Question

For a system of n particles the contribution of internal force in the motion of the center of mass is equal to-
1. infinite
2. the net external force applied
3. zero
4. none of the above

Answer 1

Correct Answer - Option 3 : zero

CONCEPT:

• Center of mass: If we take a system of n particles distributed in space. such that  \(\overrightarrow{r_{i}} = x\widehat{i}+y\widehat{j}+z\widehat{k}\) and m_i are the mass and position of i^th particle respectively then the position of the centre of mass is given by:

\(\Rightarrow \overrightarrow R = \frac{\sum m_{i} \overrightarrow r_{i}}{M}\)          ----(i)

where M is the total mass of the system

• The motion of the centre of mass: The center of mass of the system of particles moves as if all mass of the system was concentrated at the centre of mass and all the external forces were applied at that point.
• Differentiating equation (i) with respect to time we obtain:

\(\Rightarrow M\overrightarrow V = \sum m_{i} \overrightarrow v_{i}= m_{1} \overrightarrow v_{1}+ m_{2} \overrightarrow v_{2}+ m_{3} \overrightarrow v_{3}..... m_{n} \overrightarrow v_{n} \)          ----(ii)

where \(\overrightarrow V \) is the velocity of the centre of mass, \(\overrightarrow v_i\) is the velocity of the i^th particle.

• Differentiating equation (ii) with respect to time we obtain:

\(\Rightarrow M\overrightarrow A = \sum m_{i} \overrightarrow a_{i}= m_{1} \overrightarrow a_{1}+ m_{2} \overrightarrow a_{2}+ m_{3} \overrightarrow a_{3}..... m_{n} \overrightarrow a_{n} \)          ----(iii)

where \(\overrightarrow A\) is the acceleration of the centre of mass, \(\overrightarrow a_i\) is the acceleration of the ith particle.

• As Force F = mass × acceleration, from (iii) total external force on the centre of mass is given by:

\(\Rightarrow \overrightarrow F_{ext} = M \overrightarrow A= \sum \overrightarrow F_{i}= \overrightarrow F_{1}+ \overrightarrow F_{2}+ \overrightarrow F_{3}..... \overrightarrow F_{n} \)

Internal force: In a system of n particles internal force is exerted by particles on one another.

EXPLANATION:

• In a system of n particles, the internal force is exerted by particles on one another.
• According to Newton's third law these internal forces occur in equal and opposite pairs such that the total contribution of internal forces is zero.
• So the contribution of internal force in the motion of the center of mass is equal to zero.
• Hence option 2 is correct.