1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:
    Untitled_3.png

    So, with Hooke's law
    [itex]T=mg[/itex]
    [itex]ks=mg[/itex] (equation 1)
    [itex]k=mg/s[/itex]
    [itex]k=(5*9.8)/(-0.392)[/itex]
    [itex]k=-125[/itex]

    With Newton's law of motion (F=ma):
    [itex]m*y''(t)=mg-T-R[/itex]
    [itex]m*y''(t)=mg-k(s+y(t))-by'(t)[/itex]
    [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
    [itex]5*y''(t)-125y(t)+by'(t)=0[/itex]
    [itex]y''(t)+b/5y'(t)-25y(t)=0[/itex]

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
    Cheers
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Springs Question - Second Order O.D.E
  1. Another O.D.E (Replies: 6)

  2. 2nd order O.D.E. (Replies: 5)

  3. 2nd order O.D.E (Replies: 3)

  4. Bernoulli's o.d.e (Replies: 4)

Loading...