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

[Diff.Ed]

Homework Statement

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 [itex](x, y)[/itex]
(a) Write the potential energy as a function of x and y.
(b) Find the force vector for a given [itex](x, y)[/itex] pair.

Homework Equations

Hooke's Law. Potential Energy.

The Attempt at a Solution

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

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

 

[genericusrnme]

looks right to me