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Homework Help: Springs Question - Second Order O.D.E

  1. Oct 19, 2012 #1
    Hi all, I was wondering is you could help me with this springs question. We've only done springs hanging from a fixed support above being stretched, but now I've got a question where the spirng is being compressed.

    1. The problem statement, all variables and given/known data
    So, here's some basic info about the question
    • Obviously, acceleration due to gravity is 9.8 m/s^2
    • The downward direction is considered the positive direction
    • A spring is placed on a fixed support standing vertically
    • The natural length of the spring is 5 metres
    • An iron ball weighing 5kg is attached to the top of it
    • This mass compresses the spring by 0.392m
    • The ball is pulled to a distance of 0.4m above its equalibirum position then released
    • Damping force is proportional to the instantaneous velocity of the ball
    • y(t)=The displacement of the ball below the equalibrium position at t seconds after the release

    2. Relevant equations
    So, the first question is to proove that the equation of motion for the ball is...
    • [itex]0=y''(t)+(b/5)y'(t)+25y(t)[/itex], where b kg/s is the damping constant

    3. The attempt at a solution
    Here is a rough diagram of what I believe to be happening from the description of the problem:

    So, with Hooke's law
    [itex]ks=mg[/itex] (equation 1)

    With Newton's law of motion (F=ma):
    [itex]m*y''(t)=mg-ks-ky(t)-by'(t)[/itex], from equation 1: ks-mg=0 so
    [itex]m*y''(t)+ky(t)+by'(t)=0[/itex], So subbing in k=-125, m=5

    This is almost what I need, but for some reason I have -25y(t) instead of +25y(t). Can anyone see where I've gone wrong? The only thing I can think of is that k (and thus s) should be positive, but I am unsure why as it is in compression. We have been taught that stretching (extension) of the spring make "s" a postive number, so one would think that compression of the spring would make give "s" a negative value. Is k always > 0 by definition?
    Last edited: Oct 19, 2012
  2. jcsd
  3. Oct 19, 2012 #2


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    You have s- y when you should have y- s. Suppose y is very large. Then s- y will be negative so -k(s- y) will be positive and y''= -k(s- y) would be positive meaning that y will get larger. It should be the other way around- the spring should move y back toward the center. You should have either y''= k(s- y) or y''= -k(y- s).
  4. Oct 19, 2012 #3
    Woops. That was a typo actually.

    It should say [itex]my''(t)=mg-k(s+y(t))-by'(t)[/itex] instead, and this is the path that the following calculations take.

    In this sense, I am saying that the restoring force of the spring’s tension (T) is equal to the spring constant*(contraction (or extension)+y(t) (the distance below the equalibrium position)). Either I am not understanding you or the subject matter correctly or you have found the wrong error, simply a badly placed typo. Sorry on my behalf.

    I'll fix the error up in the OP.
    Last edited: Oct 19, 2012
  5. Oct 21, 2012 #4
    m*y'' = mg - T - R

    my'' = mg - k(s-y) - By'

    We write -k(s+y) because the spring is stretched but in this case spring is compressing that is why we use -k(s-y)
    Hope it makes sense.
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