SATHEE: Wave Motion 2 Question 4

Wave Motion 2 Question 4

4. A wire of length $2 L$ , is made by joining two wires $A$ and $B$ of same length but different radii $r$ and $2 r$

and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire $A$ is $p$ and that in $B$ is $q$ , then the ratio $p : q$ is

(2019 Main, 8 April I)

(a) $3 : 5$

(b) $4 : 9$

(d) $1 : 4$

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Answer:

Correct Answer: 4. (c)

Solution:

Let mass per unit length of wires are $μ_{A}$ and $μ_{B}$ , respectively.

$∵$ For same material, density is also same.

So, $μ_{A} = \frac{ρ π r^{2} L}{L} = μ$ and $μ_{B} = \frac{ρ 4 π r^{2} L}{L} = 4 μ$

Tension $(T)$ in both connected wires are same.

So, speed of wave in wires are

$\begin{aligned} v_{A} & = \sqrt{\frac{T}{μ_{A}}} = \sqrt{\frac{T}{μ}} [∵ μ_{A} = μ and μ_{B} = 4 μ] \\ v_{B} & = \sqrt{\frac{T}{μ_{B}}} = \sqrt{\frac{T}{4 μ}} \end{aligned}$

So, $n$ th harmonic in such wires system is

$\begin{matrix} f_{n t h} = \frac{p v}{2 L} \\ \Rightarrow f_{A} = \frac{p v_{A}}{2 L} = \frac{p}{2 L} \sqrt{\frac{T}{μ}} \end{matrix}$

(for $p$ antinodes)

Similarly, $f_{B} = \frac{q v_{B}}{2 L} = \frac{q}{2 L} \sqrt{\frac{T}{4 μ}} = \frac{1}{2} (\frac{q}{2 L} \sqrt{\frac{T}{μ}})$

$(for q antinodes)$

As frequencies $f_{A}$ and $f_{B}$ are given equal.

So, $f_{A} = f_{B} \Rightarrow \frac{p}{2 L} \sqrt{\frac{T}{μ}} = \frac{q}{2} [\frac{1}{2 L} \sqrt{\frac{T}{μ}}]$

$\frac{p}{q} = \frac{1}{2} \Rightarrow p : q = 1 : 2$

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Wave Motion 2 Question 4

4. A wire of length $2 L$ , is made by joining two wires $A$ and $B$ of same length but different radii $r$ and $2 r$

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

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अन्य संसाधन

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नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Wave Motion 2 Question 4

4. A wire of length 2L, is made by joining two wires A and B of same length but different radii r and 2r

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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4. A wire of length $2 L$ , is made by joining two wires $A$ and $B$ of same length but different radii $r$ and $2 r$