$A$ point object of mass $m$ is kept at $(a, 0)$ along the $x$-axis. What mass should be kept at $(-3a, 0)$ so that the center of mass lies at the origin?

The centre of mass of a thin rectangular plate (as shown in the figure) with sides of length $a$ and $b$,whose mass per unit area $(\sigma)$ varies as $\sigma = \frac{\sigma_0 x}{a b}$ (where $\sigma_0$ is a constant),would be . . . . . .

$A$ 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

The centre of mass of a system of particles does not depend on

Two objects of masses $200 \ gm$ and $500 \ gm$ possess velocities $10 \hat{i} \ m/s$ and $3 \hat{i} + 5 \hat{j} \ m/s$ respectively. The velocity of their centre of mass in $m/s$ is

Four particles of masses $m, 2m, 3m, 4m$ are placed at the corners of a square of side $a$ as shown in the figure. Find the coordinates of the center of mass.