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  4. Two springs of spring constants k1 and k2 are joined in series. The effective spring constant of the combination is given by

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Q. Two springs of spring constants $ k_1 $ and $ k_2$ are joined in series. The effective spring constant of the combination is given by

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Solution:

Two springs of spring constant $k _{1}$ and $k _{2}$ are joined in series.
The effective spring constant of the combination is -
We know using hooke's law
$\frac{ f _{1}}{ k _{1}}= x _{1}$
$\frac{ f _{2}}{ k _{2}}= x _{2}$
$X =\frac{ f _{1}}{ k _{1}}+\frac{ f _{2}}{ k _{2}}$
$X=\frac{F}{k_{1}}+\frac{F}{k_{2}}$
$X=F\left(\frac{1}{k_{1}}+\frac{1}{k_{2}}\right)$
$X=F\left[\frac{\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}\right]$
$F =\frac{ k _{1} k _{2}}{\left( k _{1}+ k _{2}\right)} X$
Therefore, $k =\frac{ k _{1} k _{2}}{\left( k _{1}+ k _{2}\right)}$