Centre of Mass

Distance of centre of mass of a thin uniform semi-circular disc of radius $R$ from its centre is

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. 

 List the conditions to increase the stability of a body.

Why is it important to find the centre of mass of a body?

Draw a diagram and explain how to find centre of mass of an irregularly shaped lamina.

A small sphere of radius slides on a smooth circular part of radius $R$on a wedge of mass $M$. The Displacement of sphere when it slides to bottom most 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.

The velocity of the centre of mass of a solid sphere of radius$R$ rotating with angular velocity about an axis passing through its centre of mass is

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

Two particles of masses, $2 \mathrm{kg}$ and $4 \mathrm{kg}$, are projected from the top of a tower simultaneously, such that $2 \mathrm{kg}$ of mass is projected with a speed $20 \mathrm{m} {\mathrm{s}}^{-1}$ at an angle ${30}^{\circ }$ above horizontal and$4 \mathrm{kg}$ is projected at $40 \mathrm{m} {\mathrm{s}}^{-1}$ horizontally. The acceleration of the centre of mass of the system of two particles will be:
 

Two balls with masses$m_1=3 kg$ and $m_2=5 kg$ have initial velocities $v_1=v_2 = 5 \mathrm{m} {\mathrm{s}}^{-1}$ in the directions as shown in the figure. They collide at the origin.
(a) Find the velocity of the centre of mass $3 \mathrm{s}$ before the collision.

(b) Find the position of the centre of mass $2 \mathrm{s}$ after the collision.

Find the velocity of centre of mass if the triangle formed is equilateral.

The centre of a disc rolling without slipping on a plane surface moves with speed $u$. A particle, on the lower half of the rim making an angle $60°$ with vertical, will be moving at speed 

In case of pure rolling, what will be the velocity of point $A$of the ring of radius $R$ ?

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