/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 123 For a particular mode of vibrati... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 123

For a particular mode of vibration of string, the distance between two consecutive nodes is \(18 \mathrm{~cm}\). For the next higher mode, the distance becomes \(16 \mathrm{~cm}\). The length of the string is (A) \(18 \mathrm{~cm}\) (B) \(16 \mathrm{~cm}\) (C) \(144 \mathrm{~cm}\) (D) \(72 \mathrm{~cm}\)

Short Answer

Expert verified
The length of the string is 144 cm, so the correct answer is (C) 144 cm.

Step by step solution

01

Derive the relationship between consecutive distances and length of the string

If we consider the nth mode, the length of the string L is given by: L = (n / 2) 位_n , where 位_n is the wavelength of the nth mode of vibration. Similarly, for the (n+1)th mode: L = ((n + 1) / 2) 位_(n+1). Since we are given the distances between consecutive nodes for two different modes, we can set up an equation relating the string length (L) and the distance between the nodes for each mode.
02

Set up the equation based on given distances

The distance between two consecutive nodes for the nth mode is given as 18 cm, which is equal to half the wavelength (位_n): 位_n = 36 cm The distance between two consecutive nodes for the (n+1)th mode is given as 16 cm, which is equal to half the wavelength (位_(n+1)): 位_(n+1) = 32 cm Now, plug in the values of 位_n and 位_(n+1) into the equations for the string length L: L = (n / 2) (36 cm) L = ((n + 1) / 2) (32 cm)
03

Solve for n

Now we have two equations with the same unknown, L. We can set them equal to each other and solve for n: ( n / 2) (36 cm) = ((n + 1) / 2) (32 cm) To eliminate the fractions, multiply both sides by 2: n (36 cm) = (n + 1) (32 cm) Distribute the expressions: 36n = 32n + 32 cm Subtract 32n from both sides: 4n = 32 cm Divide by 4: n = 8
04

Find the length of the string (L)

Now that we have the value of n, we can plug it back into one of the equations to find the length of the string, L. We can use the first equation: L = (n / 2) (36 cm) Plug in the value of n: L = (8 / 2) (36 cm) L = 4 (36 cm) Multiply to get the final answer: L = 144 cm The length of the string is 144 cm, so the correct answer is (C) 144 cm.

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