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Answer :
Final answer:
To find the difference between the transverse wave velocities calculated from the string material properties and the standing wave configuration, we need to use different formulas and substitute the given values.
Explanation:
To find the difference in transverse wave velocity, we need to first calculate the theoretical velocity based on the string's material properties and then compare it to the velocity determined from the standing wave configuration.
1. Calculate the theoretical velocity based on string properties:
The wave velocity on a string is given by the formula: v = √(T/μ), where T is the tension in the string and μ is the linear density.
Given:
- Tension (T) = weight = mass × gravity = 0.22 kg × 9.81 m/s² = 2.1622 N (rounding to four decimal places)
- Linear density (μ) = 1.3 grams/meter = 0.0013 kg/m (converting grams to kilograms)
Now, plug these values into the formula:
v_theoretical = √(2.1622 N / 0.0013 kg/m) = √1663.2308 m²/s² ≈ 40.78 m/s (rounded to two decimal places)
2. Calculate the velocity following from the standing wave configuration:
The number of antinodes (n) observed is given as 10, and the length of the string (L) is 1.70 m. In a standing wave on a string, the velocity is related to the wavelength (λ) and the frequency (f) by the formula: v = λf.
Each antinode corresponds to half a wavelength, so the total length of the string is equal to half a wavelength:
L = (1/2)λ
Rearrange to find the wavelength:
λ = 2L
Now, we need to find the frequency using the number of antinodes:
f = n × (velocity of wave on string) / (2L)
Substitute the values:
f = 10 × v_standing_wave / (2 × 1.70 m)
Solve for v_standing_wave:
v_standing_wave = f × 2L / 10
v_standing_wave = (10/10) × 1.70 m/s ≈ 1.70 m/s
3. Calculate the difference in velocities:
Difference = v_theoretical - v_standing_wave
Difference = 40.78 m/s - 1.70 m/s = 39.08 m/s
So, the difference between the transverse wave velocity calculated from the string material properties and the one following from the standing wave configuration is approximately 39.08 m/s.
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Final answer:
The velocity of a wave on a string under tension can be calculated with material properties or the standing wave pattern. We can't determine the exact difference in velocities due to unknown frequency.
Explanation:
The velocity of a wave on a string under tension can be calculated from the basic material properties such as mass of the object hung at the end, length of the string and its density or from the standing wave pattern, by observing the number of antinodes in the wave.
To calculate the velocity using the material properties, we use the formula: Vmaterial = sqrt(effect of gravity * tension / density). The tension in the string comes from a mass of 0.22 kg hanging at one end. Density needs to be converted from gram/meter to kg/meter - 1.3 g/m = 0.0013 kg/m. Substituting values, Vmaterial = sqrt(9.8 m/s2 * 0.22 kg / 0.0013 kg/m).
To calculate the velocity using the number of antinodes, we use the formula: Vantinode = 2 * frequency * wavelength. Here, the wave has 10 antinodes over a length of 1.7 m, indicating that the wavelength is 1.7 m / 10 = 0.17 m. The frequency is unknown. For standing waves, frequency must be adjusted until a stable pattern of nodes and antinodes appear. So, we don't have enough information.
The difference between the two velocities would then be Vmaterial - Vantinode.
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