   Chapter 14, Problem 48P

Chapter
Section
Textbook Problem

A standing wave is set up in a string of variable length and tension by a vibrator of variable frequency. Both ends of the string are fixed. When the vibrator has a frequency fA, in a string of length LA and under tension TA, nA antinodes are set up in the string. (a) Write an expression for the frequency fA of a standing wave in terms of the number nA, length LA, tension TA, and linear density μA. (b) If the length of the string is doubled to LB = 2LA, what frequency fB (written as a multiple of fA) will result in the same number of antinodes? Assume the tension and linear density are unchanged. Hint: Make a ratio of expressions for fB and fA. (c) If the frequency and length are held constant, what tension TB will produce nA + 1 antinodes? (d) If the frequency is tripled and the length of the string is halved, by what factor should the tension be changed so that twice as many antinodes are produced?

(a)

To determine
The expression for the frequency of standing wave.

Explanation

Given Info: Standing wave is setup in a string with both ends fixed.

Formula to calculate wave length of a wave produced in a string with both ends fixed is,

λA=2LAnA

• λA is wave length of a wave produced in a string with both ends fixed.
• LA is the length of the string.
• nA is the number of antinodes.

For a string closed at both ends and the string vibrating at fundamental frequency one anti node is present at the centre of the standing wave and two nodes at the fixed position. So, the length of the string corresponds to half the wave length λ/2 of the standing wave. For each increase in antinode, half of the wave length of the wave is increased. For example, 1 antinode is present in λ/2 , 2 antinodes are present in λ/2+λ/2=2(λ/2) , 3 antinodes are present it λ/2+λ/2+λ/2=3(λ/2) and so on.

Formula to calculate the speed of the wave in a string is,

vA=TAμA

• vA is velocity of the wave in a string.
• TA is the tension in the string

(b)

To determine
The expression for the frequency of standing wave with length of the string doubled.

(c)

To determine
The tension that will produce nA+1 antinodes.

(d)

To determine
Factor the tension should be changed to produce twice as many anti-nodes.

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