Two semicircular rings of linear mass densities $\lambda$ and $3\lambda$ and of radius $R$ each are joined to form a complete ring. The distance of the center of mass of the complete ring from its geometrical center is

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:

Three identical spheres of radius $r$ are placed on a horizontal surface such that they touch each other. These spheres are arranged such that their centers form an equilateral triangle. The center of mass of this system will be at ........

Obtain the general expression for the centre of mass of a system of $n$ distributed particles in three dimensions.

Masses $m, (1/2)(m/2), (1/2)^2(m/3), \dots, (1/2)^{N-1}(m/N), \dots \infty$ are placed at $x = 1, 2, 3, \dots, N, \dots \infty$ respectively. If the total mass is $M$,then the centre of mass of the system is:

Four particles of masses $m_1 = 2m$,$m_2 = 4m$,$m_3 = m$,and $m_4$ are placed at the four corners of a square. What should be the value of $m_4$ so that the centre of mass of all the four particles is exactly at the centre of the square?