A piano wire 50 cm long has a total mass of 10 g and its stretched with a tension of 800 N. Find the frequency…

Analyze the question: A stretched string fixed at both ends. We need the frequency of the third overtone, so we find the wave speed, then the fundamental frequency, then the overtone.

Concept:

• Speed of a wave on a string: v = √(T/μ), where μ = mass per unit length
• Fundamental frequency: f₁ = v/(2L)
• Overtones: first overtone = 2f₁, second = 3f₁, third overtone = 4f₁

Solution:

1. Convert units: • L = 50 cm = 0.5 m • m = 10 g = 0.01 kg • μ = 0.01 / 0.5 = 0.02 kg/m

2. Wave speed: v = √(800 / 0.02) = √40000 = 200 m/s

3. Fundamental frequency: f₁ = v/(2L) = 200 / (2 × 0.5) = 200 Hz

4. Third overtone = 4th harmonic: f₄ = 4 × 200 = 800 Hz

Answer: The third overtone is 800 Hz, which is option A.

Tip: For a string (or an open pipe), the nth overtone is the (n+1)th harmonic. Many students lose marks by confusing "third overtone" with "third harmonic". If it says overtone, add 1 to get the harmonic number. The 200 Hz in option D is the trap for those who stop at the fundamental.