Question

When two springs $A$ and $B$ with force constant $k_A$ and $k_B$ are stretched by the same force, then the respective ratio of the work done on them is

• A. $k_B:k_A$
• B. $k_A:k_B$
• C. $k_A:k_B:1$
• D. $\sqrt{k_B}:\sqrt{k_A}$
• E. $\sqrt{k_A}:\sqrt{k_B}$

Answer 1

Answer: (A)

$k_B:k_A$

We know that force applied,
$F=k\cdot x$
$W=\frac{F\cdot x}{2}$
$\Rightarrow W=\frac{F\cdot F}{2\cdot k}=\frac{F^2}{2k}$
In the first condition,
$W_1=\frac{F^2}{2k_A}....\left(i\right)$
In the second condition,
$W_2=\frac{F^2}{2k_B}....\left(ii\right)$
On dividing Eq. (i) by Eq. (ii), we get
$\frac{W_1}{W_2}=\frac{F^2}{2k_A}\times \frac{2k_B}{F^2}$
$\Rightarrow \frac{W_1}{W_2}=\frac{k_B}{k_A}$.