Question

Consider the following two statements:

(A) Linear momentum of the system remains constant.

(B) Centre of mass of the system remains at rest.

Options

• A implies B and B implies A. 

• A does not imply B and B does not imply A. 

• A implies B but B does not imply A. 

• B implies A but A does not imply B. 

Answer 1

B implies A but A does not imply B. 

The centre of mass of a system is given by,
\[\vec{R} = \frac{1}{M} \sum_{} m_i \vec{r}_i\]
On differentiating the above equation with respect to time, we get:
\[\frac{d \vec{R}}{dt} = \frac{1}{M} \sum_{} m_i \frac{d \vec{r}_i}{d t}\]

As the centre of mass of the system remains at rest, we have:
\[\frac{1}{M} \sum_{} m_i \frac{d \vec{r}_i}{d t} = 0\]
\[ \sum_{} m_i \vec{v}_i = 0\]
This implies that the linear momentum of the system remains constant.