# Homework Help: 2d oscillators

1. Nov 8, 2004

### matpo39

hi, I was going through my homework and i came to a problem that i cant seem to get.
Consider the mass attached to four identical spring. Each spring has the force constant k and unstreched length L_0, and the length of each spring when the mass is at the origin is a(not necessarily the same as L_0). When the mass is displaced a small distance to the point (x,y), show that its potentail energy has the form 1/2*K_prime*r^2 appropriate to an isotropic harmonic oscillator. What is the constant K_prime in terms of k? Give an expression for the corresponding force.

I started this problem by calculating the force on each spring in the x direction and got
F_1(x)= -k(x+a)+(k*L_0(x+a)/sqrt((x+a)^2+y^2))
F_2(x)= -kx + (k*L_0*x/sqrt(x^2+(y-a)^2))
F_3(x)= -k(x-a) + (k*L_0*(x-a)/sqrt((x-a)^2+y^2))
F_4(x)= -kx + (k*L_0*x/sqrt(x^2+(y+a)^2))

i tried to simplify these forces but cant seem to get any where with it, i think the fact that the displacement from (x,y) has something to do with it, but im not sure how to implement that into the problem.

anyone have any ideas?

thanks