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!

Energy Conservation for a spring

  1. Feb 28, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider a mass connected to a spring of sti ness k.
    (a) Use conservation of energy to write down a first order differential equation obeyed
    by the mass.
    (b) Find the time t for the mass to move from the origin at t = 0 out to a position
    x assuming that at t = 0 it has initial speed dx/dt = v0.
    (c) Use the above to show that the period of motion is T = 2∏sqrt(m/k)


    2. Relevant equations

    Us = .5kx^2
    K = .5mv^2

    3. The attempt at a solution

    a) I just set up a initial and final conservation of energy

    I figured that I'd just set Us = K and make v = (dx/dt). That'd give me a natural log in my answer, which doesn't seem to make much sense.

    Everything else doesn't seem to be too hard once I get part a.
     
  2. jcsd
  3. Feb 28, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    I'd just set Us = K and make v = (dx/dt)

    That would be: $$\frac{dx}{dt} = \pm \sqrt{\frac{k}{m}}x$$ ... if you are looking for x(t) then the solution should be an exponential, not a log.

    Notice that you can have a positive or a negative velocity for any given position.
     
  4. Feb 28, 2013 #3
    Wouldn't that set up yield this...

    dx/x = ±sqrt(k/m)*dt

    and if you integrate...

    ln(x) = ±sqrt(k/m)*t

    I don't know much about eulers identity, but what if I took an exponential of that answer. Then it'd be

    x = e^[±sqrt(k/m)*t]

    which (and I know this part is wrong) I'd write as

    x = cos[sqrt(k/m)*t]
    ----------------------

    If it were second order, then I could get the trig function that I think is supposed to happen from this with an ODE.

    ----------------------

    I feel like the problem lies in my first equation. Any thoughts on that?
     
    Last edited: Feb 28, 2013
  5. Feb 28, 2013 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, the problem lies in the first equation. Energy is conserved means Us+K=C where C is some constant. Write that out then take the time derivative of both sides. That will give you a second order ode.
     
    Last edited: Feb 28, 2013
  6. Feb 28, 2013 #5

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    What Dick said - the clue is that you have not accounted for the ##\pm## ... didn't work: I should have just directed you to look again at what "conservation law" means ;)
     
  7. Mar 1, 2013 #6
    The only thing I'm worried about is that the question specifically says to write down a first order ODE, but I guess it doesn't make much sense in this case.

    So how do I take a time derivative of that? I'm a little confused because of the (dx/dt)^2
    I'm guessing I want kx + ma = 0

    wouldn't the chain rule get

    d/dt(.5kx^2 + .5m(dx/dt)^2)) = kvx + mva = 0
     
    Last edited: Mar 1, 2013
  8. Mar 1, 2013 #7

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    ... well done: cancel common factor?
    Note: you have written down a 1st order ODE - this is the next step.

    You could always have done: $$\frac{dx}{dt} = \pm \sqrt{c-\frac{kx^2}{m}}$$
    ... but you'd still have to figure how the ##\pm## should be handled.
     
    Last edited: Mar 1, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Energy Conservation for a spring
  1. Conservation of Energy (Replies: 5)

  2. Conservation of Energy (Replies: 0)

Loading...