Imagine that you drill a hole through the earth along a diameter and drop a bund
ID: 1470005 • Letter: I
Question
Imagine that you drill a hole through the earth along a diameter and drop a bundle of physics solutions down the hole to your friend waiting on the other side of the planet, whom will be taking a similar exam day later than yours. Assume the earth to be a solid sphere with a uniform density. A.) Derive an expression for the gravitational force (Fg) on the bundle as a function of its distance from the earth's center. B.) If your friend is sending you cash in exchange, then how much time will it take for the bundle to complete the roundtrip?
Explanation / Answer
Let m is mass of bundle
Me is the mass of earth.
Re is the radius of earth
a) at distance r from the center of the earth.
F = G*Me*m*r/Re^3
b)
Force at any point r from the center,
F = G*M*m*r/R^3
F = k*r
==> k = G*M*m/R^3
= 6.67*10^-11*5.98*10^24*m/(6.37*10^6)^3
= m*1.543*10^-6 rad/m
angular frequency, w = sqrt(k/m)
Time periode, T = 2*pi/w
= 2*pi*sqrt(m/k)
= 2*pi*sqrt(1/(1.543*10^-6))
= 5058 s
= 5058/60
= 84.3 minutes
so, the time taken for the round trip, T = 84.3 minutes.