The variation of density of a cylindrical thick and long rod is $\rho = \rho_0 \frac{x^2}{L^2}$. The position of its centre of mass from the $x = 0$ end is:

$A$ rod of length $L$ has non-uniform linear mass density given by $\rho(x)=a+b\left(\frac{x}{L}\right)^{2}$,where $a$ and $b$ are constants and $0 \leq x \leq L$. The value of $x$ for the centre of mass of the rod is at

Four particles of masses $m_1 = 2m$,$m_2 = 4m$,$m_3 = m$,and $m_4$ are placed at the four corners of a square. What should be the value of $m_4$ so that the centre of mass of all the four particles is exactly at the centre of the square?

$A$ metre stick of mass $400 \ g$ is pivoted at one end and displaced through an angle $60^o$. The increase in its potential energy is ..... $J$.

In the case of a system of two particles of different masses,the centre of mass lies

Two particles of masses $5 \,g$ and $3 \,g$ are separated by a distance of $40 \,cm$. The centre of mass of the system of these two particles