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Simple Harmonic Motion Test Question

  1. Jul 19, 2012 #1
    I've been set this question by my tutor and I'm having difficulty doing it.

    A mass of 2 kg is hung from the lower end of a vertical spring and extends it by 40 cm. The mass is now pulled down a further 20 cm and is then released from rest so that it oscillates about the equilibrium position. Determine :

    a) the spring stiffness constant k for the spring.
    b) the time period of the oscillations.
    c) the speed and acceleration of the mass when it is 15 cm from the equilibrium position.
    d ) the maximum speed and maximum kinetic energy of the mass.
    e) the maximum accelerating force on the mass.


    I'm struggling with it as I don't really understand how to find the K. Is the displacement 20? Or would it be 60? I think K is, F/x, x being the displacement?

    If someone could please explain it to me I would be very grateful. I've been out of education for some time so I'm finding some things hard to get my head around.
     
  2. jcsd
  3. Jul 19, 2012 #2
    The distance that you would use in your k calculation is the distance which the spring has moved from equilibrium. Dont forget to multiply the weight by 9.8 to find the force in newtons
     
  4. Jul 19, 2012 #3
    Right so for K I have, 981.

    F = M.A, 2kg x 9.81 = 19620 N

    K = F/x, 19620/20 = 981.

    Is that right?
     
    Last edited: Jul 19, 2012
  5. Jul 19, 2012 #4
    Well let me put it this way. If the spring is hanging and has no weight on it, then wouldn't it be in equilibrium?
     
  6. Jul 19, 2012 #5
    Oh, so equilibrium would be 0cm? As the weight pulls it out of equilibrium, so then the displacement would be 60cm?
     
    Last edited: Jul 19, 2012
  7. Jul 19, 2012 #6
    Close, but I think that you are to assume that work from outside the mass/spring system is pulling the weight down that last 20 cm... This displacement will most likely set the spring into an oscillatory motion.
     
  8. Jul 20, 2012 #7
    Right, so the equilibrium position is 0 and the displacement is 40? So the extra 20cm is used to solve the other questions? It's not used in the string stiffness question?

    I think I'm starting to confuse myself now, lol.
     
  9. Jul 20, 2012 #8

    PeterO

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    Homework Helper

    I beg to differ with your implications.

    The equilibrium position is clearly where the mass would be if not oscillating - ie with the spring extended 40cm.
    Why didn't you take the equilibrium position to be when the spring was in the drawer, in the store-room, with all the other springs?

    The weight of the mass extends the spring by 40 cm to create that equilibrium position.

    Some one/thing THEN extends the spring a further 20cm, from where the mass oscillates.

    While oscillating, it will move between extension 20cm and extension 60cm [ie either side of the equilbrium position 40cm-extension]

    If you take the equilibrium position as extension zero, it it is very difficult to find "the speed and acceleration of the mass when it is 15 cm from the equilibrium position." as it would never gets to a position 15cm from zero.
    It does of course pass through a couple of points 15cm from the 40cm equilibrium position.
     
  10. Jul 20, 2012 #9

    PeterO

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    Homework Helper

    I am not surprised - read my previous post.
     
  11. Jul 21, 2012 #10
    1) kx=mg where x=40 cm.
    2) time period wud be 2pi(m/k)^1/2
    3) equilibrium position wud be wher kx=mg. i.e. 40 cm. n all the other parts cn be solved by using energy conservation.
     
  12. Jul 22, 2012 #11
    Yeah, that makes more sense. Thanks!
     
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