Question

Two springs 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's modulus $Y=\frac{F}{A}.\frac{L}{\Delta l}$
where $\Delta l$ is the extension in the spring of original length $L$ and cross-sectional area $A$ when a force $F=Mg$ is applied.

Force constant $k=\frac{F}{\Delta l}=\frac{YA}{L}$
Now, the time period of vertical oscillations is given by
$T=2\pi \sqrt{\frac{M}{k}}=2\pi \sqrt{\frac{ML}{YA}}$
$\therefore$ $\frac{T_1}{T_2}=\sqrt{\frac{Y_2}{Y_1}}=\sqrt{\frac{3}{2}}$
Hence, the correct choice is (a)