Question

The ratio of frequencies of oscillations of two simple pendulums is \( 3:4 \). Their lengths are in the ratio

• A. \( 16 : 9 \)
• B. \( 9 : 16 \)
• C. \( \sqrt{3} : \sqrt{4} \)
• D. \( \sqrt{4} : \sqrt{3} \)

Hint:

Frequency of a simple pendulum varies inversely as square root of length.

Answer 1

The Correct Option is A

Step 1: Write frequency formula for simple pendulum.
\[ f = \frac{1}{2\pi}\sqrt{\frac{g}{L}} \]
Step 2: Write ratio of frequencies.
\[ \frac{f_1}{f_2} = \sqrt{\frac{L_2}{L_1}} \]
Step 3: Substitute given ratio.
\[ \frac{3}{4} = \sqrt{\frac{L_2}{L_1}} \]
Step 4: Square both sides.
\[ \frac{9}{16} = \frac{L_2}{L_1} \Rightarrow \frac{L_1}{L_2} = \frac{16}{9} \]
Step 5: Conclusion.
The ratio of lengths is \( 16 : 9 \).