Question

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

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

Answer 1

The correct option is A

$\sqrt{2}:\sqrt{3}$

Force constant $k=\frac{F}{l}=\frac{YA}{L}$ where l is the extension in the spring of original length L and cross-sectional area A when a force F = Mg is applied.
Now, the time period of oscillation is given by $T=2\pi \sqrt{\frac{M}{k}}=2\pi \sqrt{\frac{ML}{YA}}$
$\therefore \frac{T_1}{T_2}=\sqrt{\frac{Y_2}{Y_1}}=\sqrt{\frac{2}{3}}$
Hence, the correct choice is (a).