The moment of inertia of a uniform cylinder of length `l and radius R` about its perpendicular bisector is `I`. What is the ratio `l//R` such that the moment of inertia is minimum ?

Options

A. `sqrt((3)/(4)`
B. `(sqr(3))/(2)`
C. 1
D. `(3)/(ssqrt(2))`

Answer

A

Explanation

We need to calculate the moment of inertia of the cylinder about its perpendicular bisector.
Consider the following figure:
Let the mass of the cylinder be M.

Let us take a segment of the cylinder of thickness dx at a distance x from the axis of rotation YY.
The segment chosen here is like a disc of radius R. As it is given that there is uniform mass distribution, hence the mass of the disc can be written as:
`dM=((M)/(piR^(2)l))(piR^(2)dx)=((M)/(l))dx`
We know that the moment of inertia of a disc of mass m and radius r about its diameter is given by
`I=(mr^(2))/(4)`
The moment of inertia of the segment disc about its diameter is `dI_(1)=(dMR^(2))/(4)`
Applying parallel axis theorem, the moment of inertia of the segment disc about the axis YY. can be calculated as:
`dI=(dMR^(2))/(4)+dMx^(2)=(MR^(2))/(4l)dx+(M)/(l)x^(2)dx`
We can calculate the moment of inertia of the cylinder by integration as follows:
`I=intdI=int_(-l//2)^(+l//2)(MR^(2))/(4l)dx+int_(-l//2)^(+l//2)(M)/(l)x^(2)dx`
`I=(MR^(2))/(4l)[(l)/(2)-(-(l)/(2))]+(M)/(3l)[((1)/(2))^(3)-(-(l)/(2))^(3)]`
`I=(MR^(2))/(4)+(Ml^(2))/(12)" "......(i)`
For the moment of inertia to be minimum, `(dI)/(dl)=0`
In the equation (i) we shall express the radius R of the cylinder in terms of its length l.
Let the mass density of the cylinder be `rho`.
That means:
`rho=(M)/(piR^(2)l)`
`impliesR^(2)=(M)/(pilrho)`
Substituting `R^(2)` in equation (i)
`I=(M)/(4)=((M)/(pilrho))+(Ml^(2))/(12)=(M^(2))/(4pilrho)+(Ml^(2))/(12)`
Differentiating both sides:
`(dI)/(dl)=(d)/(dl)((M^(2))/(4pilrho))+(d)/(dl)((Ml^(2))/(12))`
`=((M^(2))/(4pirho))(-1)l^(-2)+(Ml)/(6)`
`=(Ml)/(6)-((M^(2))/(4pirhol^(2)))`
For maxima or minima,
`(dI)/(dl)=0`
`(Ml)/(6)-((M^(2))/(4pirhol^(2)))=0`
`(l^(3))/(3)=((M)/(2pirho))`
Substituting
`rho=(M)/(piR^(2)l)`
`(l^(3))/(3)=((M)/(2pi((M)/(piR^(2)l))))`
`(l^(3))/(3)=((1)/(2((1)/(R^(2)l))))`
`(l^(3))/(3)=((R^(2)l)/(2))`
`(l^(2))/(R^(2))=(3)/(2)`
`implies(l)/(R)=sqrt((3)/(2))`