A homogeneous and solid sphere of mass M and radius R can rotate about a vertica

Question

A homogeneous and solid sphere of mass M and radius R can rotate about a vertical

axis on the frictional bearings (see the figure). A massless cord passes around the

equator of the sphere, over a pulley of rotational inertia Ip and radius r, and is

attached to a small block of mass m. No friction exists on the pulley

A homogeneous and solid sphere of mass M and radius R can rotate about a vertical axis on the frictional bearings (see the figure). A massless cord passes around the equator of the sphere, over a pulley of rotational inertia Ip and radius r, and is attached to a small block of mass m. No friction exists on the pulley's axis and the cord does not slip on the pulley. How far does block fall (i.e., distance d) when it reaches a speed v? (Please provide an algebraic expression with all the provided quantities)

Explanation / Answer

Torque = T1 R    where T1 is tension in cord connected to sphere

T1 R = I a / R = 2/5 M R a     where a is acceleration of cord

T1 = 2/5 M a

m a = m g - T2    where T2 is tension in cord connected to m

T2 = m (g - a)

T2 - T1 = 2/5 M a - m (g - a) = (2/5 M + m) a - m g

(T1 - T2) * r = Ip * a / r   acceleration of pulley

(T1 - T2) = Ip a / r^2

-Ip a / r^2 = (2/5 M + m) a - m g

(2/5 M + m + Ip / r^2) a = m g

a = m g / (2/5 M + m + Ip / r^2)

2 a s = v^2

Substitute for a

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