Find the time period of motion if a mass m is dropped in a tunnel along the diameter of earth from height h above the surface of earth. Is the motion simple harmonic?

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Solution

$T h e f o r c e d u e t o g r a v i t y F = - \frac{G M m r}{R^{3}} g r a v i t y o f t h e e a r t h s u r f a c e g = \frac{G M}{R^{2}} s o F = - \frac{m g r}{R} t h i s i s a n o r m a l r e p r e s e n t a t i o n o f S H M w h e r e f o r c e i s p r o p o r t i o n a l t o d i s p l a c e m e n t . a n d w h o s e t i m e p e r i o d i s T = 2 π \sqrt{\frac{R}{g}}$

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Find the time period of motion if a mass m is dropped in a tunnel along the diameter of earth from height h above the surface of earth. Is the motion simple harmonic?

The force due to gravity F=-GMmrR3gravity of the earth surfaceg=GMR2so F=-mgrRthis is a normal representation of SHM where force is proportional to displacement.and whose time period is T=2πRg