Correct Answer - Option 2 : Acceleration of the centre of mass must be zero
CONCEPT:
Centre of mass:
- The centre of mass of a body or system of a particle is defined as, a point at which the whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.
The motion of the centre of mass:
- Let there are n particles of masses m1, m2,..., mn.
- If all the masses are moving then,
⇒ Mv = m1v1 + m2v2 + ... + mnvn
⇒ Ma = m1a1 + m2a2 + ... + mnan
⇒ \(M\vec{a}=\vec{F_1}+\vec{F_2}+...+\vec{F_n}\)
⇒ M = m1 + m2 + ... + mn
- Thus, the total mass of a system of particles times the acceleration of its centre of mass is the vector sum of all the forces acting on the system of particles.
- The internal forces contribute nothing to the motion of the centre of mass.
EXPLANATION:
Given \(\vec{F_1}+\vec{F_2}+...+\vec{F_n}=0\)
- We know that the total mass of a system of particles times the acceleration of its centre of mass is equal to the vector sum of all the forces acting on the system of particles.
⇒ \(M\vec{a}=\vec{F_1}+\vec{F_2}+...+\vec{F_n}\) -----(1)
∴ \(M\vec{a}=0\) -----(2)
- We know that the total mass of the system of particles can't be zero.
- Therefore in this case acceleration of the centre of mass must be zero. The position of the centre of mass may change. Hence, option 2 is correct.
