Question

Two particles of masses m₁ and m₂ initially at rest start moving
towards each other under their mutual force of attraction. The speed of
the centre of mass at any time t, when they are at a distance r apart, is

(a) zero (b) ( G m₁m₂ - 1) t
( r² m₁)

(c) ( G m₁m₂ - 1) t (d) ( G m₁m₂ - 1) t
( r² m₂) ( r² m₁ + m₂)

Answer 1

Answer:

Given:

2 particles of mass m1 and m2 are initially at rest .They start moving towards each other under gravitational force.

To find:

Speed of the centre of mass at any time to when the objects are r distance apart.

Concept:

Centre is mass moves only in presence of an external force. By external force , I mean a force applied from outside the system.

But Gravitational force is an internal force developed mutually in-between the 2 masses.

So it's an Internal force.

In presence of such a force , the Centre of Mass of the system of particles will not move and stay at rest.

So velocity of centre of mass is zero.

$\boxed{ \red{ \huge{ \sf{ \bold{ \underline{Option \: :a)}}}}}}$

Answer 2

SOLUTION

Option (A) is correct

EXPLANATION

Two masses m and M start moving towards eachother by the virtue of an mutual attractive interaction which is an internal force

From Newton's Second Law of Motion,

$\sf \ F_{ext} \propto \dfrac{dp}{dt}$

But there is no external force acting on the system of masses

Thus,

$\sf 0 = \dfrac{dp}{dt} \\ \\ \leadsto \sf dp = 0 \\ \\ \leadsto \sf mdv = 0 \\ \\ \large{\leadsto \boxed{\boxed{\sf V_{CM} = 0 m{s}^{-1} }}}$

Conclusion

• When there is no external applied force acting on a given system of masses,the Momentum and Velocity of the Centre of Mass is always zero

• The total system is at rest but the constituents maybe in motion due to internal interactive forces