Question

The frequencies of three tuning forks A, B and C are related as $n_A > n_B > n_C$. When the forks A and B are sounded together, the number of beats produced per second is '$n_1$'. When forks A and C are sounded together the number of beats produced per second is '$n_2$'. How many beats are produced per second when forks B and C are sounded together?

• A. $n_1 - n_2$
• B. $\frac{n_1 + n_2}{2}$
• C. $n_2 - n_1$
• D. $n_1 + n_2$

Hint:

Think of frequencies as positions on a straight line! Since $A$ is the highest, $B$ sits below it at distance $n_1$, and $C$ sits even further below at distance $n_2$. The gap between $B$ and $C$ is simply the total distance minus the first segment: $n_2 - n_1$!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents three tuning forks with a known frequency order, $n_A > n_B > n_C$. We are given the beat frequencies produced when fork A is sounded with B, and when A is sounded with C. We need to find the beat frequency produced when B and C are sounded together.

Step 2: Key Formula or Approach:
The beat frequency between two sounding sources is equal to the absolute difference between their individual frequencies: $$ \text{Beat Frequency} = |f_1 - f_2| $$

Step 3: Detailed Explanation:
Given the frequency ordering $n_A > n_B > n_C$, we can set up the individual beat frequency equations without absolute value ambiguity:

• When A and B are sounded together: $$ n_A - n_B = n_1 \quad \text{--- (Equation 1)} $$

• When A and C are sounded together: $$ n_A - n_C = n_2 \quad \text{--- (Equation 2)} $$
We need to find the beat frequency when B and C are sounded together, which is $(n_B - n_C)$. Let's subtract Equation 1 from Equation 2: $$ (n_A - n_C) - (n_A - n_B) = n_2 - n_1 $$ The $n_A$ terms cancel out perfectly: $$ -n_C + n_B = n_2 - n_1 $$ $$ n_B - n_C = n_2 - n_1 $$

Step 4: Final Answer:
The number of beats produced per second when B and C are sounded together is $n_2 - n_1$, which corresponds to option (C).