Question

(40pt) A particle is under the influence of a force Fkk/a2 where k and a are constant and k 〉 0, Determine the potential energy U(x) and find the possible particle motions. What happens when E ka2/4?

Also, how can I generate a plot showing this on Mathematica using integration?

Answer 1

F(x) = -kx +kx³/$\alpha ^2$  

The force is 0 at x=0

=> the particle is pushed to the origin and has least PE at the origin we can take this as reference potential at set it to 0

$U(x) = \int F(x)dx = \int (-kx+kx^3/\alpha ^2) \\ = -kx^2/2 +kx^4/4\alpha ^2$

The particle has an energy E = k$\alpha ^2$/4

The particle moves in the force field until its PE is equal to k$\alpha ^2$/4 , i.e. work done against the force.

k$\alpha ^2$/4 = -kx²/2 + kx⁴/4$\alpha ^2$ k>0

solving this eq. we get

$x= \pm \alpha \sqrt{1\pm \sqrt{2}}$

out of this 2 or imaginary roots.

the real roots are  

$x= \pm \alpha \sqrt{1+ \sqrt{2}}$

The particle will oscillate between these 2 points.

To generate the plot for U(x) you must know the values for k and $\alpha$

You can put any assumed values in the expression and draw the plot , you will get the shape of the curve. The shape remains same for any values of k and $\alpha$