If linear density of a rod of length 3m varies as `lambda = 2 + x,` them the position of the centre of gravity of the rod is

Options

A. `(7)/(3)m`
B. `(12)/(7)m`
C. `(10)/(7) m`
D. `(9)/(7) m`

Answer

B

Explanation

`lamda = (2+x) , l=3m`
`x_(CM)= (int x dm)/(int dm)= (underset(0)overset(3)int x (2+x) dx)/(underset(0)^(3)int (2+ x) dx)`
`= (underset(0)overset(3)int (2x+ x^(2)) dx)/(underset(0)^(3)int (2+x) dx)= ([x^(2) + (x^(3))/(3)]_(0)^(3))/([2x+ (x^(2))/(2)]_(0)^(3))`
`= (9+ (27)/(3))/(6+ (9)/(2))= (18)/(21) xx 2= (12)/(7)m`