Question

A rod of length 'L' has the linear density given by the equation $\lambda =Kx+{\lambda }_0$ where K is the constant. Where does the centre of the mass lie?
( Origine is at the left end of the rod)

Answer 1

A rod of length $=L$

Linear density $=\lambda =Kx+{\lambda }_0$

$K=constant$

Let,

The centre of the mass lie$=m$

Total mass$=clensity\times length$

That is

$mass=\lambda \times dxdm=\lambda \times dx\frac{dm}{dx}=Kx+{\lambda }_0$

Integrating both sides

We get,

$m=\int Kxdx+{\lambda }_0\int dx=K\frac{x^2}{2}+{\lambda }_{0}x=K\frac{l^2}{2}+l{\lambda }_{0}x$

(Centre of mass)

(Put $x=l_0$)