Figure 14.26 (a) 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 14.26 (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 14.26(b) is stretched by the same force F. What is the maximum extension of the spring in the two cases?
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(i) In the first figure the maximum extension produced in the spring is
F=kx or x=F/k
In the second figure when force F,F are applied on the masses m,m in opposite directions the middle point remains stationary at its position .Therefore, the spring of length l can be considered to be made up of two parts , each of length l/2 and hence a force constant 2k , and joined to each other at the middle point.Since the extension is proportional to the length of spring , the extension can be calculated as
x/2 = F/2k
So the total extension will be equal to
x= x/2 + x/2 = F/2k + F/2k = F/k
(ii) The period of oscillation of the mass in the first figure is
T= 2π √(m/k)
The period of oscillation of the mass in the second figure is
T= 2π √(m/2k)
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