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 hertz with a 5cm amplitude and the student on the right shakes it at 3 hertz with a 7cm amplitude.
The students' waves travel at the same speed. The students agree to make a standing wave with a node centered between them, so they are the antinodes. At what frequency in hertz must they shake the slinky if the speed of a wave in the slinky is 30 m/s?
The formula for the speed of a wave is v = f * λ, where v is the speed of the wave, f is the frequency, and λ is the wavelength.
Since the waves from the two students are traveling at the same speed, the frequency of the wave they create together will be the same as the frequency of the individual waves.
To find the wavelength of each individual wave, we use the formula λ = 2 * amplitude.
For the wave with a frequency of 4 Hz and amplitude of 5 cm:
λ1 = 2 * 0.05 m = 0.1 m
For the wave with a frequency of 3 Hz and amplitude of 7 cm:
λ2 = 2 * 0.07 m = 0.14 m
Since we want to create a standing wave with a node centered between the two students (antinodes), the wavelength of the standing wave will be twice the distance between the two students. The distance between the two students is 3 m, so the wavelength of the standing wave will be 6 m.
Now we can find the frequency of the standing wave using the formula v = f * λ:
30 m/s = f * 6 m
f = 30 m/s / 6 m = 5 Hz
Therefore, the students must shake the slinky at a frequency of 5 Hz in order to create a standing wave with a node centered between them.