Mention the position of the centre of mass of particles of equal mass.

Three particles of masses $50\, g$,$100\, g$,and $150\, g$ are placed at the vertices of an equilateral triangle of side $1\, m$ (as shown in the figure). The $(x, y)$ coordinates of the centre of mass will be

$A$ uniform thin rod $AB$ of length $L$ has linear mass density $\mu(x) = a + \frac{bx}{L}$,where $x$ is measured from $A$. If the $CM$ of the rod lies at a distance of $\frac{7}{12}L$ from $A$,then $a$ and $b$ are related as

The center of mass of a system of particles with masses $1 \, g, 2 \, g$,and $3 \, g$ is at the origin. When a particle of mass $4 \, g$ with position vector $\alpha(\hat{i} + 2\hat{j} + 3\hat{k})$ is added,the center of mass of the system becomes $(1, 2, 3)$. If $\alpha$ is a constant,its value must be:

$A$ $T$-shaped object 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 translational motion without rotation. Find the location of $P$ with respect to $C$.

$A$ uniform square plate $abcd$ has a mass of $1 \, kg$. If two point masses,each of $20 \, g$,are placed at the corners $b$ and $c$ as shown,then the centre of mass shifts on the line