List of top Physics Questions on Center of Mass asked in JEE Main

Given below are two statements:
Statement I: For a mechanical system of many particles, total kinetic energy is the sum of kinetic energies of all the particles.
Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass with respect to the origin and the kinetic energy of all the particles with respect to the center of mass as reference.
In the light of the above statements, choose the correct answer from the options given below:
Consider a circular disc of radius 20 cm with center located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of the center of mass of the residual or remaining disc from the origin will be:
A rod of length $ 5L $ is bent at a right angle, keeping one side length as $ 2L $. The position of the centre of mass of the system (Consider $ L = 10 $ cm):
The amount of work done to break a big water drop of radius \( R \) into 27 small drops of equal radius is 10 J. The work done required to break the same big drop into 64 small drops of equal radius will be:
In a system, two particles of masses \( m_1 = 3 \, \text{kg} \) and \( m_2 = 2 \, \text{kg} \) are placed at a certain distance from each other. The particle of mass \( m_1 \) is moved towards the center of mass of the system through a distance \( 2 \, \text{cm} \). In order to keep the center of mass of the system at the original position, the particle of mass \( m_2 \) should move towards the center of mass by the distance ______ \( \, \text{cm} \).
A uniform thin metal plate of mass \(10 \, \text{kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of the center of mass of the plate is \(\frac{n}{9}\). The value of \(n\) is ______.
metal plate
The identical spheres each of mass 2M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is \(\frac{4 \sqrt2} x\) , where the value of x is ___________