To find the frequency of vibration required to produce a standing wave pattern with three antinodes on a rope, we can use the formula:
v = f * λ
where: v is the speed of the wave, f is the frequency of vibration, and λ is the wavelength of the wave.
In a standing wave pattern with three antinodes, we have two nodes, and the distance between each consecutive antinode and node is equal to one-fourth of a wavelength.
Let's calculate the wavelength first: Given the length of the rope, L = 6m For a standing wave pattern with three antinodes, we have 2 nodes, so the distance between consecutive antinodes is 6m - 2 * (1/4) * λ.
Since one-fourth of a wavelength is equal to the distance between consecutive antinodes, we have: 1/4 * λ = 6m - 2 * (1/4) * λ
Simplifying the equation: 1/4 * λ + 1/2 * λ = 6m 3/4 * λ = 6m λ = (4/3) * 6m λ = 8m
Now, we can substitute the given speed of the wave into the formula to find the frequency: v = f * λ 3.2m/s = f * 8m
Solving for f: f = 3.2m/s / 8m f = 0.4 Hz
Therefore, the frequency of vibration required to produce a standing wave pattern with three antinodes on a 6m rope is 0.4 Hz.