Two Springs Displaced Horizontally

[Septim]

1. 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. $$\vec{F}$$=-k$$\vec{x}$$(Hookes Law) $$\vec{x}$$ is the displacement along the spring from equilibrium point. Magnitude wise l=L-L_{0}
3. First I assumed for the sake of simplicity potential energy as $$U(l)=(1/2)l^2C$$and summed them up since U is a scalar quantity. Then to find the force I differentiated U with respect to x and y respectively. But I am very curious if this solution is right cause a friend of mine has provided another solution which does not correspond with mine. I am including both of the solutions as attachments in jpeg format. By the way I need a vectorial solution for this thanks for your contribution in advance.

 

[berkeman]

1. 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. $$\vec{F}$$=-k$$\vec{x}$$(Hookes Law) $$\vec{x}$$ is the displacement along the spring from equilibrium point. Magnitude wise l=L-L_{0}



3. First I assumed for the sake of simplicity potential energy as $$U(l)=(1/2)l^2C$$and summed them up since U is a scalar quantity. Then to find the force I differentiated U with respect to x and y respectively. But I am very curious if this solution is right cause a friend of mine has provided another solution which does not correspond with mine. I am including both of the solutions as attachments in jpeg format. By the way I need a vectorial solution for this thanks for your contribution in advance.

The attachments are pretty hard to read. Can you re-scan them? Why are they so dark?

 

[Septim]

They are taken with a cell phone unfortunately I do not have the oppurtunity to scan it but I will do my best to provide better photos.

 

[Septim]

I think these photos taken without a flash is better. Our main problem is potential energy and force is dependent on the y component of the displacement or not?