Student observed ten antinodes on a string of length 1.70 m under tension produced by a mass of 0.22 kg. The string linear density is 1.3 gram/meter. What is the difference between the transverse wave velocity calculated from the string material properties and the one following from the standing wave configuration?
To find the difference between the transverse wave velocity calculated from the string material properties and the one following from the standing wave configuration, we need to calculate the transverse wave velocity using both approaches and then find their difference.
1. Transverse wave velocity calculated from the string material properties:
The transverse wave velocity (v) of a wave on a string can be calculated using the formula: v = √(T/μ), where T is the tension in the string and μ is the linear density of the string.
Given:
Tension (T) = mass * acceleration due to gravity (g) = 0.22 kg * 9.8 m/s^2 = 2.156 N
Linear density (μ) = 1.3 grams/meter = 0.0013 kg/m
Using the formula, we can calculate the transverse wave velocity:
v1 = √(T/μ) = √(2.156 N / 0.0013 kg/m) = √1663.08 m^2/s ≈ 40.8 m/s
2. Transverse wave velocity following from the standing wave configuration:
The formula for the transverse wave velocity in a standing wave configuration depends on the length of the string (L) and the frequency of the standing wave (f). The relationship is given by: v2 = 2Lf.
Given:
Length of the string (L) = 1.70 m
Number of antinodes (n) = 10
To find the frequency (f), we need to know the number of complete waves (λ) that fit in the length of the string. In a standing wave, there is 1 complete wave between two consecutive antinodes. So, the total number of complete waves in the string will be half the number of antinodes:
Number of complete waves (λ) = n/2 = 10/2 = 5
The wavelength of a wave is given by: λ = 2L/f. Rearranging the formula, we can solve for the frequency (f):
f = 2L / λ = 2L / (n/2) = (2L*n) / n = 2Ln / n = 2L
Using the formula, we can calculate the transverse wave velocity:
v2 = 2Lf = 2 * 1.70 m * 2 = 6.8 m/s
Now, we can find the difference between the two velocities:
Difference = v1 - v2 = 40.8 m/s - 6.8 m/s = 34 m/s
Therefore, the difference between the transverse wave velocity calculated from the string material properties and the one following from the standing wave configuration is 34 m/s.