I am given that a mass $m$ is connected to a spring such that when it is at position $x$ the force acting on it is $F=-kx^3$. The mass is then moved to position $x_0$ and released from rest.
What is the velocity at $x=0?$
My thoughts so far are as follows:
$$F=-kx^3=-\frac{d}{dx}\bigg(\frac{1}{4}kx^4\bigg)$$
So the potential $V$ is then $V=\frac{1}{4}kx^4.$
So then set $\frac{1}{4}kx_0^4=\frac{1}{2}mv^2$ which then gives $v=\sqrt{\frac{kx_0^4}{2m}}$
Is this correct?
The next part of the question then asks me to find the coefficient of friction required to bring the mass to rest at $x=0$ given that the surface is now rough.
To do this I have set $$\frac{1}{4}kx_0^4=\frac{1}{2}mv^2+\mu mgx_0$$
Then rearranging this I get $$\mu =\frac{kx_0^4-2mv^2}{4mgx_0}$$