Centre Of Mass

The ABC is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. I , I and I are the moments of inertia of the plate about AB, BC and CA respectively. Which one of the following relations one is correct ?

Two persons of masses and respectively, are at the opposite ends of a boat. The length of the boat is and weighs . The man walks up to the man and sits with him. If the boat is in still water the center of mass of the system shift by:

Which of the following statements are correct?

1. Centre of mass of a body always coincides with the centre of gravity of the body.
2. Centre of mass of a body is the point at which the total gravitational torque on the body is zero.
3. A couple on a body produce both translational and rotational motion in a body.
4. Mechanical advantage greater than one means that small effort can be used to lift a large load.

Two particles of mass and respectively are attached to the two ends of a rigid rod of length 1m with negligible mass. The centre of mass of the system from the particle is nearly at a distance of :

Three identical spheres, each of mass , are placed at the corners of a right angle triangle with mutually perpendicular sides equal to (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of centre of mass