First slide

Applications of SHM

Question

Springs of constants K, 2K, 4K, 8K, 16K,… are connected in series. The mass ‘m’ kg is attached to the lower end of the last spring and the system is allowed to vibrate. The frequency of oscillation is

Moderate

A

$\frac{1}{π} \sqrt{\frac{m}{2K}}$
B

$\frac{1}{2π} \sqrt{\frac{K}{2m}}$
C

$\frac{1}{2π} \sqrt{\frac{2m}{2K}}$
D

$\frac{1}{2π} \sqrt{\frac{4m}{K}}$

Solution

K, 2K, 4K, 8K, 16K,… are connected in series
$\frac{1}{K_{effective}} = \frac{1}{K} + \frac{1}{2K} + \frac{1}{4K} + - - \to - g e o m e t r i c p r o g r e s s i o n (g p) s u m t o i n f i n i t e s e r i e s o f g p = \frac{a}{1 - r}; a i s f i r s t t e r m; r i s c o m m o n r a t i o \frac{1}{K_{eff}} = \frac{1}{K} [1 + \frac{1}{2} + \frac{1}{2^{2}} + ...] h e r e f i r s t t e r m a = 1 a n d c o m m o n r a t i o r = \frac{1}{2} \frac{1}{K_{eff}} = \frac{1}{K} [\frac{a}{1-r}] s u b s t i t u t e t h e v a l u e s \frac{1}{K_{eff}} = \frac{1}{K} [\frac{1}{1- \frac{1}{2}}] \frac{1}{K_{eff}} = \frac{1}{K} [\frac{2}{1}]$
$K_{e f f} = \frac{K}{2}$
$Frequency n= \frac{1}{2π} \sqrt{\frac{{K_{ef}}_{f}}{m}} {; substitute obtained value of K}_{e f f} n = \frac{1}{2π} \sqrt{\frac{\frac{K}{2}}{m}}$
$n = \frac{1}{2π} \sqrt{\frac{K}{2m}}$

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