Q. 14.13 (b) 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.

                   

(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?

Answers (1)
S Sayak

(b).(a) In Fig, (a) we have

F=-kx

ma=-kx

a=-\frac{k}{m}x

\\\omega ^{2}=\frac{k}{m}\\ T=\frac{2\pi }{\omega }\\ T=2\pi \sqrt{\frac{m}{k}}

(b) In fig (b) the two equal masses will be executing SHM about their centre of mass. The time  period of the system would be equal to a single object of same mass m attached to a spring of half the length of the given spring (or undergoing half the extension of the given spring while applied with the same force)

Spring constant of such a spring would be 2k

F=-2kx

ma=-2kx

\\a=-\frac{2k}{m}x\\ \omega ^{2}=\frac{2k}{m}\\ T=\frac{2\pi }{\omega }\\ T=2\pi \sqrt{\frac{m}{2k}}\\ T=\pi \sqrt{\frac{2m}{k}}

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