A uniform rod of length $l$ is placed symmetrically on two walls as shown in figure. The rod is in equilibrium. If $N_1 \text{and} N_2$are the normal forces exerted by the walls on the rod then 

A uniform wire of length $l$ is bent into the V shape as shown in the figure below. The distance of its centre of mass from the vertex $A$ is

The coordinates of the centre of mass of the three particles of masses $m_1=m_2=1 \mathrm{kg}$ and $m_3=2 \mathrm{kg}$placed at the corners of an equilateral triangle of side $1 \mathrm{m}$ will be

A uniform square wooden sheet of side a has its centre of mass located at point $O$ as shown in the figure on the left. A square portion of side $b$ of this sheet is cut out to produce and $\mathrm{L}$-shaped sheet as shown in the figure on the right.

The centre of mass of the L-shaped sheet lies at the point $P$ (in the diagram) when,

Two uniform plates of the same thickness and area but of different materials, one shaped like an isosceles triangle and the other shaped like a rectangle are joined together to form a composite body as shown in the figure. If the centre of mass of the composite body is located at the midpoint of their common side, the ratio between masses of the triangle to that of the rectangle is,

A small sphere of radius $R$ is held against the inner surface of a smooth spherical shell of radius $6R$ as shown in the figure. The masses of the shell and the small sphere are $4M$ and $M$, respectively.

This arrangement is placed on a smooth horizontal table. The small sphere is now released. The $x$-coordinate of the center of the shell when the smaller sphere reaches the other extreme position is,