The fundamental frequencies of two wires P and Q are 200 Hz and 335 Hz respectively. How many beats could be heard by 5th harmonic of P and third harmonic of Q in 2 s ?

Answer

To solve the problem, we need to determine how many beats can be heard between the 5th harmonic of wire P and the 3rd harmonic of wire Q in 2 seconds. Let's break it down step by step.

### Step 1: Identify the Fundamental Frequencies
The fundamental frequencies of the wires are given as:
- For wire P, \( f_P = 200 \, \text{Hz} \)
- For wire Q, \( f_Q = 335 \, \text{Hz} \)

### Step 2: Calculate the Harmonics
Next, we need to calculate the frequencies of the 5th harmonic of wire P and the 3rd harmonic of wire Q.

- The frequency of the 5th harmonic of wire P is given by:
\[ f_{5P} = 5 \times f_P = 5 \times 200 \, \text{Hz} = 1000 \, \text{Hz} \]

- The frequency of the 3rd harmonic of wire Q is given by:
\[ f_{3Q} = 3 \times f_Q = 3 \times 335 \, \text{Hz} = 1005 \, \text{Hz} \]

### Step 3: Calculate the Beat Frequency
The beat frequency is the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_{3Q} - f_{5P}| = |1005 \, \text{Hz} - 1000 \, \text{Hz}| = 5 \, \text{Hz} \]

### Step 4: Calculate the Number of Beats in 2 Seconds
The number of beats heard in a given time can be calculated by multiplying the beat frequency by the time duration. Given that the beat frequency is 5 Hz (5 beats per second), we can find the total beats in 2 seconds:
\[ \text{Total beats} = f_{\text{beat}} \times \text{time} = 5 \, \text{Hz} \times 2 \, \text{s} = 10 \, \text{beats} \]

### Final Answer
Thus, the total number of beats that could be heard in 2 seconds is **10 beats**.

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