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.

Question

The student's waves travel at the same speed and the maximum deflection of the string, from its resting position, between the students is 12 cm. 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?

Answer 1

First, we need to determine the wavelength of the waves being produced by each student.

For the student on the left:
Amplitude = 5 cm = 0.05 m
Frequency = 4 Hz
Speed of wave = 30 m/s

Using the formula: Speed = Frequency * Wavelength
Wavelength = Speed / Frequency = 30 / 4 = 7.5 m

For the student on the right:
Amplitude = 7 cm = 0.07 m
Frequency = 3 Hz
Speed of wave = 30 m/s

Wavelength = Speed / Frequency = 30 / 3 = 10 m

Since the maximum deflection of the string between the students is 12 cm = 0.12 m, which equals half a wavelength for the standing wave, the total wavelength of the standing wave is 0.24 m.

Since the node is at the center, the distance between the node and each antinode (student) is half the wavelength, so each student is 0.12 m away from the node.

For the left student:
0.12 m = n * (wavelength / 2)
0.12 = n * 7.5 / 2
n = 0.032

For the right student:
0.12 m = m * (wavelength / 2)
0.12 = m * 10 / 2
m = 0.024

Since n and m must be whole numbers for standing waves, take the least common multiple:
n = 8
m = 6

Now, the frequency of the standing wave is:
Frequency = Speed / Wavelength
Frequency = 30 / 0.24 = 125 Hz

Therefore, the students must shake the slinky at 125 Hz to create a standing wave with a node centered between them.