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Question

The particles A and B of mass m each are separated by a distance r. Another particle C of mass M is placed at the midpoint of A and B. Find the work done in taking C to a point equidistant r from A and B without acceleration (G= Gravitational constant and only gravitational interaction between A, B and C is considered)

A
2GmM×(2R1r)
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B
2GmM×(2R+1r)
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C
3GMmr
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D
4GMmr
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Solution

The correct option is A 2GmM×(2R1r)
Work done by external agent (without changing kinetic energy of the system of particles)
=Final potential energy of the systemInitial petential energy of the system
=UfUi
Ui=GmMR2+GmMR2+GmmR
Uf=GmMr+GmMr+Gmmr
Wexternalagent=(dfracGmMr+GmMr+Gmmr)⎜ ⎜ ⎜GmMR2+GmMR2+GmmR⎟ ⎟ ⎟
=(dfrac2GmMr+Gmmr)⎜ ⎜ ⎜2GmMR2+GmmR⎟ ⎟ ⎟
=2GmMr+4GmMR
=2GmM×(2R1r)

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