Suppose you decide to go ice fishing. You use powerful strong Auger drill to cut

Question

Suppose you decide to go ice fishing. You use powerful strong Auger drill to cut through the ice. You drill a hole with smooth sides straight through. As a physics student, suddenly you w ant to find out whether one can drill straight through the center of the earth a hole into the earth as a tube to find (assuming If the air is removed from this tube and the tube doesn't fill up with water, liquid rock or iron from the core), whether an object dropped into one end of the tube w ill have enough energy to just exit the other end and if so after an interval of time. Make the assumption that the earth has uniform mass density. While calculating the time for go through the steps What is the gravitational force as a function of the distance r from the center of the earth? Express your answer in terms of the gravitational acceleration at the surface of the earth g and r^e. What is the potential energy inside the earth as a function of r for the object-earth system? Can you think of a natural point to choose a zero point for the potential energy? Be careful because you will need relation between work and potential energy when the object moves inside the earth and with appropriate gravitation force when the object is inside the earth. Use energy considerations to find the velocity of the object when it passes through the center of the earth

Explanation / Answer

a)    Gravitational force as a function of r is as below,

F = -mgr/Re ------------(1)

b) Put k= mg/Re in above equation

Thus

F= -kr

This is equation of spring mass system. So object will perform SHM.

Here k = mg/Re

period of oscillation is,

T = 2*sqrt(m/k) = 2*sqrt(mRe/mg) = 2*sqrt(Re/g) = 2*3.14*sqrt(6.378*10^-6/9.8) = 5068s = 84.5min

To reach other side it takes half the period = T/2 = 5068/2 = 2534s = 42.25min

c) Use equation

W = -PE = 1/2kr^2 = 1/2*mg/Re* r2

d) Use equation

PEi + KEi = PEf + KEf

1/2*mg/Re* ri2 + 0 = 1/2*mg/Re* rf2 + 1/2mv^2

ri= Re , rf= 0

Thus,

1/2*mg/Re* Re2 + 0 = 0 + 1/2mv^2

1/2*mgRe =1/2mv^2

v= sqrt(gRe ) = sqrt(9.8*6.378*10^6) = 7900m/s

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