First slide

Applications of SHM

Question

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

Moderate

A

$2 π \sqrt{\frac{m}{2k}}$
B

$2 π \sqrt{\frac{2m}{k}}$
C

$2 π \sqrt{\frac{m}{4k}}$
D

$2 π \sqrt{\frac{4m}{k}}$

Solution

springs of force constants $k,2k,4k,8k, ...,$ are in series
$f o r s p r i n g s i n s e r i e s, e f f e c t i v e f o r c e c o n s \tan t k_{effective} {=k}_{eff} is obtained from \frac{1}{k_{eff}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + ... . . . . . \frac{1}{k_{eff}} = \frac{1}{k} (1 + \frac{1}{2} + \frac{1}{2^{2}} + ... . . . .) t h e p a r e n t h e s i s t e r m i s i n g e o m e t r i c p r o g r e s s i o n, w h o s e f i r s t t e r m a = 1 a n d c o m m o n r a t i o r = \frac{1}{2} s u m t o i n f i n i t y t e r m s i n g e o m e t r i c p r o g r e s s i o n i s S_{\infty} = \frac{a}{1-r} \frac{1}{k_{eff}} = \frac{1}{k} (\frac{a}{1-r}) \frac{1}{k_{eff}} = \frac{1}{k} (\frac{1}{1- \frac{1}{2}}) \frac{1}{k_{eff}} = \frac{1}{k} (\frac{2}{1}) k_{eff} = \frac{k}{2}$
$time period of mass attached to a spring is T=2π \sqrt{\frac{m}{k_{eff}}} {substituting value of k}_{eff} = \frac{k}{2} T =2 π \sqrt{\frac{m}{k/2}} T =2 π \sqrt{\frac{2m}{k}}$

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