Physics · Waves · NEET
To √T, not T. The wave speed on a string is v = √(T/μ), so v ∝ √T. Since f = v/(2L), frequency also goes as √T. So if you double the tension, frequency rises by √2 (about 1.41 times), not 2 times. This is a very common NEET slip.
A thicker string of the same material has more mass per unit length, so μ is larger. Because f ∝ 1/√μ, a larger μ means a lower frequency. That is why the low-pitch strings on a guitar are the thick ones. If you compare two wires of the same material, μ ∝ r² (radius squared), so a wire of double the radius has 4 times the μ and half the frequency.
μ is the mass per unit length of the wire, in kg per metre. If a wire has mass m and length L, then μ = m/L. For a solid wire μ can also be written as μ = ρ·A = ρ·πr², where ρ is the material density, A is the cross-section area and r is the radius. Never plug in the full mass; always use mass per metre.
It goes down. f ∝ 1/L, so a longer vibrating segment gives a lower frequency. On a sonometer you move the bridges to change L. Halving L doubles the frequency. This is the law of length and is the easiest one to test in the lab because L is easy to measure.
The string is fixed at both ends, so it forms standing waves. The fundamental has one loop of length L = λ/2, giving λ = 2L. Using f = v/λ with v = √(T/μ) gives f = (1/2L)√(T/μ). Every law of vibrating strings is just this one equation with the other two quantities held constant.
In a guitar, two strings A and B of the same material are slightly out of tune and produce beats of frequency 6 Hz. When the tension in B is slightly decreased, the beat frequency increases to 7 Hz. If the frequency of A is 530 Hz, the original frequency of B is:
If the tension in a stretched string is doubled, the ratio of the initial speed to the final speed of a transverse wave along the string is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
f = (1/2L)√(T/μ), where L is the vibrating length in metres, T is the tension in newtons and μ is the linear mass density in kg/m. This is the frequency of the first harmonic (one loop).
Law of length: f ∝ 1/L (T and μ constant). Law of tension: f ∝ √T (L and μ constant). Law of mass: f ∝ 1/√μ (L and T constant). All three come from the single formula f = (1/2L)√(T/μ).
A sonometer is used to verify the laws of vibrating strings and to compare or find an unknown frequency of a tuning fork. By adjusting the vibrating length until the wire resonates with a fork (shown by a small paper rider flying off), you find L and then use the formula.
For a wire μ = ρπr², so doubling r makes μ four times larger. Since f ∝ 1/√μ, the frequency becomes 1/√4 = 1/2 of the original. So the note drops by one octave.
Dimensional and derivation reasons: tension provides the restoring force and μ provides the inertia. The wave speed comes out as v = √(T/μ), so speed grows with the square root of tension. This square root is the source of every 'root of T' answer in NEET string problems.
