A hole is drilled all the way through the center of the earth , from one side to

Question

A hole is drilled all the way through the center of the earth , from one side to the other, and a particle of mass m is dropped in. Recall that the gravitational attraction of the earth for a particle a distance r from the center of the earth really arises entirely from that portion of the matter of the earth internal to the position of the particle, and the external shells exert no force at all. a) Assuming that the Earth has uniform density , find the force exerted on the particle as a function of r. b) How long does it take to get to the other end? c) What Happens then?comment intelligently. d) Actually this result is true for any chord, not just one through the center of the earth.

Explanation / Answer

a) force on a body of mass m at distance r from the centre of earth such that r<R = F
Density of earth, rho = M/(4piR^3/3) [M ios mass of earth, R is radius of earth]
Mass of earth responsible for the force = rho*(4pir^3/3) = M(r/R)^3
F = GmMr^3/r^2R^3 = GMmr/R^3 = (GMm/R^3) * r = kr
b) This can be compared to SHM
Time period of SHM of Spring mass system, T = 2pisqroot(m/k)
Time taken to reach other end = T/2 = 3.14*sqroot(mR^3/GMm) = 3.14*sqroot(6371000^3/6.674*10^-11*5.972*10^24) = 2529.797s = 42.16342 min
c) The ball will osscilate about the centre of the earh in linear osscilations with amplitude is equal to radius of earth

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