This might be better suited for a physics forum. If you can set up the integral, I or someone else can help you, but other than that you might not get the help you're looking for.
A spring has a natural length of 28 cm. If a 28 N force is required to keep it stretched to a length of 31 cm, how much work W is required to stretch it from 28 cm to 38 cm?
W= ___ J
I know Work= Force * Distance but i dont know how to go about solving this i know it will be an integration of some sort
- When the extension in the spring is , the tension, , is given by
- When the spring is stretched from an extension to an extension , the work done, , is given by
So here, m. And when (because the total length is m), .
So, using the first of the two formulae above:
And we need to find when and
, for some constant .
Now if we define the modulus (or modulus of elasticity), , to be the force required to double the length of the spring, then when . So , which in turn gives the first of the two formulae that I quoted, namely
The second formula is derived by integration, as follows:
Suppose the spring is extended from to . Then the work done is given by
, where is the work done extending the spring from to .
So there's nothing other than Hooke's Law here. However, it's much easier to learn and apply these two well-known formulae, than to derive the answers from first principles each time!