If all the particle of a system lie in X-Y plane is it necessary that the centre of mass be in X-Y plane?

If all the particle of a system lie n a cube is it necessary that the cente of mas be in the cube?

The centre of mass is defined as vecR=1/Msum_im_ivecr_i . Suppose we define centre of charge as vecR_c=1/Qsum_iq_Ivecr_i where q_i represents the ith charge placed at vecr_i and Q is the total charge of the sytem a. Can a centre of charge of a two charge system be outside the line segment joining the charges? b. If all the charges of a system are in X-Y plane? c. ILf all the charges f a system lie in a cube, is it necessary that the centre of charge be in the cube?

If all the particles of a system lie in y-z plane, the x-coordinate of the centre of mass will be zero. Is this statement true or false?

If large number of particles are distributed on XY plane and their centre of mass is at origin of coordinates, then

If a large number of particles are distributed on YZ plane and their centre of mass is at origin of coordinates, then

The sum of moments of masses of all the particle in a system about the centre of mass is always

The potential energy of a particle of mass 1 kg moving in X-Y plane is given by U=(12x+5y) joules, where x an y are in meters. If the particle is initially at rest at origin, then select incorrect alternative :-

A particle of mass m is placed at a distance x from the centre of ring along the line through the centre of the ring and perpendicular to its plane. Gravitational potential energy of this system.

A particle of mass m is placed at a distance x from the centre of ring along the line through the centre of the ring and perpendicular to its plane. if xlt lta , the particle of mass m will:

On which line does the centre of the circle lie? x+y=0