Through the Fancy Tunnel

Classical Mechanics Level 2

Imagine you are fetching an object from infinity towards the surface of the earth. If you have to do work that amounts to 50J while making it appear on surface in constant velocity, how much work will you have to do if the same object has to be taken to the center of the earth from the surface (through a fancy tunnel if you could dig it) ?

100J 25J 40J 200J

Anik Saha by Anik Saha , 19, Bangladesh

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1 solution

Hosam Hajjir
Dec 6, 2020

The force that has to be applied from infinity to the surface of Earth is

F = G M m r 2 F = \dfrac{G M m}{r^2}

So that the work is

∫ R ∞ G M m r 2 d r = G M m R = 50 J \displaystyle \int_{R}^{\infty} \dfrac{G M m }{ r^2} dr = \dfrac{G M m}{R} = 50 J

Inside Earth, the force acting on the object is

F = G M m r 2 ⋅ ( r R ) 3 = G M m R 3 r F = \dfrac{G M m}{r^2} \cdot \left( \dfrac{r}{R} \right)^3 = \dfrac{G M m }{R^3} r

So that the second work from the surface of Earth to the center is

∫ 0 R G M m R 3 r d r = 1 2 G M m R = 1 2 ( 50 ) = 25 J \displaystyle \int_{0}^{R} \dfrac{G M m }{R^3} r dr = \dfrac{1}{2} \dfrac{G M m }{R} = \dfrac{1}{2} (50) = 25 J

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