Centre of Mass of Isolated and Continuous Body

Mass is distributed uniformly over a thin triangular plate and positions of two vertices are given by $(1, 3)$and $(2, -4)$. What is the position of $3\mathrm{rd}$ vertex if the centre of mass of the plate lies at the centre of origin?

The ball is released from rest. Find centre of mass of large sphere when small sphere reaches the extreme position. 

The density of a linear rod of length $L$ varies as $\lambda =\frac{{\lambda }_{0}x}{L}$, $x$ is the distance from the left end of rod. Location of centre of mass of the rod, is

Locate centre of mass of a system of three particles, as shown in the figure.

What is centre of gravity of semi-circular disc of radius $\left(R\right)$?

What is the distance of centre of mass from$4 \mathrm{kg}$ mass in the following arrangement:

The particles of mass $1 \mathrm{kg}$, $2 \mathrm{kg}$ and $3 \mathrm{kg}$ are placed at the vertices of an equilateral triangle of side $1 \mathrm{m}$ as shown in the figure. The co-ordinates of centre of mass will be

Four particles having masses $2 \mathrm{kg}$, $3 \mathrm{kg}$, $4 \mathrm{kg}$ and$6 \mathrm{kg}$ are placed at four vertices $A$, $B$, $C$ and $D$ respectively of square of side $1 \mathrm{m}$ as shown in the figure. The position of centre of mass of the system of particles is

A disc of radius$r$is cut from a larger disc of radius$4r$ in such a way that the edge of the hole touches the edge of the disc. Locate the centre of mass of the residual disc.

In the figure shown a small hole of radius $3 \mathrm{cm}$ is made in a semicircular disc of radius $9 \mathrm{cm}$ а distance $12 \mathrm{cm}$ from the centre C of the disc. The distance (in $\mathrm{cm}$) of centre of mass of this system from point C is :-

 Two rods each of mass $m$ and length$L$ are placed along $X$and $Y$ axis as shown in figure. Coordinates of the centre of mass of the combination is

Linear mass density of rod of length $l$ is directly proportional to $x^3$, where $x$ is distance from one end of rod. Centre of mass of rod lies at a distance

In the figure shown a small hole of radius $3 \mathrm{cm}$ is made in a semicircular disc of radius $9\pi \mathrm{cm}$ а distance $12 \mathrm{cm}$ from the centre C of the disc. The distance (in cm) of centre of mass of this system from point C is :-

A uniform chain of length $L$ is placed on a smooth table of height $h \left(h>L\right)$ with a length $l$ hanging from the edge of the table. The chain begins to slide down the table. When the end of the chain is about to leave the edge of the table its velocity is

Find the centre of mass from the origin, if the disc and the square plate have same surface density.

Figure shows a composite object made of same material shaped as a square plate joined with a semicircular disc. Side length of the square is same as the diameter of the semicircle and the dashed line is the interface between the square and semicircle. The centre of mass for the system:

From a sphere of radius $1 \mathrm{m}$, a sphere of radius $0.5 \mathrm{m}$ is removed from the edge. The position of centre of mass of the remaining portion from centre of the sphere is

The mass per unit length of a non-uniform rod of length $l$ is given by $\lambda =A\cos \left(\frac{\pi x}{2l}\right)$, where $x$ is position along the rod, $0\le x\le l$.

(a) What is the mass $M$ of the rod?

(b) What is the coordinate $X$ of the center of mass?

A smooth chain $AB$ of mass $m$ rests against a surface in the form of a quarter of a circle of radius $R$. If it is released from rest, the velocity of the chain after it comes over the horizontal part of the surface is