A rigid body consists of a $3 \mathrm{kg}$ mass connected to a $2 \mathrm{kg}$ mass by a massless rod. The $3 \mathrm{kg}$ mass is located at ${\overrightarrow{r}}_1=(2\hat{i}+5\hat{j})m$ and the $2 \mathrm{kg}$ mass at ${\overrightarrow{r}}_2=(4\hat{i}+2\hat{j})m$. If coordinates of the centre of mass is $(a, b)$ then $\frac{5}{11}(a+b)$ will be :

Three rods of the same mass are placed as shown in the figure. If sum of the coordinates of the centre of mass of the system is $\frac{ka}{3}$ then $k$ is :

A man has constructed a toy as shown in figure. If density of the material of the sphere is $12$ times of the cone, if position of the centre of mass is at $\alpha R$ from $O$, then $\alpha =$.......
[Centre of mass of a cone of height $h$ is at a height of $\frac{h}{4}$ from its base]

If the centre of gravity of a thin homogeneous plate having the form of a rectangle with sides $r$ and $2r$ from which a semicircle with a radius $r$is cut out of figure is $\frac{2r}{\alpha (4-\pi )}$ from $O$, then $\alpha =$ ?

A circular plate of uniform thickness has a diameter of $56 \mathrm{cm}$. A circular portion of diameter $42 \mathrm{cm}$ is removed from one edge of the plate as shown in figure. Find the position of the centre of mass of the remaining portion from centre of bigger circle.

A thin sheet of metal of uniform thickness is cut into the shape bounded by the line $x=a$ and $y=\pm k{x}^2$, as shown. If sum of the coordinates of the centre of mass is $\frac{\alpha a}{4}$ then $\alpha$ will be?

A ball of mass $100 \mathrm{g}$ is projected vertically upwards from the ground with a velocity of $49 \mathrm{m} {\mathrm{s}}^{-1}$. At the same time another identical ball is dropped from a height of $98 \mathrm{m}$ to fall freely along the same path as that followed by the first ball. After some time the two balls collide and stick together and finally fall to the ground. The time of flight of the masses is $\frac{5+\sqrt{x}}{2} \mathrm{s}$. Write the value of $x$.

(Take $g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$)

A simple pendulum is suspended from a peg on a vertical wall. The pendulum is pulled away from the wall to, a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{\sqrt{5}}$. What is the minimum number of collisions after which the amplitude of oscillations becomes less than $60$ degrees?