# Thread: Single particle with conservative force

1. ## 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.

2. Originally Posted by zorop
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:

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

Hence:

$\displaystyle mx''+x^3-x=0$

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

CB

3. Originally Posted by zorop
(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

4. yeah get the formulas and the method, thx