The centre of mass of a non-uniform rod of length $L$ whose mass per unit length $\lambda$ varies as $\lambda = \frac{k x^3}{L^3}$ (where $k$ is a constant and $x$ is the distance of any point on the rod from one end) is at a distance from the same end equal to:

Distinguish between the center of mass and the center of gravity.

Look at the figure drawn with ink of uniform linear density. $A$ mass $m$ of ink is used to draw each of the two inner circles and each of the two line segments. $A$ mass $6m$ of ink is used to draw the outer circle. The coordinates of the centers of the different parts are: outer circle $(0, 0)$,left inner circle $(-a, a)$,right inner circle $(a, a)$,and the horizontal line segment $(0, -a)$. Find the $y$-coordinate of the center of mass of the ink in the figure.

Two point masses of $1.5 \ g$ and $2.5 \ g$ are placed at a distance of $16 \ cm$ from each other. If the center of mass is at a distance $x$ from the $1.5 \ g$ mass,then $x = \dots \ cm$.

An isolated particle of mass $m$ is moving in a horizontal plane $(x-y)$ along the $x$-axis,at a certain height above the ground. It suddenly explodes into two fragments of masses $m/4$ and $3m/4$. An instant later,the smaller fragment is at $y = +15 \ cm$. The larger fragment at this instant is at

Consider a system of two particles of masses $m_1$ and $m_2$. If mass $m_1$ is pushed towards the center of mass of the system by a distance $d$,by what distance should the mass $m_2$ be displaced so that the center of mass of the system remains unchanged?