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Question:

A uniform string is vibrating with a fundamental frequency 'f'. The new frequency, if radius and length both are doubled would be:

2f

f/4

f/3

3f

Solution:

Wavelength, λ=2L
Speed of wave, v=√(T/m)
v=fλ ⇒ f ∝ 1/√(m/L)
If the radius is doubled, then the mass becomes 8 times the initial mass (since volume is increased to 8 times).
If length is doubled, L2 = 2L1 ⇒ f2 = f/4