Two springs having spring constant $k_1$ and $k_2$ is connected in series, its resultant spring constant will be $2\,unit$. Now if they connected in parallel its resultant spring constant will be $9\,unit$, then find the value of $k_1$ and $k_2$.
Two springs having spring constant $k_1$ and $k_2$ is connected in series, its resultant spring constant will be $2\,unit$. Now if they connected in parallel its resultant spring constant will be $9\,unit$, then find the value of $k_1$ and $k_2$.
For parallel connection $k_{1}+k_{2}=9$
$\ldots(1)$
For series connection $\frac{k_{1} k_{2}}{k_{1}+k_{2}}=2$
...$(2)$
$\therefore$ From equation (1) and (2) $=\frac{k_{1} k_{2}}{9}$
$\therefore k_{1}+k_{2}=18$
$\therefore k_{2}=\frac{18}{k_{1}}$ putting in equation
$(1)$
$\ldots(3)$
$\therefore k_{1}^{2}+18=9 k_{1}$
$\therefore k_{1}^{2}-9 k_{1}+18=0$
$\therefore\left(k_{1}-6\right)\left(k_{1}-3\right)=0$
$\therefore k_{1}=6$ unit or $k_{1}=3$ unit
$\therefore k_{2}=3$ unit or $k_{2}=6$ unit
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