Three identical spheres of mass $1 \ kg$ each are arranged as shown in the figure. The centers of the spheres,which are touching each other,lie on a straight line. If their centers are denoted by $P, Q,$ and $R$,what is the distance of the center of mass of the system from $P$?

What is the torque acting on the center of mass of an object due to the gravitational force?

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 mid-point of their common side,then the ratio between the masses of the triangle to that of the rectangle is:

Match Column-$I$ with Column-$II$.

Column-$I$	Column-$II$
$(1)$ $\frac{{{m_1}{m_2}}}{{{m_1} + {m_2}}}$	$(a)$ Reduced mass of a two-particle system
$(2)$ $\frac{{{r_1} + {r_2}}}{2}$	$(b)$ Position vector of the center of mass for a system of two equal masses

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of the position vector of the centre of mass of this system will be equal to the magnitude of which of the following vectors?

The linear mass density of a rod of length $L$ varies as $\lambda = A + Bx$. Find the center of mass.