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!

Motion of a point is equal to dynamic harmonic oscilation

  1. Apr 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Point with mass is moving along the positive direction of x axis, its velocity is described by (A-Bx^2)^(1/2). Show that its equation of motion describes dynamic harmonic oscillation and find period (T) of this oscillation.

    2. Relevant equations
    v=(A-Bx^2)^(1/2)
    A and B is known constants

    3. The attempt at a solution

    Hi, guys!
    My first instinct is to try to get something from F=m*a=-k*x.
    Further more m*(dv/dt)=-k*x, but immediately i see that its wrong because v=v(x) not v(t).

    So ,please, could someone give me some advice.
    Thank you!
     
  2. jcsd
  3. Apr 2, 2016 #2
    Hint short:

    If energy is conserved for a SHO along the lines of
    $$ stuff1 \times v^2 + stuff2 \times x^2 = constant$$
    no?

    Hint, long:

    $$ F = m {dv \over dt} = m {dv \over dx} {dx \over dt} = m v {dv \over dx} = -kx$$

    Integrate dx both sides, what do you get?
     
  4. Apr 2, 2016 #3
    hello!
    Thank you for your hints, I had forgotten connections between derivatives.

    If I take the equation mv (dv/dx)=-kx
    I obtain m*v*dv=-k*x*dx
    By integrating m *(v^2)/2+C1=-k*(x^2)/2+C2
    I suppose I could now substitute v with given equation but if I do so, the equation becomes very messy and I do not think that it could be reduced to something useful.

    What do you suggest?
     
  5. Apr 2, 2016 #4
    Does your equation with the squares look like a conservation equation to you? :D (protip, you could haved used conservation of energy, but instead you derived it like a beast!)

    Anyways, if you solve for v from x from that equation, what do you get? What is the equivalent of
    $$\omega = \sqrt{k \over m}$$ when you try to match your equation to the given equation $$v = (A-Bx^2)^{1 \over 2}$$
     
    Last edited: Apr 2, 2016
  6. Apr 2, 2016 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    There is a simpler alternative to showing it satisfies the usual SHM ODE: show it satisfies the solution to an SHM ODE.
     
  7. Apr 3, 2016 #6
    If I solve for v, I obtain [itex]v = ({\frac{\ 2(C1-C2)}{m}} - {\frac{\ k x^2}{m}})^{(1/2)} [/itex]
    which looks similar to given equation and if I assume that
    [itex]A≡{\frac{\ 2(C1-C2)}{m}} [/itex]
    it can be rewritten
    [itex]v = (A - ω^2x^2)^{(1/2)} [/itex]
    which means that
    [itex] B≡ ω^2 [/itex]
     
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: Motion of a point is equal to dynamic harmonic oscilation
Loading...