# Two connected springs and potential energy as a function of x and y

1. Dec 11, 2011

### Diff.Ed

1. The problem statement, all variables and given/known data
Two springs each of natural length a and spring constant C are connected at one end
(see figure). Consider a two dimensional displacement given by $(x, y)$
(a) Write the potential energy as a function of x and y.
(b) Find the force vector for a given $(x, y)$ pair.

2. Relevant equations
Hooke's Law. Potential Energy.

3. The attempt at a solution

a) The stretch lengths of A and B springs Da and Db are
$$D_A = \sqrt{(a+x)^2 + y^2} - a$$
$$D_B = \sqrt{(a-x)^2 + y^2} - a$$
Since potential energy of a spring is
$$U_{spring} = 1/2kx^2$$
The total potential energy U can be written by
$$U(x,y) = U_A + U_B = C/2(D_A^2+D_B^2) = c/2((\sqrt{(a+x)^2 + y^2} - a)^2 + (\sqrt{(a-x)^2 + y^2} - a )^2)$$
b) $$\vec{F} = -\vec{\nabla} U$$ and etc

Would you check my solution ? Is my answer correct ? Thanks for help in advance.

2. Dec 11, 2011

### genericusrnme

looks right to me