derive an expression for the gravitational potential energy of body mass m located at distance r from centre of the earth.

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Solution

$C o n s i d e r t w o p a r t i c l e s o f m a s s e s m_{1} a n d m_{2} a r e a t d i s \tan c e r . N o w o n e o f t h e p a r t i c l e i s t a k e n a t s o m e o t h e r p o i n t N o w t h e i r s e p a r a t i o n b e c o m e s r' F o r c e o n s e c o n d p a r t i c l e, F = \frac{G m_{1} m_{2}}{r^{2}} T h e W o r k d o n e b y t h e g r a v i t a t i o n a l f o r c e i n d i s p l a c e m e n t d W = - \frac{G m_{1} m_{2}}{r^{2}} d r I n c r e a s e i n P o t e n t i a l E n e r g y, d U = - d W = \frac{G m_{1} m_{2}}{r^{2}} I n c r e a s e i n E n e r g y, = \int d U i n m o v i n g f r o m r t o r', d U = G m_{1} m_{2} [\frac{1}{r} - \frac{1}{r'}] L e t d i s \tan c e b e t w e e n t h e m i s i n f i n i t y, u (r) = - \frac{G m_{1} m_{2}}{r}$

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