Q4.3 Work as an integral (S)
A certain type of spring generates a restoring force of the form F=-λx3, where x is the change in length of the spring. For a distortion of 1 cm, the magnitude of the restoring force is 2 N.- What is the value of λ?
- How much work is done by the restoring force when the spring is compressed a distance 2 cm?
Hint
Reveal
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Hint
Remember the force changes as the spring is compressed so you will need to integrate the force to get the work.Solution
Reveal
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(see Equation 3.2).

Solution
In this problem, the force varies with the displacement. Therefore to find the work, we will have to integrate the force along the path. We must assume that both F and x are parallel, so that the problem is one dimensional. i.e.
- If F=-2 N when x=0.01 m, then λ=-F/x3=2/(0.01)3 = 2×106 Nm-3.
- The work done by the restoring force is then