If a rope with linear mass density .167kg/m and is 4m long, what are the lowest four frequencies of standing waves possible on such a rope? What is the fundamental frequency?
what does linear mass density have to do with wavelength?
wavlength=1/2 length if fixed on both ends.
Harmonics? see this...
http://www.physics.isu.edu/~hackmart/spl1sws.pdf
To find the lowest four frequencies of standing waves on a rope with a linear mass density of 0.167 kg/m and a length of 4 m, we can use the formula for the frequencies of standing waves on a stretched string:
f = n/(2L) * √(T/μ)
Where:
- f is the frequency of the standing wave
- n is the mode number (1, 2, 3, ...)
- L is the length of the rope
- T is the tension in the rope
- μ is the linear mass density of the rope
In this case, we are given the linear mass density (μ = 0.167 kg/m) and the length of the rope (L = 4 m). However, to determine the frequencies, we also need the tension in the rope (T).
The fundamental frequency (n = 1) is the lowest frequency for a standing wave, so let's find that first.
Fundamental Frequency:
For n = 1, the formula simplifies to:
f1 = 1/(2L) * √(T/μ)
Now, let's solve for the tension (T) using the given information:
μ = 0.167 kg/m
L = 4 m
T = ? (unknown)
To find T, we need to use an additional equation involving the linear mass density and tension:
T = μ * g * L
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the given values, we get:
T = (0.167 kg/m) * (9.8 m/s²) * (4 m)
T ≈ 6.54 N
Now we have the tension (T), so we can calculate the fundamental frequency (f1):
f1 = 1/(2 * 4 m) * √(6.54 N / 0.167 kg/m)
f1 ≈ 0.125 Hz
The fundamental frequency (n = 1) is approximately 0.125 Hz.
To find the other three lowest frequencies (n = 2, 3, 4), we can use the same formula and substitute the respective values for n:
f2 = 2/(2 * 4 m) * √(6.54 N / 0.167 kg/m)
f3 = 3/(2 * 4 m) * √(6.54 N / 0.167 kg/m)
f4 = 4/(2 * 4 m) * √(6.54 N / 0.167 kg/m)
Calculating these values will give you the frequencies for the other three modes of the standing waves on the rope.