Problem with power and potential energy

[costas]

Homework Statement

Particle mass m is moving along the x-axis.
The particle is attracted to the principle axis O to measure power mk/x^2 when x>a
and repels the O with measure power mka/x^3 when 0<x<a, where a,k>0

i) Find the potential energy of the particle
ii)The particle is released from rest at 2a from O. Describe the movement to perform the particle and show that the particle will stand briefly when x=a/sqrt2

The Attempt at a Solution

i) I tried to find the potential energy by integrating the function of power:

....mk / x ^ 2 , x> a
F (x) =
....mka / x ^ 3 , 0 <x <a

using the relation F (x) = - dU / dx, but the integral constant C if assume that U (O) = 0, goes infinity (oo).

I still can not design the function of potential energy which found that likely have the form:

.....-mk / x , x> a
U (x) =
...mka / x ^ 2 , 0 <x <a

ii) For the second question I can not think of anything.

 

[tiny-tim]

Hi costas!

(try using the X² tag just above the Reply box )

using the relation F (x) = - dU / dx, but the integral constant C if assume that U (O) = 0, goes infinity (oo).

Then don't assume U(0) = 0 …

assume U(a) = 0, or U(∞) = 0. ​