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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.
Solution
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.
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