First slide

Spring forces

Question

Two springs of spring constants $k_{1} and k_{2}$ are joined in series. The effective spring constant of the-combination is given by:

Easy

A

$\sqrt{k_{1} k_{2}}$
B

$\frac{(k_{1} + k_{2})}{2}$
C

$k_{1} + k_{2}$
D

$\frac{k_{1} k_{2}}{(k_{1} + k_{2})}$

Solution

Let us consider two springs of spring constants $k_{1} and k_{2}$ joined in series as shown in the figure.
Under a force F, they will stretch by $y_{1} and y_{2}$ .
So $y = y_{1} + y_{2}$
or $\frac{F}{k} = \frac{F_{1}}{k_{1}} + \frac{F_{2}}{k_{2}}$
but as springs are massless, so force on them must be the same, i,e., $F_{1} = F_{2} = F$
So, $\frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}} or k = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$
Aliter: In series combination,
$\frac{1}{k_{S}} = \frac{1}{k_{1}} + \frac{1}{k_{2}} = \frac{k_{2} + k_{1}}{k_{1} k_{2}} \Rightarrow k_{S} = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$

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