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Question

A solid hemisphere of radius R is mounted on a solid cylinder of same radius and density as shown. The height of the cylinder is such that resulting center of mass is O. If the radius of the sphere is 22m and the height of the cylinder is n meter. Find n.
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Solution

Let the point O be the origin.

Mass of cylinder = πR2Hρ with center of mass (0,H2)

Mass of hemisphere = Mh=2πR3ρ3

Center of mass of hemisphere calculation
Consider a disc of thickness dy at height y from O
radius of this disc will be given by r2=R2y2
y co-ordinate of center of mass = 1MhR0yπ(R2y2)ρy
= 1Mh[πρR44]
= 3R8

So co-ordinate of center of mass of hemisphere = (0,3R8)

As effective center of mass = (0,0)

For the COM of the system to be at O, m1×OS=M2×OC

=>πR2HρH2=2πR3ρ33R8

=> H=R2=22m2=2m

417754_432047_ans.png

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