Question 338: Oscillations

Question

'$$n$$' waves are produced on a string in 1 second. When the radius of the string is doubled, keeping tension same, the number of waves produced in 1 second for the same harmonic will be

Accepted Answer

The frequency of a wave on a string depends on the tension in the string and the linear mass density (mass per unit length) of the string. The formula for the frequency, when we consider the tension $T$ and the linear mass density $\mu$, is given by:
$$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$
where:
$f$ is the frequency of the wave,
$L$ is the length of the vibrating segment of the string,
$T$ is the tension in the string,
$\mu$ is the linear mass density of the string ($\mu = \frac{m}{L}$, with $m$ being the mass and $L$ the length).
When the radius of the string is doubled, its cross-sectional area becomes four times larger (since the area of a circle is $A = \pi r^2$, doubling $r$ increases $A$ by a factor of 4). This implies that the mass of a given length of the string also becomes four times larger, and therefore, the linear mass density ($\mu$) increases by a factor of 4, because the length of the string we are examining hasn't changed.
Now, taking this into account, the new linear mass density is $4\mu$, and substituting this into the frequency formula gives us:
$$f_{new} = \frac{1}{2L} \sqrt{\frac{T}{4\mu}} = \frac{1}{2L} \cdot \frac{1}{2} \sqrt{\frac{T}{\mu}} = \frac{1}{2} f$$
This shows that the frequency of the wave, and therefore the number of waves produced per second (since frequency and the number of waves per second are equivalent), will be halved if the radius of the string is doubled while keeping the tension the same. The correct answer is thus:
Option B $$\frac{\mathrm{n}}{2}$$

Answer

$$2 \mathrm{n}$$

Answer

$$\frac{\mathrm{n}}{2}$$

Answer

$$\frac{\mathrm{n}}{\sqrt{2}}$$

Answer

$$\sqrt{2} \mathrm{n}$$

Oscillations

'$n$' waves are produced on a string in 1 second. When the radius of the string is doubled, keeping tension same, the number of waves produced in 1 second for the same harmonic will be

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