If the linear mass density of a rod of length $L$ varies as $\lambda = kx^3$,determine the position of its centre of mass (where $x$ is the distance from one of its ends and $k$ is a constant):

Two smooth and similar right-angled prisms are arranged on a smooth horizontal plane as shown in the figure. The lower prism has a mass $3$ times the upper prism. The prisms are held in an initial position as shown and are then released. As the upper prism touches the horizontal plane,the distance moved by the lower prism is

The centre of mass of a system of particles does not depend on

Two objects of masses $200 \text{ g}$ and $500 \text{ g}$ possess velocities $10 \hat{i} \text{ m/s}$ and $(3 \hat{i} + 5 \hat{j}) \text{ m/s}$ respectively. The velocity of their centre of mass in $\text{m/s}$ is:

In the figure shown,$ABC$ is a uniform wire. If the center of mass of the wire lies vertically below point $A$,then $\frac{BC}{AB}$ is close to:

$A$ square plate and a circular disc of uniform density are arranged as shown in the figure. The center of mass of this combined system will be .........