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Q.

Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig-1 shows one of them and Fig-2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{T_{b}}{T_{a}} = \sqrt{x}$ , where value of x is______. (Round off to the Nearest Integer)

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Detailed Solution

In figure (1) force constant $i s k_{a} = k$
In figure (2) force constant is $k_{b} = \frac{k k}{k + k} = \frac{k}{2} [∵ t h e y a r e j o i n e d i n s e r i e s]$
Time period of spring $T = 2 π \sqrt{\frac{M}{K}} \Rightarrow T \propto \frac{1}{\sqrt{k}}$

$∴ \frac{T_{b}}{T_{a}} = \sqrt{\frac{k_{a}}{k_{b}}}$ $= \sqrt{\frac{k}{\frac{k}{2}}} = \sqrt{2}$

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Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig-1 shows one of them and Fig-2 shows their series combination. The ratios of time period of oscillation of the two SHM is TbTa=x , where value of x is______. (Round off to the Nearest Integer)