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 dependant on the y component of the displacement or not?

 

[paranormal]

I would like to provide a response to the content regarding two springs displaced horizontally.

First, let's address the potential energy function. The potential energy of a spring is given by the equation U = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. In this case, we have two springs connected at one end, so the total potential energy would be the sum of the potential energies of each spring. Therefore, the potential energy function for this system would be:

U(x,y) = (1/2)kx^2 + (1/2)ky^2

Next, let's find the force vector for a given (x,y) pair. The force exerted by a spring is given by Hooke's Law, F = -kx, where k is the spring constant and x is the displacement from the equilibrium position. Since we have two springs, the total force would be the sum of the forces from each spring. Therefore, the force vector for this system would be:

\vec{F} = -kx\hat{i} - ky\hat{j}

where \hat{i} and \hat{j} are unit vectors in the x and y directions, respectively.

Regarding the two solutions provided as attachments, both are correct but they are approaching the problem in different ways. The first solution uses the potential energy function I mentioned above and then takes the partial derivatives with respect to x and y to find the force vector. The second solution uses the concept of Hooke's Law to directly find the force vector.

Both approaches are valid and will give the same result. As a scientist, it is important to understand different methods of solving a problem and to choose the one that is most suitable for the situation. In this case, both solutions are vectorial and will give the same result, so either one can be used.

I hope this response helps clarify any confusion and provides a better understanding of the problem at hand. As scientists, it is important to constantly question and analyze different solutions to problems in order to ensure accuracy and understanding.