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Two objects of masses m and 4m are at re...

Two objects of masses `m` and `4m` are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If `G` is the universal gravitaitonal constant, then at separation `r`

A

the total energy of the two objects is zero

B

net angular momentum of both the objects is zero about any point

C

the total `K.E.` of the objects is `4Gm^(2)//r`

D

their relative velocity of approach is
`((8Gm)/(r ))^(1//2)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
Initially both the objects are at rest at an infinite separation, hence their initial `PE` and `KE` are zero. Therefore, following law of conservation of machanical energy, total energy of the two objects at separtion `r` will be zero.
There will be two equal and opposite forces acting on two objects. The net torque on two onjects is zero. So, the angular momentum will remain conserved. Initially, both the objects were at rest, therefore, the angular momentum about any point is zero.
The reduced mass of the two objects,
`mu = (m xx 4m)/(m + 4m) = (4m)/(5)`
If `upsilon_(r )` is the relative velocity of approach and `r` is the distance between the two objects, then total `KE` of objects = loss in `PE` of objects
`= (Gm xx 4m)/(r ) = (4Gm^(2))/(r )`
As total `KE` of objects = loss in `PE` of objects, so
`(1)/(2) mu upsilon_(r )^(2) = (4Gm^(2))/(r )`
or `(1)/(2) xx (4m)/(5) upsilon_(r )^(2) = (4Gm^(2))/(r )` or `upsilon_(r ) = sqrt((10Gm)/(r ))`
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