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

Question

a solid homogeneous sphere of mass M and radius R can rotate about a vertical axis on the friction bearings (see the figure). A massless cord passes around the equator of the sphere, over a pulley of rotational inertia.

How far does blockfall. Please provide algebraic expression

a solid homogeneous sphere of mass M and radius R can rotate about a vertical axis on the friction bearings (see the figure). A massless cord passes around the equator of the sphere, over a pulley of rotational inertia. How far does blockfall. Please provide algebraic expression

Explanation / Answer

mg - T1 = ma

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

T2.R = (2MR^2 /5 ) (a/R) => T2 = 2Ma/5

add them

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

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

v^2 = 2ad

d = ( m + Ip/r^2 + 2M/5) .v^2 / 2mg

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