Potential energy equilibrium positions

[frostking]

1. The problem statement, all variables and given/known data
A particle has potential energy

U(x) = x + sin ((2 rad/m) x)

over the range of x greater or equal to 0 meters and less than or equal to pi meters
Where are the equilibrium positions in this range and for each is it a point of stable or unstable equilibrium?

2. Relevant equations
derv of u(x) set = to 0 and then the second derv of u(x)


3. The attempt at a solution


I solved derv of u(x) = 1 + 2 cos(2x) then set = to 0

so 2 cos(2x) = -1 divide by 2 and cos(2x) = -1/2
2x = arch cos of (-1/2) = 2pi/3 rad

x = 1 pi/3 rad

I get this part but the answer says that x can = 2pi/3 rad as well and I do not understand why.

To determine if equilibrium is unstable or stable I took the second derv and at x = pi/3 and second derv of - 4 sin(2x) I got less than 0 so a maximum and therefore unstable equilibrium

Can someone please help me understand why I should have known to consider 2pi /3??? Thanks for your efforts, Frostking

[diazona]

There are an infinite number of solutions to \cos(2x) = -1/2. You just found one - there's another solution which will give you 2\pi/3 as the final answer.

[frostking]

Thanks, yes I should have realized I needed to check for other values less than or equal to pi!!!!