Question 428: Waves

Question

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $$9 \mathrm{~kg}$$ is suspended from the wire. When this mass is replaced by a mass $$\mathrm{M}$$, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of '$$M$$' is

Accepted Answer

To solve this problem, we need to use the relationship between the frequency of standing waves on a string and the tension in the string. The formula for the fundamental frequency of a stretched string is given by:
$$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$
where:
$$f$$ is the frequency of the standing wave
$$L$$ is the length of the string
$$T$$ is the tension in the string
$$\mu$$ is the linear mass density of the string
The tension $$T$$ in the wire due to a suspended mass $$m$$ is given by:
$$T = mg$$
where $$g$$ is the acceleration due to gravity.
In the first case, the wire resonates with five antinodes. For a wire with $$n$$ antinodes, the length of the wire $$L$$ is an integer multiple of half the wavelength $$\lambda / 2$$:
$$L = \frac{n \lambda}{2}$$
Thus, if there are five antinodes (n=5):
$$L = \frac{5\lambda}{2}$$
Similarly, for three antinodes (n=3):
$$L = \frac{3\lambda}{2}$$
Since the frequency is the same in both cases, we equate the frequencies using the formula for the fundamental frequency:
$$f_1 = \frac{5}{2L} \sqrt{\frac{T_1}{\mu}}$$
$$f_2 = \frac{3}{2L} \sqrt{\frac{T_2}{\mu}}$$
But since $$f$$ remains constant and the length $$L$$ of the wire remains constant between the two cases, we have:
$$\frac{5}{2L} \sqrt{\frac{T_1}{\mu}} = \frac{3}{2L} \sqrt{\frac{T_2}{\mu}}$$
Simplifying this, we get:
$$5 \sqrt{T_1} = 3 \sqrt{T_2}$$
Square both sides:
$$25 T_1 = 9 T_2$$
Recall that the tension $$T$$ is given by $$T = mg$$:
$$25 \cdot 9g = 9 \cdot Mg$$
Cancel out $$g$$:
$$225 = 9M$$
Solve for $$M$$:
$$M = 25 \mathrm{~kg}$$
Therefore, the value of '$$M$$' is $$\boxed{25 \ \mathrm{kg}}$$.

Answer

$$5 \mathrm{~kg}$$

Answer

$$12.5 \mathrm{~kg}$$

Answer

$$\frac{1}{25} \mathrm{~kg}$$

Answer

$$25 \mathrm{~kg}$$

Waves

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