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!

Mechanics Question

  1. Sep 9, 2007 #1
    1. The problem statement, all variables and given/known data
    Hi, I'd just like to know if I'm on the right track or a hint or something to help. The problem is: "A particle of mass m is released from rest a distance b from a fixed origin of force that attracts the particle according to the inverse square law:

    F(x) = -kx^(-2). Show that the time required for the particle to reach the origin is
    [tex] \pi \sqrt{ \frac{mb^3}{8k}} [/tex]

    2. Relevant equations
    F = ma, dV/dx = -F

    3. The attempt at a solution
    Initially, I rather hoped to solve the ODE [tex] m \ddot{x} + kx^{-2} = 0 [/tex]
    However, this was nonlinear and I didn't know how to solve it analytically.

    So, I decided to look at the potential. I know the negative derivative of the potential with respect to x is the force. So, I separated and integrated and got
    [tex] V(x) = \int kx^{-2} dx = \frac{-1}{x} + K = \frac{mv^2}{2} [/tex]. I know that when x = b, my velocity is zero (released from rest), so I get the K, the constant of integration, should be 1/b.

    So, my idea now was to solve for velocity as a function of x. I then could call velocity dx/dt, separate and integrate hopefully. I then could set my x(t) = 0 and solve for t hopefully. This, however, did not work well. I got

    [tex] v = \sqrt{ \frac{2k}{m} * ( 1/b - 1/x)) }[/tex] . Separation of variables (calling v = dx/dt) yielded the recipriocal of that fraction being integrated, which I was unaware of how to do, and did not look simple.

    I'm pretty sure I'm approaching this problem in a far too difficult manner. Any hints or suggestions would be appreciated, I don't want a full solution, just a hint in the right direction. Thank you.
     
    Last edited: Sep 9, 2007
  2. jcsd
  3. Sep 9, 2007 #2

    learningphysics

    User Avatar
    Homework Helper

    Note that you forgot the k factor when you did the integral.

    I think you're almost there... use separation of variables in the last part as you were going to... I think a variable substitution will allow you to solve that integral. Might be a trigonometric substitution since the answer has a [tex]\pi[/tex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Mechanics Question
  1. Mechanics Question (Replies: 1)

  2. Mechanics Question (Replies: 2)

  3. Mechanics question (Replies: 8)

  4. Mechanics question (Replies: 22)

  5. Mechanics Question (Replies: 6)

Loading...