A uniform rod of mass $M$ and length $L$ is bent into a semicircular ring. The coordinates of the center of mass of the semicircular ring are:

Two particles of masses 2 kg and 4 kg are moving with velocities 4 m/s and 2 m/s respectively along the same direction. What is the velocity of the centre of mass?

If linear density of a rod of length 3 m varies as $\lambda = 2 + x$, then the position of the centre of gravity of the rod is

Two particles of masses $m_1$ and $m_2$ are connected by a light inextensible string passing over a smooth pulley. If the acceleration of the center of mass of the system is $g/4$, what is the ratio of the masses $m_1/m_2$?

A cannon shell lands 2 km away from the cannon. A second shell, fired identically, breaks into two equal parts at the highest point. One part falls vertically. How far from the cannon will the other land ?

An isolated particle of mass m is moving in a horizontal plane (x-y), along the x-axis, at a certain height above the ground. It suddenly explodes into two fragments of masses m/4 and 3m/4. An instant later, the smaller fragment is at $y = + 15$ cm. The larger fragment at the instant is at

A system consists of three particles, each of mass m and located at (1, 1) (2, 2) and (3, 3). The coordinates of the centre of mass are

A uniform rod of mass M and length L is pivoted at one end and rotates with angular velocity ω. The linear velocity of its center of mass is:

Two particles of masses 1 kg and 2 kg are located at ${x_1} = 0,\;{y_1} = 0\,and\,{x_2} = 1,\;{y_2} = 0$ respectively. The centre of mass of the system is at

A circular disc of radius R is removed from a bigger circular disc of radius 2R, such that the circumference of the discs coincide. The center of mass of the new disc is $\frac{\alpha }{R}\;$ from the center of the bigger disc the value of $\alpha$ is

Circular hole of radius 1 cm is cut off from a disc of radius 6 cm. The centre of hole is 3 m from the centre of the disc. The position of centre of mass of the remaining disc from the centre of disc is