Centre of Mass

An object of mass $4\text{kg}$ and another object of mass $1\text{kg}$ are moving such that they have equal kinetic energy. The ratio of the magnitudes of their linear momentum is

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

Two particles of masses $\text{'}m\text{'}$ and$\text{'}2m\text{'}$ are connected by massless string of length 3 meter and whole system is rotating about their centre of mass with angular speed$\text{'}\omega \text{'}$. The kinetic energy of system 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

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 

Two masses $2 \mathrm{kg}$ and $5 \mathrm{kg}$ are placed at points 'A' and 'B' as shown in the figure.

$$\begin{gathered} \overrightarrow{OA}= 2\hat{i}+3\hat{j} \\ \overrightarrow{AB}= 5\hat{i}+8\hat{j} \end{gathered}$$

Find the coordinates of the centre of mass.

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

$\mathrm{P}$ is the centre of mass of four point masses $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$, which are coplanar but not collinear.

Both are same material Find COM from C.

A $\left(2\times 2\right)$ square sheet is cut out from a $\left(4\times 4\right)$ square sheet as shown in the figure. Shift in centre of mass will be (take length and breadth in metre)

Two bodies of $2 \mathrm{kg}$ and $4 \mathrm{kg}$ are moving with velocities $20 \mathrm{m}/\mathrm{s}$ and $10 \mathrm{m}/\mathrm{s}$ respectively towards each other under mutual gravitational attraction. Find the velocity of their centre of mass in $\mathrm{m}/\mathrm{s}$.

In free space, a shell moving with velocity $60 \mathrm{m} {\mathrm{s}}^{-1}$ along the positive $x$-axis. When it passes through origin, it explodes into two peices of mass ratio $1:2$. After the explosion, the velocity of the centre of mass is :-

A $10 \mathrm{g}$ bullet is fired from a rifle horizontally into a $5\mathrm{kg}$ block of wood suspended by a string and the bullet gets embedded in the block. The impact causes the block to swing to a height of $5\mathrm{cm}$ above its initial level. Calculate the velocity of the bullet.

A uniform circular disc has a radius of $20\mathrm{cm}$. A circular disc of radius $10\mathrm{cm}$ is cut concentrically from the original disc. The shift in centre of mass is

A bob of mass $200 \mathrm{gram}$ is released from rest when string makes an angle ${37}^{°}$ with vertical. Sleeve has mass $50 \mathrm{gram}$ and there is no friction anywhere. Displacement of sleeve when bob reaches to the other extreme position will be:

Two bodies of mass $1 \mathrm{kg}$ and $2 \mathrm{kg}$ are located at $\left(1,2\right)\text{ and }\left(-1,3\right)$ respectively. Calculate the coordinates of centre of mass.