Three resonant frequencies of a string are observed to be 90 Hz ,...

For a string fixed at both ends, the resonant frequencies are given by:

$f_n = n \dfrac{v}{2L}$

where $n = 1, 2, 3, \dots$ and $v$ is the speed of the transverse wave.

The given observed resonant frequencies are $90 \text{ Hz}$, $150 \text{ Hz}$, and $210 \text{ Hz}$. These frequencies must be integer multiples of the fundamental frequency $f_1$.

To find the fundamental frequency, we take the highest common factor (HCF) of the given frequencies:

$f_1 = \text{HCF}(90, 150, 210) = 30 \text{ Hz}$

The observed frequencies correspond to the 3rd, 5th, and 7th harmonics ($3f_1, 5f_1, 7f_1$), which are typically the modes excited when a string is plucked at its center (where even harmonics are absent).

Using the formula for the fundamental frequency:

$f_1 = \dfrac{v}{2L}$

Substituting the given values ($f_1 = 30 \text{ Hz}$ and $L = 80 \text{ cm} = 0.8 \text{ m}$):

$30 = \dfrac{v}{2 \times 0.8}$

$30 = \dfrac{v}{1.6}$

$v = 30 \times 1.6 = 48 \text{ m s}^{-1}$

Answer: $48 \text{ m s}^{-1}$