While determining the Young modulus of a metal wire, a student...

The given graph is a straight line passing through the origin, which implies that the quantity on the Y-axis is directly proportional to the quantity on the X-axis ($Y \propto X$).

Young's modulus is given by $Y = \dfrac{\text{Stress}}{\text{Strain}} = \dfrac{W/A}{\Delta L/L}$, where $W$ is the weight hung, $A$ is the cross-sectional area, $\Delta L$ is the increase in length, and $L$ is the original length.

Let us analyze the given options:

For option (1): X-axis = $\Delta L$, Y-axis = $W^2$. Since $W \propto \Delta L$, we have $W^2 \propto (\Delta L)^2$. This would result in a parabolic graph, not a straight line.

For option (2): X-axis = $W$, Y-axis = $\Delta L$. We know $\Delta L = \left(\dfrac{L}{YA}\right) W$. Since $L, Y,$ and $A$ are constants for a given wire, $\Delta L \propto W$. This gives a straight line passing through the origin.

For option (3): X-axis = Stress, Y-axis = Strain. We know $\text{Strain} = \left(\dfrac{1}{Y}\right) \text{Stress}$. Since $Y$ is constant, $\text{Strain} \propto \text{Stress}$. This gives a straight line passing through the origin.

For option (4): X-axis = Stress, Y-axis = $\Delta L$. We know $\Delta L = L \times \text{Strain} = \left(\dfrac{L}{Y}\right) \text{Stress}$. Since $L$ and $Y$ are constants, $\Delta L \propto \text{Stress}$. This gives a straight line passing through the origin.

Thus, the quantities on the X and Y-axes could be those in options (2), (3), and (4).