The centre of mass of a homogeneous semi-circular plate of radius $r$ is located at $A$ as shown in the figure. The distance $OA$ is

Define the position vector of the centre of mass.

$A$ 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 mass per unit length of a rod of length $3 \ m$ varies in proportion to the distance $x$ from one of its ends. The center of mass of this rod from that end is at a distance of ........ $m$.

Center of mass of a system of three particles of masses $1 \, kg, 2 \, kg, 3 \, kg$ is at the point $(1 \, m, 2 \, m, 3 \, m)$ and center of mass of another group of two particles of masses $2 \, kg$ and $3 \, kg$ is at point $(-1 \, m, 3 \, m, -2 \, m)$. Where should a $5 \, kg$ particle be placed so that the center of mass of the system of all these six particles shifts to the center of mass of the first system?

The sum of moments of all the particles in a system about its center of mass is always . . . . . .