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.

Question

Philoid · Accepted Answer

A. 10 Hz, 30 Hz, 50 Hz

B. 800 m, 267 m, 160 m

Mass per unit length,

T=256 N

We know that,

So,

For fundamental frequency, (in case of fixed at one end)

So, m (putting the value of L, from question)

So,

Now, in case of one end fixed and wave generation from one end, we only get odd no of overtones.

So, frequencies of 1^st and 2^nd overtones, are 3f and 5f, i.e. 30 Hz and 50 Hz (A)

(proved previously)

Now, for each overtone must be equal to velocity divided by frequency of that overtone.

We calculated that,

for 1st overtone, f ‘=30 Hz

for 2nd overtone, f ‘‘=50 Hz

Now, (for 1^st overtone)

Also, (for 2^nd overtone) (B)

A 200 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.