# 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?

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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