The centre of mass of a system of two particles of masses $m_1$ and $m_2$ separated by a distance $d$ is:

$A$ $T$-shaped object of uniform thickness and same material with dimensions shown in the figure,is lying on a smooth floor. $A$ force $\vec F$ is applied at the point $P$ parallel to $AB$,such that the object has only the translation motion without rotation. Find the location of $P$ with respect to $C$.

Which of the points is the likely position of the centre of mass of the system shown in the figure?

The distance of the centre of mass from end $A$ of a one-dimensional rod $(AB)$ having mass density $\rho = \rho_{0} \left(1 - \frac{x^{2}}{L^{2}}\right) \text{ kg/m}$ and length $L$ (in meters) is $\frac{3L}{\alpha} \text{ m}$. The value of $\alpha$ is $\ldots \ldots \ldots$ (where $x$ is the distance from end $A$).

The position of the centre of mass of a uniform semi-circular wire of radius $R$ placed in the $x-y$ plane with its centre at the origin and the line joining its ends as the $x$-axis is given by $\left(0, \frac{x R}{\pi}\right)$.
Then,the value of $|x|$ is ...... .

Two particles of mass $5\, kg$ and $10\, kg$ respectively are attached to the two ends of a rigid rod of length $1\, m$ with negligible mass. The centre of mass of the system from the $5\, kg$ particle is nearly at a distance of $..........\, cm$.