Two particles A and B initially at rest, move towards each other, under mutual force of attraction. At an instance when the speed of A is $\mathrm{v}$ and speed of B is $2\mathrm{v}$, the speed of centre of mass (CM) is

A circular disc of radius $\mathrm{R}$  is removed from a bigger circular disc of radius $2\mathrm{R}$ such that the circumferences of the discs coincide. The centre of mass of the new disc is $\mathrm{\alpha R}$  from the centre of the bigger disc. The value of α is.

Two blocks of masses $10 \mathrm{kg}, 4 \mathrm{kg}$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An Impulse gives a velocity of $14 \mathrm{m}/\mathrm{s}$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is:

From a uniform circular disc of radius R, a coaxial smaller disc of radius $\frac{\mathrm{r}}{4}$ is moved. If  $O$ is the centre of whole disc, then the centre of mass of the remaining disc from $O$ will be will be

A system of particles contains three masses of $5 \mathrm{g}, 2\mathrm{g}$ and $3\mathrm{g}$ respectively. The respective coordinates of these particles are $\left(1, 0,-1\right), \left(1, 2,1\right),and \left(1,1,-3\right).$ The centre of mass of the system is at:

The drawing in the figure is drawn with the ink of uniform thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $\mathrm{m}$. The mass of the ink used to draw the outer circle is $6 \mathrm{m}$. The coordinates of the centres of the different parts are: outer circle $(0,0)$, left circle $\left(-\mathrm{a}.\mathrm{a}\right)$, right inner circle $\left(\mathrm{a},\mathrm{a}\right)$ vertical line $\left(0.0\right)$ and horizontal line $\left(0.-a\right)$. The y-coordinate of the centre of mass of the ink in this drawing is