Kibble problem (linear motion chap. 2)

[levilevi]

Homework Statement

7. A particle of mass m moves (in the region x > 0) under a force F =
−kx+c/x, where k and c are positive constants. Find the corresponding
potential energy function. Determine the position of equilibrium, and
the frequency of small oscillations about it.

Homework Equations

dx/dt = [(2/m)(E - V(x)]^1/2

The Attempt at a Solution

dx/[(2/m)(E+ Clnx - (1/2)kx^2]^1/2 = dt
V = -Clnx + (1/2)kx^2]^1/2
i think at equilibrum point V(x) must be zero but then how can i continue? This is en exam question and i must solve it completely correct. Thank you for your answers. (my firt language is not english. Sorry for mistakes.)

 

[kuruman]

i think at equilibrum point V(x) must be zero but then how can i continue?

V(x) is not zero at equilibrium. What does equilibrium mean to you?

 

[levilevi]

Thanks, you are right. Total force, i think is zero at equilibrium point. So;
-ka + $$\frac{c}{a}$$ = 0 (a = equilibrium point.) k = c/a²
Is it true?

 

[kuruman]

Thanks, you are right. Total force, i think is zero at equilibrium point. So;
-ka + $$\frac{c}{a}$$ = 0 (a = equilibrium point.) k = c/a²
Is it true?

It is true. If this is an exam question, you have to finish it by yourself. I will not tell you how to solve it, but I can tell you if something is correct or not and the rest is up to you.

 

[levilevi]

The exam was three days ago. Nobody could solve this question and a similar one (in which F = -kx +a/x^3) The teacher gave us as a homework. If i will solve maybe my exam mark will increase but i am not sure this. Maybe not. If you don not want to help it is up to you. I will study on it again and again...

 

[levilevi]

I solved at last. a=(c/k)^1/2 and w=(2k/m)^1/2 , w=(V''(a) /m)^1/2. Thanks.