First slide

Applications of SHM

Question

Springs of force 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
effective spring constant is $K_{e f f e c t i v e} = K_{e f f}$ is
$\frac{1}{K_{eff}} = \frac{1}{K} + \frac{1}{2K} + \frac{1}{4K} + - - -$
$\frac{1}{K_{eff}} = \frac{1}{K} [1 + \frac{1}{2} + \frac{1}{2^{2}} + ...]$ , the paranthesis part is in geometric progression
whose first term =1 and common ratio is r= $\frac{1}{2}$
$s u m t o i n f i n i t e t e r m s i s S_{\infty} = \frac{a}{1 - r} w h e r e a i s f i r s t t e r m, r i s c o m m o n r a t i o$
$\frac{1}{K_{e f f}} = \frac{1}{K} [\frac{a}{1-r}] = \frac{1}{K} [\frac{1}{1- \frac{1}{2}}]$
$\frac{1}{K_{eff}} = \frac{1}{K} [\frac{2}{1}]$
$K_{e f f} = \frac{K}{2}$
Frequency $η = \frac{1}{2π} \sqrt{\frac{{K_{ef}}_{f}}{m}} = \frac{1}{2π} \sqrt{\frac{\frac{K}{2}}{m}}$

$d u e t o s e r i e s o f s p r i n g s n e w f r e q u e n c y i s = \frac{1}{2π} \sqrt{\frac{K}{2m}}$

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