A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz. The length of the string producing a fundamental frequency of 180 Hz will be _________ cm.

Solution

Physics sir · 3 steps

Same string, same tension: the fundamental frequency goes as one over the length, so set up a ratio.

1. Fundamental mode. A string fixed at both ends vibrates in its fundamental with nodes only at the ends, so half a wavelength fits in the length: λ = 2L. With wave speed v, the frequency is f = v/2L.
2. Speed stays fixed. The tension and the mass per unit length are unchanged, so v = √(T/μ) is the same. That means f ∝ 1/L.

   L₂/L₁ = f₁/f₂ ⇒ L₂ = 90 × 120/180

3. So. L₂ = 90 × 2/3 = 60 cm. A higher note needs a shorter string, so the answer is 60.

The common slip: 135. Treating frequency as proportional to length gives 90 × 180/120 = 135 cm, but a higher frequency needs a shorter string.

The idea to keep. For a string under fixed tension, f = v/2L with v constant, so frequency and length are inversely related. Check your answer: a higher pitch must come from a shorter string.

Answer from the NTA final answer key. Solution by PrepWiser.