If the linear density (mass per unit length) of a rod of length `3 m` is proportional to `x`, where `x`, where `x` is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is.

Options

A. `1.5 m`
B. `2 m`
C. `2.5 m`
D. `3.0 m`

Answer

B

Explanation

Mass per unit length , i.e. linear density
`mu prop x rArr prop lambda x , lambda : constant`
`dm = mu dx = lambda x dx`
`x_(c.m.) = (int x dm)/(int dm) = (int_(0)^(3) x lambda x dx)/(int_(0)^(3) lambda xdx)`
`=(int x^(2) dm)/(int x dx) = (|(x^(3))/(3)|_(0)^(3))/(|(x^(2))/(2)|_(0)^(3)) = ((3^(3))/(3))/(3^(2)/(2)) = 2m`