- #1

Ebby

- 41

- 14

- Homework Statement
- How much work must you do to push the midpoint of the string up or down a distance y?

- Relevant Equations
- F = -kx

W = Fy

My question is whether I've formed the integral for the work done correctly? It just seems a bit unwieldy to me...

If I call the extension of the spring ## x ##, I can see that ## z = \frac l 2 + x ## and ## z^2 = \left( \frac {l} {2} \right)^2 + y^2 ##. Combining them gives: $$ x = \sqrt {y^2 - l} $$

Since the restoring force generated by one spring is ## F_{res} = -k \sqrt {y^2 - l} ## along its axis, the force that must be exerted by me to overcome

*both*springs is: $$ F_{me} = 2k \sqrt {y^2 - l} $$

Now, using ## W = \int |\vec F| \, \cos \theta \, |d \vec r| ## where ## cos \theta = \frac {y} {z} = \frac {y} {\frac {l} {2} + x} ## we can say that: $$ W_{me} = 2k \int_a^b \frac {y \sqrt {y^2 - l}} {\frac {l} {2} + \sqrt {y^2 - l}} \, dy $$ $$ = 2k \int_a^b \frac {y} {\frac {l} {2 \sqrt {y^2 - l}} + 1} \, dy $$

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