If the linear density of a rod of length $3 \ m$ varies as $\lambda = 2 + x$,then the position of the centre of mass of the rod is:

The position vectors of two particles of masses $1 \ kg$ and $3 \ kg$ are $\hat{i} + 2\hat{j} + \hat{k}$ and $-3\hat{i} - 2\hat{j} + \hat{k}$ respectively. The position vector of the center of mass of this system is:

$A$ uniform square wooden sheet of side $a$ has its centre of mass located at point $O$ as shown in the figure below on the left. $A$ square portion of side $b$ of this sheet is cut out to produce an $L$-shaped sheet as shown in the figure on the right. The centre of mass of the $L$-shaped sheet lies at the point $P$ (in the above diagram),when

Four particles of masses $m, 2m, 3m$ and $4m$ are arranged at the corners of a parallelogram with each side equal to $a$ and one of the angles between two adjacent sides is $60^o$. The parallelogram lies in the $x-y$ plane with mass $m$ at the origin and $4m$ on the $x$-axis. The centre of mass of the arrangement will be located at

$A$ $T$-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 $C$.

In the $HCl$ molecule,the separation between the nuclei of the two atoms is about $1.27 \; \mathring{A} \; (1 \; \mathring{A} = 10^{-10} \; m)$. Find the approximate location of the center of mass $(CM)$ of the molecule,given that a chlorine atom is about $35.5$ times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.