Question 86: Waves

Question

The fundamental frequency of vibration of a string stretched between two rigid support is $50 \mathrm{~Hz}$. The mass of the string is $18 \mathrm{~g}$ and its linear mass density is $20 \mathrm{~g} / \mathrm{m}$. The speed of the transverse waves so produced in the string is ___________ $\mathrm{ms}^{-1}$

Accepted Answer

To find the speed of the transverse waves produced in the string, we can use the formula for the fundamental frequency of a vibrating string:
$f = \frac{1}{2L} \cdot v$
where $f$ is the fundamental frequency, $L$ is the length of the string, and $v$ is the speed of the transverse waves.
First, we are given the mass of the string ($m = 18g$) and the linear mass density ($\mu = 20g/m$). We can find the length of the string by dividing the mass by the linear mass density:
$L = \frac{m}{\mu} = \frac{18g}{20g/m} = 0.9m$
Now we can plug in the values for the fundamental frequency ($f = 50Hz$) and the length of the string ($L = 0.9m$) into the formula:
$50Hz = \frac{1}{2(0.9m)} \cdot v$
To isolate $v$, we multiply both sides by $2(0.9m)$:
$v = 50Hz \cdot 2(0.9m) = 90 \mathrm{ms}^{-1}$
The speed of the transverse waves produced in the string is $90 ~\mathrm{ms}^{-1}$.
Alternate Method:
To find the speed of the transverse waves produced in the string, we can use the formula for the fundamental frequency of a vibrating string:
$f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$
where $f_1$ is the fundamental frequency, $L$ is the length of the string, $T$ is the tension in the string, and $\mu$ is the linear mass density of the string.
We're given that the fundamental frequency $f_1 = 50 ,	ext{Hz}$, the mass of the string $m = 18 ,	ext{g}$, and the linear mass density $\mu = 20 ,	ext{g/m}$. To find the speed of the transverse waves, we need to find the tension $T$ and the length $L$ of the string.
First, let's find the length $L$ of the string using the mass and linear mass density:
$L = \frac{m}{\mu} = \frac{18 ,	ext{g}}{20 ,	ext{g/m}} = 0.9 ~	ext{m}$
Now, we can rearrange the formula for the fundamental frequency to solve for the tension $T$:
$T = \mu \left(\frac{2Lf_1}{1}ight)^2$
Substitute the known values:
$T = 20 ,	ext{g/m} \cdot \left(\frac{2 \cdot 0.9 ~	ext{m} \cdot 50 ~	ext{Hz}}{1}ight)^2$
$T = 20 ,	ext{g/m} \cdot (90 ~	ext{m/s})^2$
$T = 20 ,	ext{g/m} \cdot 8100 ~	ext{m}^2/	ext{s}^2$
$T = 162000 ,	ext{g m}/	ext{s}^2$
Now, we can find the speed of the transverse waves $v$ using the formula:
$v = \sqrt{\frac{T}{\mu}}$
Substitute the known values:
$v = \sqrt{\frac{162000}{20}}$
$v = \sqrt{8100} = 90~ 	ext{m/s}$
The speed of the transverse waves produced in the string is $90 ~	ext{m/s}$.

Waves

The fundamental frequency of vibration of a string stretched between two rigid support is 50 \mathrm{~Hz}. The mass of the string is 18 \mathrm{~g} and its linear mass density is 20 \mathrm{~g} / \mathrm{m}. The speed of the transverse waves so produced in the string is ___________ \mathrm{ms}^{-1}

Choose the correct answer: