Question

The ratio of the frequencies of two simple pendulums is 4 : 3 at the same place. The ratio of their respective lengths is ______.

• A. 3 : 4
• B. 4 : 3
• C. 9 : 16
• D. 16 : 9

Hint:

Frequency is inversely proportional to the square root of length. So, if a pendulum swings FASTER (higher frequency), it must be SHORTER. Pendulum 1 is faster (4:3), so it must be shorter (9:16).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The frequency $n$ of a simple pendulum is inversely proportional to the square root of its length: $n = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$.

Step 2: Formula Application:
$n \propto \frac{1}{\sqrt{L}} \implies L \propto \frac{1}{n^2}$.

Step 3: Explanation:
Given $\frac{n_1}{n_2} = \frac{4}{3}$. $\frac{L_1}{L_2} = \left( \frac{n_2}{n_1} \right)^2 = \left( \frac{3}{4} \right)^2 = \frac{9}{16}$.

Step 4: Final Answer:
The ratio of their lengths is 9 : 16.