Question

Springs of spring constants K, 2K, 4K, 8K, 2048 K are connected in series. A mass 'm' is attached to one end the system is allowed to oscillation. The time period is approximately :

• A. $2\pi \sqrt{\frac{m}{2K}}$
• B. $2\pi \sqrt{\frac{2m}{2K}}$
• C. $2\pi \sqrt{\frac{2m}{K}}$
• D. none of these

Answer 1

The correct option is C $2\pi \sqrt{\frac{2m}{K}}$

In series the equation spring constant is given by:-

$\frac{1}{k^{\prime }}=\sum \frac{1}{k^{\prime }}\Rightarrow \frac{1}{k^{\prime }}+\frac{1}{k}+\frac{1}{2k}+\frac{1}{4k}+\frac{1}{8k}+\frac{1}{2048k}$

$\Rightarrow \frac{1}{k^{\prime }}=\frac{3841}{2048k}$

or approximately $k^{\prime }\simeq \frac{2048k}{3841}\simeq \frac{k}{2}$

$\therefore$ New time period, $T^{\prime }=2\pi \sqrt{\frac{m}{k^{\prime }}}=2\pi \sqrt{\frac{2m}{k}}$