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Chapter 10: Problem 1519

The equation \(\mathrm{y}=10 \sin (\pi \mathrm{x} / 4) \cos 10 \pi t\) represents a stationary wave where \(\mathrm{x}\) and \(\mathrm{y}\) are in centimeter and \(\mathrm{t}\) is in seconds. The separation between two consecutive nodes is (A) \(2 \mathrm{~cm}\) (B) \(4 \mathrm{~cm}\) (C) \(5 \mathrm{~cm}\) (D) \(8 \mathrm{~cm}\) Copyright \(\odot\) StemEZ.com. All rights reserved.

Short Answer

Expert verified
The separation between two consecutive nodes is 4 cm. Therefore, the correct answer is (B) 4 cm.

Step by step solution

01

Identify the spatial part of the wave equation

In the given equation, y = 10 * sin(πx/4) * cos(10πt), the function of x is sin(πx/4). We will focus on this part to find the distance between nodes.
02

Set the spatial function equal to zero

To find the nodes, we set the displacement y to zero. This means we need to find the values of x for which sin(πx/4) = 0. We will find two consecutive x values (x1 and x2) such that sin(πx1/4) = 0 and sin(πx2/4) = 0.
03

Solve for the consecutive x-values

Since sin(πx/4) = 0: \( \mathrm{n} \pi = \frac{\pi x}{4} \) for any integer n. Solving for x, we get: \( \mathrm{x} = 4\mathrm{n} \) Now we need to find two consecutive x values for which sin(πx/4) = 0. For n=1, x1 = 4 For n=2, x2 = 8
04

Calculate the separation between consecutive nodes

The distance between two consecutive nodes is the absolute difference between their x values. \( d = \vert x_2 - x_1 \vert \) Substitute x1 = 4 and x2 = 8: \( d = \vert 8 - 4 \vert \) \( d = 4 \mathrm{~cm} \) So, the separation between two consecutive nodes is 4 cm. Therefore, the correct answer is (B) 4 cm.

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