$A$ 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?

Give the location of the centre of mass of a $(i)$ sphere,$(ii)$ cylinder,$(iii)$ ring,and $(iv)$ cube,each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?

$A$ carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is $12$ times that of the cone,the position of the centre of mass of the toy is given by:

The integral $\int \vec{r} dm = 0$ for a homogeneous body is defined for which point?

The linear mass density of a $3 \ m$ long rod varies as $\lambda = 2 + x$. The position of the center of mass of the rod is at:

The coordinates of the positions of particles of mass $7 \, g$,$4 \, g$,and $10 \, g$ are $(1, 5, -3) \, cm$,$(2, 5, 7) \, cm$,and $(3, 3, -1) \, cm$ respectively. The position of the centre of mass of the system would be: