A uniform thin rod AB of length L has linear mass density `mu(x) = a + (bx)/(L)`, where x is measured from A. If the CM of the rod lies at a distance of `((7)/(12)L)` from A, then a and b are related as :_

Options

A. `a=2b`
B. `2a=b`
C. `a=b`
D. `3a=2b`

Answer

B

Explanation

Centre of mass of the rod is given by:
`x_(cm)=(underset(0)overset(L)int(ax+(bx^(2))/(L))dx)/(underset(0)overset(L)int(a+(bx)/(L))dx)=(L((a)/(2)+(b)/(3)))/(a+(b)/(2))`
Now `(7L)/(12)=((a)/(2)+(b)/(3))/(a+(b)/(2))` On solving we get, `b=2a`