Question

Two springs A and B are identical except that A is stiffer than B i.e, ${{k}_{A}}\rangle {{k}_{B}}$. If the two springs are stretched by the same force then
A. More work is done on B, i.e, ${{W}_{B}}\rangle {{W}_{A}}$
B. More work is done on A, i.e, ${{W}_{A}}\rangle {{W}_{B}}$
C. Work done on A and B are equal
D. Work done depends upon the way in which they are stretched

Answer 1

Hint: As a first step, one could read the question well and hence note down the given points. Then you could recall Hooke's law and the relation for work done in terms of displacement in the string. After that, use them accordingly to find the answer.

Formula used:
Hooke’s law,
$F=-kx$
Work done,
$W=\dfrac{1}{2}k{{x}^{2}}$

Complete step-by-step solution:
In the question we are given two springs A and B. We are also said that spring A is stiffer than spring B and we know that stiffness is determined by the spring constant and hence, ${{k}_{A}}\rangle {{k}_{B}}$. We are supposed to find the correct condition for the springs being stretched by the same force.
So let us recall a relation that relates, force with spring constant and work done.
We have the Hooke’s law that is given as,
$F=-kx$
$\Rightarrow x=\dfrac{-F}{k}$…………………………………. (1)
Now we have the work done given by,
$W=\dfrac{1}{2}k{{x}^{2}}$
Substituting (1),
$W=\dfrac{1}{2}k{{\left( -\dfrac{F}{k} \right)}^{2}}=\dfrac{{{F}^{2}}}{2k}$
So, for spring A,
${{W}_{A}}=\dfrac{{{F}^{2}}}{2{{k}_{A}}}$……………………………… (2)
And for spring B,
${{W}_{B}}=\dfrac{{{F}^{2}}}{2{{k}_{B}}}$……………………………….. (3)
From the question,
${{k}_{A}}\rangle {{k}_{B}}$……………………………………….. (4)
From equations (2), (3) and (4), we have,
${{W}_{B}}\rangle {{W}_{A}}$
Hence, we found that the work done on the spring B is greater than work done spring A; option A is correct.

Note: Most cases we won't be able to recall the direct relation between the given quantities and so we have to go for other relations to get help to achieve the required relations. Also, the negative sign in Hooke's law simply means that the displacement is in the direction opposite to the direction of restoring force.