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Question

If g is the acceleration due to gravity on the Earth' surface, find the gain in potential energy of a body of mass m when taken from the surface of Earth at a height equal to the radius R of the Earth.

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Solution

Let

R Radius of the earth

M Mass of the earth

m Mass of the body

g Acceleration due to gravity on earth
when the body is on earth surface, considering the force of attraction of earth on it we can write

mg=GMmR2 Where, G= is gravitational constant.

Es=GmMR(2)

Again the PE of the system when the body is at height h from the the earth surface is given by

Eh=GmMR+h

If h=R then

ER=GmMR+R=GmM2R(3)

So increase in PE due to shift of the body of mass m from surface to the height equal to the radius (R) of the earth is given by-

ΔEp=EREs=GmM2R+GmMR=GmM2R=mgR22R=12mgR

Hence, increase in potential energy is 12mgR


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