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

A implies B and B implies A

Answer 2

(A) Linear momentum of the system remains constant.

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

On considering the above statement,

The correct one is Option(d) B implies A but A does not imply B.

Explanation:

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

The center of mass position is given by $R \equiv(1 / M) \Sigma i m i r i$

When we distinguish between the two sides as regards time t,

we get,  

$\frac{d R}{d t}=\frac{1}{M} \Sigma m \text { i. } \frac{d r}{d t} v \mathrm{cm}=\frac{1}{M} \Sigma_{i} \mathrm{m} i \mathrm{v}_{i}$

Now if  (B) is true, it means vcm = 0,  → Σimivi = 0 = Constant.  

So (B) implies (A).  Now let us check if (A) is true but not equal to zero $\rightarrow \Sigma m v_{i}=k \text { Then } \mathrm{vcm}=\left(\frac{1}{M}\right) \mathrm{k}=\frac{K}{M}=$ not equal to zero. i.e The mass core of the body is not in repose. And (A) doesn't mean (B).                        Therefore the correct answer is Option (d) B implies A but A does not imply B.