Spring equation proof and work question

[gaborfk]

A particle is attached between two identical springs on a horizontal frictionless table. Both springs have spring constant k and are initially unstressed.

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a) If the particle is pulled a distance x along a direction perpendicular to the initial configuration of the spring, show that the force exerted by the springs on the particle F=-2kx(1-L/(sqrt(x^2+L^2)))i (where L is the vertical distance of each spring at rest)

b) Determine the amount of work done by this force in moving the particle from x=A to x=0


On part a)

I got that the 2 is there since there are two springs. The L/(sqrt(x^2+L^2) is the sin of the angle between the a spring and the vertical. Where does the (1-) comes from? Also how do I tie the equation together, after figuring out all the parts?

On part b)

Do I have to integrate f(x) as x=A goes to x=0?

Thank you

 

[robphy]

Calculate the stretch of one spring from its equilibrium length of L. (This is where you'll see the "1-" term.)
Only the horizontal component of that force survives in the net force.

 

[gaborfk]

Thank You ! I did not realize that the vertical components cancel out!

Bye

 

[presage]

Need Explanation

I still don't understand how the problem will work... could you explain with a few steps how to solve both parts?