Question

Two spring of equal lengths and equal cross sectional areas are made of materials whose Young's modulii are in the ratio of 2:3. They are suspended and loaded with the same mass. When stretched and released, they will oscillate with time periods in the ratio of

• A. $\sqrt{3}:\sqrt{2}$
• B. 3 : 2
• C. $3\sqrt{3}:2\sqrt{2}$
• D. 9 : 4

Answer 1

The correct option is A $\sqrt{3}:\sqrt{2}$

Young modulus $Y=\frac{\left(F/A\right)}{l/L}=\frac{FL}{lA}$ where $l$ is the extension of spring of original length L and A be the cross-sectional area and $F=mg$ , force applied.

now force constant $k=\frac{F}{l}=\frac{AY}{L}$
Time period $T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{mL}{AY}}$
thus, $\frac{T_1}{T_2}=\sqrt{\frac{Y_2}{Y_1}}=\sqrt{\frac{3}{2}}$