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Question

Find the moment of inertia of a solid cylinder of length l, radius R, about the central axis parallel to the height of the cylinder. The density of the cylinder varies with radius as ρ=2r.

A
35πlR5
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B
45πlR5
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C
πlR5
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D
πlR52
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Solution

The correct option is B 45πlR5
Moment of inertia about the central axis passing through the centre of the cylinder as shown in figure.


Considering solid cylinder to be made up of concentic hollow cylinders, let us take a small element (hollow cylinder) of thickness dr at distance r from the centre of the cylinder.

For elemental hollow cylinder as shown in figure, the moment of inertia about central axis is given as
dI=dm r2 ...(i)
where dm=mass of hollow cylinder


Volume of elemental hollow cylinder is given by,
dV=2πrl×dr ....(ii)
where 2πrl is the surface area of hollow cylinder and dr is the thickness

Mass of element (hollow cylinder) is given as:
dm=ρ×dV=ρ(2πrldr) ...(iii)

From Eq (i), (ii), (iii)
dI=ρ(2πrldr)r2
dI=2πρlr3dr

Putting equation for variable density ρ=2r
dI=2π(2r)lr3dr=4πlr4dr

The cylinder's moment of inertia is found by integrating the expression, with limits of r from 0 to R.
I0dI=R04πlr4dr
I=4πl[r55]R0
I=45πlR5

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