Question

Consider the following two statements:
(A) Linear momentum of the system remains constant.
(B) Centre of mass of the system remains at rest.
(a) A implies B and B implies A.
(b) A does not imply B and B does not imply A.
(c) A implies B but B does not imply A.
(d) B implies A but A does not imply B.

Answer 1

(d) B implies A but A does not imply B.

The centre of mass of a system is given by,
$\overrightarrow{R}=\frac{1}{M}\sum _{}{m}_i{\overrightarrow{r}}_i$

On differentiating the above equation with respect to time, we get:
$\frac{\mathrm{d}\overrightarrow{R}}{\mathrm{d}t}=\frac{1}{M}\sum _{}{m}_i\frac{d{\overrightarrow{r}}_i}{dt}$

As the centre of mass of the system remains at rest, we have:
$$\begin{gathered} \frac{1}{M}\sum _{}{m}_i\frac{d{\overrightarrow{r}}_i}{dt}=0 \\ \sum _{}{m}_i{\overrightarrow{v}}_i=0 \end{gathered}$$

This implies that the linear momentum of the system remains constant.