The centre of mass of three particles of masses $1 \ kg, 2 \ kg$ and $3 \ kg$ is at $(2, 2, 2)$. The position of the fourth mass of $4 \ kg$ to be placed in the system so that the new centre of mass is at $(0, 0, 0)$ is:

Write a short note on the centre of gravity.

The mass per unit length of a rod of length $l$ is given by $\lambda = \frac{M_0 x}{l}$,where $M_0$ is a constant and $x$ is the distance from one end of the rod. The position of the center of mass of the rod is:

$A$ solid cylinder of mass $2 \ kg$,length $40 \ cm$ and radius $10 \ cm$ is placed in contact with a solid sphere of mass $0.5 \ kg$ and radius $10 \ cm$ such that the centres of the two bodies lie along the geometrical axis of the cylinder. The distance of the centre of mass of the system of two bodies from the centre of the sphere is (in $cm$)

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of the position vector of the centre of mass of this system will be equal to the magnitude of which of the following vectors?

$A$ uniform rod of length $L$ is placed along the $x$-axis with one end at the origin. If the linear mass density of the rod varies as $\lambda(x) = ax$,where $a$ is a constant,find the position of the centre of mass of the rod.