Question

The figure shows stress versus strain graphs of two materials A and B. If $Y_A$, $Y_B$ are the Young's moduli of materials respectively, then:

• A. $Y_A = \sqrt{2} Y_B$
• B. $Y_B = \sqrt{3} Y_A$
• C. $Y_B = \sqrt{2} Y_A$
• D. $Y_A = \sqrt{3} Y_B$

Hint:

On a Stress-Strain graph, a steeper slope indicates a more rigid material with a larger Young's Modulus.
Since line A is steeper than line B, $Y_A$ must be larger than $Y_B$, quickly eliminating options (B) and (C).

Answer 1

The Correct Option is D

Step 1: Recall what the graph's slope means.
On a stress versus strain plot, Young's modulus is the tangent of the angle the line makes with the strain axis, $Y=\tan\theta$.
Step 2: Read the two angles from similar right triangles.
Material A's line sits at $45^\circ$, so for every unit along the strain axis it rises by an equal unit, giving $\tan45^\circ = 1$. Material B's line sits at $30^\circ$, a shallower rise of $\frac{1}{\sqrt{3}}$ per unit strain.
Step 3: Compare the two moduli as a ratio. \[ \frac{Y_A}{Y_B} = \frac{\tan45^\circ}{\tan30^\circ} = \frac{1}{1/\sqrt{3}} = \sqrt{3} \]
\[ \boxed{Y_A = \sqrt{3}\,Y_B} \]