Question

Two springs of equal lengths and equal cross – sectional areas are made of material 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 $Y=\frac{F}{A}.\frac{L}{l}$
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 vertical oscillations is given by
$T=2\pi \sqrt{\frac{M}{k}}=2\pi \sqrt{\frac{ML}{YA}}\therefore \frac{T_1}{T_2}=\sqrt{\frac{Y_2}{Y_1}}=\sqrt{\frac{3}{2}}$