$(a)\;\large\frac{1}{2}mv^{2} \qquad(b)\;mv^{2} \qquad(c)\;\large\frac{1}{3}mv^{2} \qquad(d)\;\large\frac{1}{6}mv^{2} $

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i was having problem with the smae question since yesterday.... i found it here...

We can find the effective mass of the spring by finding its kinetic energy. This requires adding all the length elements' kinetic energy, and requires the following integral:

Since the spring is uniform, , where is the length of the spring. Hence,

The velocity of each mass element of the spring is directly proportional to its length, i.e. , from which it follows:

- hence the answer

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