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Question: If g is acceleration due to gravity on the earth's surface the gain in the potential energy of an ob...

If g is acceleration due to gravity on the earth's surface the gain in the potential energy of an object of mass m raised from the surface of the earth to a height equal to the radius R of the earth is -

A

12\frac { 1 } { 2 } mgR

B

2 mgR

C

mgR

D

14\frac { 1 } { 4 } mgR

Answer

2 mgR

Explanation

Solution

At a distance x, from the centre of the earth, gravitational force :

F = GMmx2\frac { \mathrm { GMm } } { \mathrm { x } ^ { 2 } }

dw = fdx = dx

w = dx = GMm

= gR1m [1R1R+h]\left[ \frac { 1 } { \mathrm { R } } - \frac { 1 } { \mathrm { R } + \mathrm { h } } \right]

\ Gain in potential energy

= gR2m mgR2\frac { \mathrm { mgR } } { 2 }