Define and dereive a relation for the gravitional potential energy of an object in the gravitional field of the earth.

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Solution

Gravitational potential energy at a point in gravitational field of another body is defined as the amount of work done in bringing the given body from infinity to that point without acceleration.

$F o r c e o f a t t r a c t i o n b e t w e e n t h e e a r t h a n d t h e o b j e c t w h e n a n o b j e c t i s a t d i s \tan c e a f r o m t h e c e n t r e o f t h e e a r t h . F = \frac{G m M}{x^{2}} l e t d W i s t h e s m a l l a m o u n t o f w o r k d o n e i n b r i n g i n g t h e b o d y w i t h o u t a c c e l e r a t i o n t h r o u g h t h e v e r y s m a l l d i s \tan c e d x i s g i v e n b y : d W = F d x t o t a l w o r k d o n e t o b r i n g t h e b o d y f r o m i n f i n i t y t o p o i n t p w h i c h i s a t d i s \tan c e r f r o m t h e c e n t r e o f t h e e a r t h . W = \int_{\infty}^{r} \frac{G m M}{x^{2}} d x = - \frac{G M m}{r} s o g r a v i t a t i o n a l p o t e n t i a l e n e r g y = - \frac{G M m}{r}$

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We say that gravitational potential energy is zero when the object is on the ground(earth surface). My doubt is that can gravitational potential energy be considered zero anywhere (at any altitude) when object is kept on the surface of earth? And also there is distance between the object and the centre of the earth, how can the gravitational potential energy be considered zero?

Gravitational potential energy of an object will:

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