A dog of mass $m$ stands at one end of a flat boat of mass $M$ and length $L$ at rest on a frictionless surface of a lake. The dog walks to the other end of the boat and stops. The displacement of the center of mass of the boat-dog system is:

A non-uniform thin rod of length L is placed along X-axis as such its one of ends is at the origin. The linear mass density of rod is $l = {l_0}x$. The distance of centre of mass of rod from the origin is

Two masses ${m_1}\,and\,{m_2}$ \left( {{m_1} > {m_2}} \right) are connected by massless flexible and inextensible string passed over massless and frictionless pulley. The acceleration of centre of mass is

A 2 kg body and a 3 kg body are moving along the x-axis. At a particular instant the 2 kg body has a velocity of $3\;m{s^{ - 1}}$ and the 3 kg body has the velocity of $2m{s^{ - 1}}$. The velocity of the centre of mass at that instant is

Two objects of equal mass are moving towards each other with equal speeds. The velocity of their centre of mass is:

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

A man of mass M stands at one end of a plank of mass m and length L, lying on a smooth horizontal surface. The man walks to the other end of the plank. The distance moved by the plank with respect to the ground is:

For an isolated system, the velocity of the centre of mass:

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?

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

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