Question

A sonometer wire resonates with 4 antinodes between two bridges for a given tuning fork, when 1 kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates producing 2 antinodes between the two bridges (distance between two bridges is as before). The value of M is

• A. 2.5 kg
• B. 3.5 kg
• C. 4 kg
• D. 1 kg

Hint:

The number of loops is inversely proportional to the square root of tension ($p \propto 1/\sqrt{T}$). Halving the number of loops ($4 \rightarrow 2$) requires the internal tension to increase by a factor of $2^2 = 4$. Since the initial mass was 1 kg, the new mass must be $1 \times 4 = 4\ \text{kg}$ immediately!

Answer 1

The Correct Option is C

Step 1: Recall the sonometer rule.
With frequency and length fixed, the number of loops $p$ satisfies $p\sqrt T=$ constant, so $p^2T$ is constant.

Step 2: Set up the two states.
First: $p_1=4$, $T_1=1$ kg. Second: $p_2=2$, $T_2=M$. So $p_1^2T_1=p_2^2T_2$.

Step 3: Solve for $M$.
\[ M=T_1\left(\frac{p_1}{p_2}\right)^2=1\cdot\left(\frac{4}{2}\right)^2=4 \]
\[ \boxed{4\text{ kg}} \]