A 3 m 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 is 30 m/s?
The frequency of the standing wave can be calculated using the equation:
f = v / λ
Where:
f = frequency of the standing wave
v = speed of the wave in the slinky (30 m/s)
λ = wavelength of the standing wave
The wavelength of the standing wave can be calculated as twice the distance between the antinodes:
λ = 2 * (distance between antinodes)
The distance between antinodes can be calculated as the sum of the amplitudes of the two individual waves:
distance between antinodes = amplitude of left wave + amplitude of right wave
Plugging in the values:
distance between antinodes = 5 cm + 7 cm = 12 cm = 0.12 m
Now, calculate the wavelength:
λ = 2 * (0.12 m) = 0.24 m
Now calculate the frequency:
f = 30 m/s / 0.24 m = 125 Hz
Therefore, the two students must shake the slinky at 125 Hz to create a standing wave with a node centered between them.