Copper of fixed volume $V$ is drawn into a wire of length $l$. When this wire is subjected to a constant force

Question

Copper of fixed volume $V$ is drawn into a wire of length $l$. When this wire is subjected to a constant force $F$, the extension produced in the wire is $\Delta l$. Which of the following graphs is a straight line?

Accepted Answer

1. **Young's Modulus Formula:** Young's modulus ($Y$) is defined as the ratio of longitudinal stress to longitudinal strain: $Y = \frac{F/A}{\Delta l/l}$ .
2. **Rearranging for Extension:** $\Delta l = \frac{Fl}{AY}$.
3. **Volume Constraint:** Since the volume ($V$) of the wire is fixed and $V = A 	imes l$, we can substitute the area $A$ with $V/l$.
4. **Substitution:**
   $$ \Delta l = \frac{Fl}{(V/l)Y} = \frac{F l^2}{VY} $$
5. **Analysis:** Since $F$, $V$, and $Y$ are constants for the given copper wire, $\Delta l$ is directly proportional to the square of the length ($\
   $$ \Delta l \propto l^2 $$
6. **Graphical Representation:** To obtain a straight line graph ($y = mx$), we must plot $\Delta l$ on the y-axis and $l^2$ on the x-axis. Therefore, the graph of $\Delta l$ versus $l^2$ is a straight line.

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