$A$ uniform square plate is shown in the figure. Four identical squares are removed from its corners. If squares $1$ and $3$ are removed,where will the center of mass $(C.M.)$ be located?

$A$ 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

The masses and positions (in rectangular coordinates) of four particles are as follows: $1 \ kg$ at $(a, a)$,$2 \ kg$ at $(-a, a)$,$3 \ kg$ at $(-a, -a)$,and $4 \ kg$ at $(a, -a)$. The position vector of the centre of mass of the system of four particles is:

Two particles of masses $m_1$ and $m_2$ are separated by a distance $d$. The distance of the centre of mass of the system from the particle of mass $m_1$ is:

The figure below shows a shampoo bottle in a perfect cylindrical shape. In a simple experiment,the stability of the bottle filled with different amounts of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle $\theta$ depict the critical angular displacement resulting in the bottle losing its stability and tipping over. Choose the graph that correctly depicts the fraction $f$ of shampoo filled ($f=1$ corresponds to completely filled) versus the tipping angle $\theta$.

Four particles of masses $m, 2m, 3m, 4m$ are placed at the corners of a square of side $a$ as shown in the figure. Find the coordinates of the center of mass.