley) = m * g * h

0.5 * 4.1 * v^2 + 0.5 * (2/3) * 8.5 * 0.670^2 * v^2/0.670^2 + 0.5 * 0.190 * v^2/0.670^2 = 4.1 * 9.8 * 0.924

solving for v

v = 2.7 m/s

the speed of the block is 2.7 m/s

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