Question 10: Waves

Question

When a string is divided into four segments of $$l_1, l_2, l_3$$ and $$l_4$$. The fundamental frequencies of these three segments are $$v_1, v_2, v_3$$ and $$v_4$$, respectively. The original fundamental frequency $$(v)$$ of the string is

Accepted Answer

The fundamental frequency of a string is inversely proportional to its length. So, if the string is divided into four segments of lengths $$l_1, l_2, l_3$$ and $$l_4$$, then the fundamental frequencies of these segments will be inversely proportional to their lengths.
Let the original length of the string be $$L$$. Then, we have:
$$l_1 + l_2 + l_3 + l_4 = L$$
The fundamental frequency of the original string is given by:
$$v = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$
where $$T$$ is the tension in the string and $$\mu$$ is the mass per unit length of the string.
The fundamental frequencies of the four segments are given by:
$$v_1 = \frac{1}{2l_1} \sqrt{\frac{T}{\mu}}$$
$$v_2 = \frac{1}{2l_2} \sqrt{\frac{T}{\mu}}$$
$$v_3 = \frac{1}{2l_3} \sqrt{\frac{T}{\mu}}$$
$$v_4 = \frac{1}{2l_4} \sqrt{\frac{T}{\mu}}$$
Adding these equations, we get:
$$v_1 + v_2 + v_3 + v_4 = \frac{1}{2} \sqrt{\frac{T}{\mu}} \left(\frac{1}{l_1} + \frac{1}{l_2} + \frac{1}{l_3} + \frac{1}{l_4} \right)$$
Now, using the equation $$l_1 + l_2 + l_3 + l_4 = L$$, we can write:
$$\frac{1}{l_1} + \frac{1}{l_2} + \frac{1}{l_3} + \frac{1}{l_4} = \frac{L}{l_1 l_2 l_3 l_4}$$
Substituting this into the previous equation, we get:
$$v_1 + v_2 + v_3 + v_4 = \frac{1}{2} \sqrt{\frac{T}{\mu}} \frac{L}{l_1 l_2 l_3 l_4}$$
Since $$v = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$, we can write:
$$v_1 + v_2 + v_3 + v_4 = \frac{vL}{l_1 l_2 l_3 l_4}$$
Therefore, the correct answer is:
$$\boxed{\frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} + \frac{1}{v_4}}$$
This is option
C
.

Answer

$$\sqrt{v}=\sqrt{v_1}+\sqrt{v_2}+\sqrt{v_3}+\sqrt{4}$$

Answer

$$v=v_1+v_2+v_3+v_4$$

Answer

$$1 / v=1 / v_1+1 / v_2+1 / v_3+1 / v_4$$

Answer

$$\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{v_1}}+\frac{1}{\sqrt{v_2}}+\frac{1}{\sqrt{v_3}}+\frac{1}{\sqrt{v_4}}$$

Waves

When a string is divided into four segments of $l_1, l_2, l_3 and l_4. The fundamental frequencies of these three segments are v_1, v_2, v_3 and v_4, respectively. The original fundamental frequency (v)$ of the string is

Choose the correct answer: