Question

A uniform cylinder of radius R and mass M is mounted so as to rotate freely about a horizontal axis that is parallel to, and a distance h from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position? State your answers in terms of the given variables, using g when appropriate.

Answer 1

If I is the M.I about an axis passing through its CENTER OF MASS, then I(r) its M.I about a parlles axis distant d from the initial axis then

I(r) = I + Md^2 where M is its mass.
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Using the given values I about the C.M, which is deeived using calculus and is expected that every one knows that formula. If one wants he should derive it.
There are standard list of M.I of regular object about its center of mass.

a)

In the given problem the M.I of a uniform solid cylinder about central longitudinal axis is mr^2/2
Its M.I about an axis d m from this axis is
m(r^2/2 + d^2)
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b)
Now the cylinder (think as if a mass point is ) is rotating aout an vertical circle of radius 0.55m
At the top point its p.e = mgh.
At the lowest postion is has rotational kinetic energy 0.5 I?^2 and a linear k.e of 0.5 mv^2.
Total energy is 0.5 m(r^2/2 + d^2) ?^2 +0.5 mv^2 = mg( 2d) ( intial p.e)
(r^2/2 + d^2) ?^2 + v^2 = 4gd
(r^2/ 2d^2 + 1) v^2 + v^2 = 4gd
v^2 (r^2 + 4d^2 ) = 4gd*2d^2
v^2 = 8gd^3/ (r^2 + 4d^2 )