Question

A tuning fork gives 5 beats per second with 40 cm length of sonometer wire. If the length of the wire is shortened by 1 cm, the number of beats is still the same. The frequency of the fork is ______.

• A. 390 Hz
• B. 395 Hz
• C. 400 Hz
• D. 405 Hz

Hint:

When a tuning fork produces the exact same number of beats with two different wire lengths, its frequency is always mathematically bracketed between the two wire frequencies. The longer wire gives $(N - \text{beats})$ and the shorter wire gives $(N + \text{beats})$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A tuning fork of unknown frequency $N$ produces 5 beats/sec with two different lengths of the same sonometer wire. Shortening the wire increases its frequency.

Step 2: Key Formula or Approach:
The frequency of a sonometer wire is inversely proportional to its resonating length:
$$n \propto \frac{1}{l} \implies n_1 l_1 = n_2 l_2$$
Since shortening the wire increases its frequency ($n_2 > n_1$), and both produce 5 beats with the tuning fork $N$, the tuning fork's frequency must lie exactly halfway between the two wire frequencies.
Thus, $n_1 = N - 5$ and $n_2 = N + 5$.

Step 3: Detailed Explanation:
Given:
Initial length $l_1 = 40 \text{ cm}$. Frequency $n_1 = N - 5$.
Final length $l_2 = 40 - 1 = 39 \text{ cm}$. Frequency $n_2 = N + 5$.
Using the inverse relationship:
$$\frac{n_1}{n_2} = \frac{l_2}{l_1}$$
$$\frac{N - 5}{N + 5} = \frac{39}{40}$$
Cross-multiply to solve for $N$:
$$40(N - 5) = 39(N + 5)$$
$$40N - 200 = 39N + 195$$
Rearrange the terms:
$$40N - 39N = 195 + 200$$
$$N = 395 \text{ Hz}$$

Step 4: Final Answer:
The frequency of the tuning fork is 395 Hz, matching option (b).