Gravity Train 2

Classical Mechanics Level 3

Two cities are situated at an angular distance of $\theta$ with respect to Earth's center.

There is a straight tunnel through Earth’s crust connecting the two cities.

What is the time $T$ to commute between the cities when the train moves only under the influence of gravity?

Details and Assumptions:

Assume Earth to be a sphere of uniform mass density with radius $R$ .
Ignore friction.

2\pi\sqrt{\frac Rg}

\frac \pi2\sqrt{\frac Rg}

\pi\sqrt{\frac Rg}

It depends on

\theta

1 solution

Parth Sankhe
Oct 12, 2018

The answer does not depend on $\theta$ , the time to cover that distance will be the same for every angle. Therefore, it will take the same amount of time as it takes to fall to the centre of the earth and back up to the other end.( $\theta=π$ )

Time taken to fall to the centre of the earth = $\frac {π}{2}√\frac {R}{g}$

Hence the total time would be double of that.

How do you know that the answer does not depend on theta????

Aaghaz Mahajan - 2 years, 7 months ago

You can calculate it at any $\theta$ , and all the $\theta$ terms will get cancelled. This is because as you keep on decreasing the angle, the distance to cover decreases, but so does the magnitude of gravitational acceleration along that line, so they perfectly cancel each other out.

Parth Sankhe - 2 years, 7 months ago

How do you know that t=pi/2*sqrt(R/g)? If that's given and you only have to double it it's easy but where does it come from?

Sebastian Pochert - 2 years, 7 months ago

You'll have to use integration for that, take the particle to be at distance x from the centre, and calculate its acceleration at that point, integrate that to give velocity at that point, and write velocity as dx/dt and integrate it once again, and put limits on x from 0 to R to determine the time. The constants of integration in all these steps can be found by putting x=R or 0. ( g at a distance x from the centre of a solid sphere = $\frac {GMx}{R^3}$ )

Parth Sankhe - 2 years, 7 months ago

Gravity Train 2

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