Tunneling

Classical Mechanics Level 2

A straight tunnel is drilled between two points (not necessarily diametrically opposite) on the Earth. An object is dropped into the tunnel. How much time (in minutes) does it take to reach the other end?

Ignore air resistance and/or friction, and assume that the density of the Earth is constant.

41 39 43 42 44 45 40

Denton Young by Denton Young , 47, USA

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2 solutions

42: just because it's the answer to life, universe and everything else(couldn't resist referring to hitchhikers guide to the galaxy).

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And the question ended up being, "How many roads must a man walk down?"

Denton Young - 4 years, 3 months ago
Denton Young
Mar 11, 2017

Let the Earth’s mass be M M , and let its radius be R R . Consider the object when it is a distance x x from the center of the tunnel, at radius r . r.

The gravitational force on the object is due to the mass of the earth that is inside the radius r r . Since mass is proportional to volume, the mass inside the radius r r is M ( ( r / R ) 3 ) M((r/R)^3)

The gravitational force exerted is therefore ( ( G M m r ) / R 3 ) ((GMmr)/R^3)

The component exerted along the tunnel is thus − ( ( G M m x ) / R 3 ) -((GMmx)/R^3)

That makes the movement a simple harmonic with frequency f = ( ( ( G M ) / R 3 ) ) 1 / 2 f = (((GM)/R^3))^{1/2}

Substituting in the values for G, M and R yields a round-trip time of about 84 minutes, or a one way time of 42 minutes. .

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