Question

When the load on a wire is slowly increased from 3 to 5kgwt, the elongation increases from 0.61 to 1.02mm. The work done during the extension of wire is:
A) 0.16J
B) 0.016J
C) 1.6J
D) 16J

Answer 1

Hint: Here, we need to form a relation between work, stress and strain. The general formula for work which is W =F.d, will not work as the work is being applied to the wire and not on a particular object. The wire will get stress and strain.

Complete step by step solution:
Find the work done during extension of wire:
Here, the wire is experiencing stress and strain, so the work done for the initial elongation would be:
${W_i} = \dfrac{1}{2}\left( {\dfrac{F}{A} \times \dfrac{{\Delta l}}{l}} \right) \times A \times l$;
Here:
$\dfrac{F}{A}$= Stress;
F = Force;
$\dfrac{{\Delta l}}{l}$= Strain;
$\Delta l$= Change in length;
$A$= Area;
$l$= Length;
Cancel out the common and put F = mg;
 $ \Rightarrow {W_i} = \dfrac{1}{2}\left( {mg \times \Delta l} \right)$;
Now, put the given value in the above equation:
$ \Rightarrow {W_i} = \dfrac{1}{2}\left( {3 \times 9.8 \times 0.61 \times {{10}^{ - 3}}J} \right)$;
$ \Rightarrow {W_i} = \dfrac{1}{2}\left( {3 \times 0.61 \times 9.8 \times {{10}^{ - 3}}J} \right)$;
Do the needed calculation:
$ \Rightarrow {W_i} = 8.965 \times {10^{ - 3}}J$;
Now, we can do this for the final elongation by applying the formula for work $W = \dfrac{1}{2}\left( {mg \times \Delta l} \right):$
$\Rightarrow {W_f} = \dfrac{1}{2}\left( {mg \times \Delta l} \right)$;
$\Rightarrow {W_f} = \dfrac{1}{2}\left( {5 \times 9.8 \times 1.02 \times {{10}^{ - 3}}J} \right)$;
Now, do the needed calculation:
$\Rightarrow {W_f} = 24.99 \times {10^{ - 3}}J;$
The Net work done on the wire is: ${W_f} - {W_i}$
$\Rightarrow {W_f} - {W_i} = 24.99 \times {10^{ - 3}} - 8.965 \times {10^{ - 3}}$;
Do, the needed calculation:
$ \Rightarrow {W_{net}} = 16.025 \times {10^{ - 3}}J$;
$ \Rightarrow {W_{net}} = 0.016J$;

Final answer is option B. The work done during the extension of wire is 0.016J.

Note: Here, the work done on the wire will include the (1/2) times the stress and strain on the wire times the area and length. Here the stress on the wire would be forced upon the area and the strain on the wire will be changed of length upon original length. After calculating the initial work done and the final work done, subtract the final work done from the initial work done and that would be the net or the required work done. Be careful when converting the units, here the units are given in mm, kg-wt.