Aparticle of mass m moves under a conservative force with potential energy v ( x )= cx/(x2+a2),where c and a are positive constants. find the position of stable equilibrium and the period of small oscillations about it. (b) if the particle starts from this point with velocity v, find the r

The position of stable equilibrium is -a

And the period of small oscillations  must be: c/(ma^3)

Explanation:

Since the potential is:

We first look for a position of stable equilibrium. This posiiton must satisfy two considtions, that the first derivative of the potential must vanish at this point and the second derivative must be positive.

Which vanish for

x = a   ; x =-a

The second derivative of V(x) is:

And:

Therefore:

a)

The position of stable equilibrium is -a

And the period of small oscillations  must be:

(c/(ma^3))^1/2

b)

Let's find the maximum amplitude if the particle starts at this point with velocity v

If this is the case, the total energy will be:

(mv^2)/2

And the maximum amplitude will be

x = a^3/c mv^2 = (m v^2 a^3)/ c

cant without the picture! ?