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Period of motion of an object dropped through the Earth

  1. Oct 25, 2016 #1
    1. The problem statement, all variables and given/known data
    An object of mass ##m## moves in a smooth, straight tunnel dug between two points on the Earth’s surface. Show that the object moves with simple harmonic motion, ##a = - ω^2 x##. Find the period of this motion. You can assume that the Earth’s density is uniform.
    2. Relevant equations
    $$F=G\frac{Mm}{r^2}$$
    $$V_{sphere}=\frac{4}{3}πr^3$$
    $$ρ = \frac{M}{V}$$
    3. The attempt at a solution
    I have been staring at this problem for the past 30 minutes. I would be very appreciative if someone could give me a hint on how to start this. I intuitively understand why the object would experience simple harmonic motion, I just can't figure out how to express it mathematically, especially since the hole does not have to go through the center of Earth.
     
    Last edited: Oct 25, 2016
  2. jcsd
  3. Oct 25, 2016 #2

    Simon Bridge

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    For SHM, ##F\propto r## when r is the displacement from an equilibrium point.
    Notice that the two points on the surface of the earth do not have to be diametrically opposite each other.
    However - you have ##F\propto r^{-2}## for the case that r>R. But that is not the case here.

    How does M vary with r when r<R (R=radius of the earth)?
     
  4. Oct 25, 2016 #3
    ##M=\frac{4}{3}πr^3ρ=(\frac{4}{3}πr^3)(\frac{M_{Earth}}{\frac{4}{3}πR^3})=\frac{r^3M_{Earth}}{R^3}##
    ##a=G\frac{M}{r^2}=G\frac{\frac{r^3M_{Earth}}{R^3}}{r^2}=G\frac{rM_{Earth}}{R^3}##
    Is this correct so far?
     
  5. Oct 26, 2016 #4

    SammyS

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    That is the magnitude of radial acceleration if the object were not constrained to move in the tunnel.

    Multiply by mass, m, to find the magnitude of the gravitational force. What component of that force is in the direction of the tunnel?
     
  6. Oct 26, 2016 #5
    ##F = G\frac{rmM_{Earth}}{R^3}\cosθ## where ##θ## is the angle between ##r## and the direction of motion. ##\cosθ = \frac{x?}{r}##.
     
  7. Oct 26, 2016 #6

    haruspex

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    If you fall into a vertical hole, what is the angle between your direction of motion and the line joining you to the centre of the Earth?
     
  8. Oct 26, 2016 #7
    The angle would be ##0°## so then ##\cos(0°)=1##. That leaves me with ##F = G\frac{rmM_{Earth}}{R^3}##. How would that account for the component of the force if the tunnel was not through the center of the Earth?
     
  9. Oct 26, 2016 #8

    haruspex

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    I'm sorry - I misread the question.
    So let x be the distance from the midpoint of the tunnel. Express theta as a function of x and r.
     
  10. Oct 26, 2016 #9
    So ##\cos(θ)=\frac{x}{r}## and then ##F = G\frac{mMx}{R^3}##?
     
  11. Oct 26, 2016 #10

    haruspex

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    Yes.
     
  12. Oct 26, 2016 #11
    Okay, so then
    ##a=ω^2x=G\frac{Mx}{R^3}##
    ##ω=\sqrt{\frac{GM}{R^3}}=\frac{2πr}{T}##
    ##T=\frac{2πr}{\sqrt{\frac{GM}{R^3}}}##
    Is this correct?
     
    Last edited: Oct 26, 2016
  13. Oct 26, 2016 #12

    haruspex

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    What is r there? Check the dimensional consistency.
     
  14. Oct 26, 2016 #13
    Oh, ##ω=\frac{2π}{T}\neq\frac{2πr}{T}##
    Then this should be the answer:
    $$T=\frac{2π}{\sqrt{\frac{GM}{R^3}}}$$
    $$T=\frac{2π}{\sqrt{\frac{(6.67⋅10^{-11})(5.96⋅10^{24})}{(6.37⋅10^6)^3}}}=5066 s=84.44 min$$
     
    Last edited: Oct 26, 2016
  15. Oct 26, 2016 #14

    haruspex

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    Looks right.
     
  16. Oct 26, 2016 #15
    Thank you so much for your help!
     
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