Oscillations

Question : 13

Total: 25

Figure (i) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure (ii) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in figure (ii) is stretched by the same force F.
(a) What is the maximum extension of the spring in the two cases?
(b) If the mass in figure (i) and the two masses in figure (ii) are released, what is the period of oscillation in each case?

Solution:

a) Maximum extension of the spring :
(i) Suppose the maximum extension produced in the spring is y. Then,
F=k y (in magnitude)
or y=

(ii) In this case, each mass relative to the other behaves as if the other mass is fixed. In other words, force F on each mass acts as the force of reaction developed due to force F on the other mass. Therefore, in this case also, maximum extension is given by
y=

ii) In this case, each mass relative to the other behaves as if the other mass is fixed. In other words, force F on each mass acts as the force of reaction developed due to force F on the other mass. Therefore, in this case also, maximum extension is given by
y=

(b) Period of oscillation:
In figure (i), T=2π√

To calculate the period of oscillation, the spring in figure (ii) can be considered as to be equivalent to the two springs, each of length

and joined at the point O, the centre of the string as shown in the figure.

If k′ is force constant of each half, then k′ = 2k (Because, if a spring is cut to half of its length, its force constant becomes double. Therefore,
T=2π√

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