Question

A 2.00 m-long rope, having a mass of 80 g, is fixed at one end and is tied to a light string at the other end. The tension in the string is 256 N. Find the frequencies of the fundamental and the first two overtones.

• A. 20 Hz, 40 Hz, 80 Hz
• B. 10 Hz, 20 Hz, 30 Hz
• C. 10 Hz, 30 Hz, 50 Hz
• D. None of these

Answer 1

The correct option is C

10 Hz, 30 Hz, 50 Hz

Since one end is tied to a light string on the other side. At this point an antinode will be formed and a node will be formed at the fixed end.

given $m=80g=0.08kg$

$L=2m$

$\mu =\frac{m}{L}=\frac{0.08}{2}=0.04\frac{kg}{m}$

tension,$T=256$

$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{256}{0.04}}=\frac{16}{0.2}=80m{s}^{-1}$

fundamental frequency is given by $\frac{v}{4L}$

$f_0=\frac{80}{4\times 2}=10Hz$

The first overtone is the 3rd harmonic and is given by 30 Hz.

The second overtone is the 5th harmonic and is given by 50 Hz