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

Question

A rod of length L and mass Mr is attached to a solid sphere of radius R and mass Ms , and the system is rotated about a vertical axis at an angular velocity. (a) What is the moment of inertia of only the rod using the Parallel Axis Theorem: Iz = Icm + Mh^2 given that the moment of inertia a rod for rotations about its center of mass is Icm = 1/12Mr^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 moment of inertia about the end

d = L/2

Using parallel axis theorum

Iz = Icm + M * d^2

Iz = M * L^2/12 + M * (L/2)^2

Iz = M * L^2/3

the moment of inertia of rod is M * L^2/3

b)

Now , for the moemnt of inertia of sphere

Using parallel axis theorum

Iz = Icm + M * L^2

Iz = 0.4 * M * R^2 + M * L^2

Now , for the total moment of inertia of system

total moment of inertia of system = 0.4 * M * R^2 + M * L^2 + M * L^2/3

total moment of inertia of system = M * (0.4 * R^2 + 4L^2/3)

the total moment of inertia of system is M * (0.4 * R^2 + 4L^2/3)

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