A rod of length L and mass M_r is attached to a solid sphere of radius R and mas

Question

A rod of length L and mass M_r is attached to a solid sphere of radius R and mass M_s, and the system is rotated about a vertical axis at an angular velocity omega. (a) What is the moment of inertia of only the rod using the Parallel Axis Theorem: I_2 = I_CM + Mh_^2 given that the moment of inertia a rod for rotations about its center of mass is I_CM = 1/12M_rr^2 (b) Use the Parallel Axis Theorem to find the total moment of inertia of the rod-sphere system for rotations about the vertical axis shown above.

Explanation / Answer

a)

For the rod , Icm = MrL2/12

h = L/2

so I = (MrL2/12) + Mr (L/2)2

I = (MrL2/12) + (MrL2/4)

Ir = MrL2/3

b)

for sphere , Icm = (0.4) MsR2

moment of inertia about the axis = Is = (0.4) MsR2 + Ms (L + R/2)2

Net moment of inertia = Inet = Is + Ir = (0.4) MsR2 + Ms (L + R/2)2 + MrL2/3

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