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Math Help - Single particle with conservative force

  1. #1
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    Single particle with conservative force

    A particle of mass 1 moves under the influence of a conservative force such that its potential energy function V (x) = 0.25 x^4 -0.5 x^2 + 0.25

    (a) Describe (qualitatively) the possible motions of the particle for all values of the total energy E.
    comment: E = T + U = 0.5mv^2 + V(x) looking for v?

    (b) Find and classify the equilibrium points of V (x).
    comment: V(x) = 0?

    (c) Find the angular frequency and period of small oscillations about the stable equilibria.
    comment: how to find the frequency and period? depends on the function of sine or cos?

    (d) Suppose that at t = 0 the particle is at Xo= sqrt(2) and its speed there is v = 0. Find the position of the particle for all t > 0 (i.e., find x(t)). What happens as t approaches to infinity?
    comment: dx/dt = v(t) from question (a)?

    (e) For the initial conditions of part (d), find the velocity of the particle for all t > 0. Prove that the speed of the particle v approaches to 0 as t approaches to infinity .
    comment: v(t)? then |v(t)|?

    I have some ideas, but still get confusion between each questions. It seems that there are so many variables and functions. Suggestions and help would be awesome. Thanks a lot with my appreciation.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by zorop View Post
    A particle of mass 1 moves under the influence of a conservative force such that its potential energy function V (x) = 0.25 x^4 -0.5 x^2 + 0.25

    (a) Describe (qualitatively) the possible motions of the particle for all values of the total energy E.
    comment: E = T + U = 0.5mv^2 + V(x) looking for v?
    Well you have a conservative field so energy is conserved:

    \frac{dE}{dt}=mv v'+x^3x'-xx=mx'x''+x^3x'-xx'=0

    Hence:

    mx''+x^3-x=0

    (this is OK as long as x' is continuous even when it goes through 0)


    CB
    Last edited by CaptainBlack; June 26th 2009 at 11:42 PM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by zorop View Post
    (c) Find the angular frequency and period of small oscillations about the stable equilibria.
    comment: how to find the frequency and period? depends on the function of sine or cos?
    Having found the stable equilibrium you linearize the equation of motion about the equilibrium/a and solve the charateristic equation for the angular frequencies as usual.

    CB
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  4. #4
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    yeah get the formulas and the method, thx
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