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. (a) Find the frequencies of the fundamental and the first two overtones. (b) Find the wavelength in the fundamental and the first two overtones.

Answer 1

L = length of rope = 2m,

M = mass = 80 gm

= 0.08 kg

mass per unit length = m

$=\frac{0.08}{2}=0.04kg/m$

Tension, T = 256 N

Velocity, $V=\sqrt{\left(\frac{T}{m}\right)}$

$=\sqrt{\left(\frac{25600}{4}\right)}=\frac{160}{2}=80m/s$

For fundamental frequency,

$I=\frac{\lambda }{4}$

$\Rightarrow \lambda =4I=4\times 2=8m$

$\Rightarrow f=\frac{80}{8}=10Hz$

(a) Therefore, the frequency of 1st two overtones are

1st overtone = 3f = 30 Hz

2nd overtone = 5f = 50 Hz

(b) $\lambda =4I=4\times 2=8m$

$\therefore {\lambda }_1=\frac{V}{f_1}=\frac{80}{30}=2.67m$

${\lambda }_2=\frac{V}{f_2}=\frac{80}{50}=1.6m$

So, the wavelengths are 8m, 2.67 m and 1.6 m respectively.