Q. If $k_{s}$ and $k_{p}$ respectively are effective spring constant in series and parallel combination of springs as shown in the figure, find $\frac{ k _{ s ⁡}}{ k ⁡_{ p ⁡}}$

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A

$\frac{9}{2}$

B

$\frac{3}{7}$

C

$\frac{2}{9}$

D

$\frac{7}{3}$

Solution:

The effective spring constant k_s of this arrangement is
$\frac{1}{ k _{ s ⁡}} = \frac{1}{ k ⁡} + \frac{1}{2 k ⁡}$
$\frac{1}{ k _{ s ⁡}} = \frac{2 + 1}{2 k ⁡} = \frac{3}{2 k ⁡}$
$ k _{ s ⁡} = \frac{2 k ⁡}{3}$
The effective spring constant k_p of this arrangement is
$ k _{ p ⁡} = k ⁡_{1} + k ⁡_{2}$
$= k + 2 k ⁡ = 3 k ⁡$
$∴ \frac{ k _{ s ⁡}}{ k ⁡_{ p ⁡}} = \frac{2 k ⁡ / 3}{3 k ⁡} = \frac{2}{9}$

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