Question

The frequency of sonometer wire is \(f\), but when the weights producing the tensions are completely immersed in water, the frequency becomes \(f/2\) and on immersing the weights in a certain liquid, the frequency becomes \(f/3\). The specific gravity of the liquid is

• A. 4/3
• B. 16/9
• C. 15/12
• D. 32/27

Hint:

Since $f \propto \sqrt{T}$, a change in frequency ratio directly gives the tension ratio, which then gives information about the buoyant force.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Frequency \(\propto \sqrt{T}\) for a sonometer. When weights are immersed, apparent weight (tension) decreases by buoyancy.

Step 2: Detailed Explanation:
\(f \propto \sqrt{T}\). In water: \(f/2 = f\sqrt{T_w/T}\), so \(T_w = T/4\). Thus upthrust in water = \(3T/4\), giving density ratio \(\rho/\rho_w = T/(3T/4) = 4/3\). In liquid with specific gravity \(S\): \(f/3 = f\sqrt{T_L/T}\), so \(T_L = T/9\). Upthrust in liquid = \(8T/9 = V\rho_L g\). Since upthrust in water = \(3T/4 = V\rho_w g\): \[ S = \frac{\rho_L}{\rho_w} = \frac{8T/9}{3T/4} = \frac{8}{9}\times\frac{4}{3} = \frac{32}{27} \]

Step 3: Final Answer:
Specific gravity of liquid \(= \dfrac{32}{27}\).