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Chapter 9: Problem 90

The length of a sonometer wire \(A B\) is \(110 \mathrm{~cm}\). Where should the two bridges be placed from \(A\) to divide the wire in three segments whose fundamental frequencies are in the ratio of \(1: 2: 3\) ? (A) \(30 \mathrm{~cm}, 90 \mathrm{~cm}\) (B) \(60 \mathrm{~cm}, 90 \mathrm{~cm}\) (C) \(40 \mathrm{~cm}, 70 \mathrm{~cm}\) (D) None of these

Short Answer

Expert verified
The positions of the bridges should be at \(60 \mathrm{~cm}\) and \(90 \mathrm{~cm}\) from A, which corresponds to option (B).

Step by step solution

01

Recall the formula for the fundamental frequency of a sonometer wire.

The fundamental frequency of a wire is given by the formula: \[f = \frac{v}{2L}\] where \(f\) is the frequency, \(v\) is the speed of the wave in the wire, and \(L\) is the length of the wire.
02

Understand the given frequency ratio.

If the fundamental frequencies are in the ratio of \(1:2:3\), we can write them as \(f_1 = k\), \(f_2 = 2k\), and \(f_3 = 3k\) for some constant \(k\).
03

Recognize that the total length of the wire is 110 cm.

We know that the lengths of the three segments of the wire must add up to 110 cm, i.e., \(L_1 + L_2 + L_3 = 110\).
04

Determine the relationship between segment lengths and frequencies.

Using the formula for the fundamental frequency of a sonometer wire from Step 1, we can write the lengths of the segments as follows: \[L_1 = \frac{v}{2f_1} = \frac{v}{2k}\] \[L_2 = \frac{v}{2f_2} = \frac{v}{4k}\] \[L_3 = \frac{v}{2f_3} = \frac{v}{6k}\]
05

Express the total length of the wire in terms of the constant k.

Using the expressions for the lengths of the segments from Step 4, we can write the total length of the wire as: \[L_1 + L_2 + L_3 = \frac{v}{2k} + \frac{v}{4k} + \frac{v}{6k}\]
06

Solve for the constant k using the total length of the wire.

Substitute the total length of the wire (110 cm) into the expression for the sum of the segment lengths: \[110 = \frac{v}{2k} + \frac{v}{4k} + \frac{v}{6k}\] Multiplying both sides by \(12k\): \[1320k = 6v + 3v + 2v\] Simplify and divide by 11: \[120k = 11v\] So, \(k = \frac{11v}{120}\)
07

Calculate the positions of the bridges.

Substitute the value of \(k\) back into the expressions for \(L_1\) and \(L_2\), and then find the positions of the bridges by adding the lengths of the segments. \(L_1 = \frac{v}{2k} = \frac{v}{2 \times \frac{11v}{120}} = 60 \mathrm{~cm}\) \(L_2 = \frac{v}{4k} = \frac{v}{4 \times \frac{11v}{120}} = 30 \mathrm{~cm}\) The first bridge should be placed at \(60 \mathrm{~cm}\) from A and the second bridge should be placed at \(60 + 30 = 90 \mathrm{~cm}\) from A. This corresponds to option (B).

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Most popular questions from this chapter

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