A 3m slinky rests on a table with a student holding each end. The students laterally shake the ends of the slinky to generate transverse waves. The student on the left shakes the slinky at 4 Hz with a 5 cm amplitude and the student on the right shakes it at 3 Hz with a 7 cm amplitude.
The students agree to make a standing wave with a node centered between them, so they are the antinodes. at what frequency in Hz must they shake the slinky if the speed of a wave in the slinky in 30 m/s?
To form a standing wave with a node centered between the two students, there should be half of a wavelength between them. Since the slinky is 3 meters long, this means the wavelength of the standing wave is 2 * 3 = 6 meters.
Now we can use the wave speed equation to find the frequency they must shake the slinky:
v = f * λ
Where v is the wave speed, f is the frequency, and λ is the wavelength. Plugging in the values, the equation becomes:
30 m/s = f * 6 m
Now, we can solve for the frequency:
f = (30 m/s) / (6 m) = 5 Hz
So, the students must shake the slinky at a frequency of 5 Hz to create the desired standing wave with a node centered between them.
To find the frequency at which the students must shake the slinky to create a standing wave with a node in the center and an antinode at each end, we need to consider the principles of standing waves.
In a simple standing wave, the distance between a node and an adjacent antinode (or vice versa) represents half of a wavelength (λ/2). The wavelength of the wave can be expressed as the length of the slinky (L).
To calculate the frequency (f) of the standing wave, we can use the formula:
v = fλ
where v is the speed of the wave. In this case, the wave speed is given as 30 m/s.
We know that the distance between the node and an antinode is L/4. So, we have:
L/4 = λ/2
Simplifying the equation, we find:
L = 2λ/4 = λ/2
Substituting the value of v into the formula, we get:
30 m/s = f(λ/2)
Given that the student on the left shakes the slinky at 4 Hz and the student on the right shakes it at 3 Hz, we can consider the frequency of the standing wave to be the average of these two frequencies.
Thus, the frequency at which the students must shake the slinky to create the desired standing wave is:
f = (4 Hz + 3 Hz)/2 = 3.5 Hz
Therefore, they must shake the slinky at a frequency of 3.5 Hz to achieve the desired standing wave configuration.
To create a standing wave with a node centered between them, the students need to make sure that the distance between adjacent nodes or antinodes is equal to half the wavelength of the wave.
The frequency of a wave is related to its wavelength and the speed of the wave through the formula:
frequency = speed / wavelength
Since the wavelength is twice the distance between a node and its adjacent antinode, we can write:
wavelength = 2 * (distance between a node and its adjacent antinode)
We need to find the distance between a node and its adjacent antinode.
The amplitude of the wave is the maximum displacement from the equilibrium position. In this case, the maximum displacement is the amplitude of either of the students' shaking.
Given:
Amplitude of left student = 5 cm = 0.05 m
Amplitude of right student = 7 cm = 0.07 m
Since the distance between adjacent nodes is equal to the sum of the amplitudes, we have:
distance between nodes = amplitude of left student + amplitude of right student
= 0.05 m + 0.07 m
= 0.12 m
Now, we can calculate the wavelength:
wavelength = 2 * (distance between nodes)
= 2 * 0.12 m
= 0.24 m
The speed of the wave in the slinky is given as 30 m/s.
Using the formula:
frequency = speed / wavelength
we can substitute the values:
frequency = 30 m/s / 0.24 m
= 125 Hz
Therefore, the students need to shake the slinky at a frequency of 125 Hz to create a standing wave with a node centered between them.