Question

A rope attached at both ends has consecutive standing wave modes for which the approximate distances between two adjacent nodes are 215 mm and 184 mm, respectively.

(a) What is the length of the rope?
(b) If the tensile modulus is 12 N and the linear mass density is 4 g/m, what is the fundamental frequency?

Answer 1

The length of the rope is approximately 0.798 m, and the fundamental frequency is 4 Hz.

Step-by-step explanation:

To solve this problem, we can use the formula:

Speed of waves = frequency × wavelength

Given the approximate distances between two adjacent nodes, we can calculate the wavelengths:

1. Wavelength for the first mode = 2 × 215 mm = 430 mm = 0.43 m
2. Wavelength for the second mode = 2 × 184 mm = 368 mm = 0.368 m

Since the length of the rope is the sum of the distances between the two adjacent nodes, we can calculate the length:

Total length of the rope = 0.43 m + 0.368 m = 0.798 m

To find the fundamental frequency, we can rearrange the formula to:

Frequency = Speed of waves / wavelength

Given the tensile modulus (12 N) and linear mass density (4 g/m), we can calculate the speed of waves:

Speed of waves = √(tensile modulus / linear mass density) = √(12 N / 4 g/m) = √(3 N/m) = 1.73 m/s

Using the speed of waves and the wavelength we calculated earlier, we can find the fundamental frequency:

Fundamental frequency = 1.73 m/s / 0.43 m = 4 Hz.