Question

A student sets up an experiment to investigate interference and standing waves in a string. The student fixes one end of the string to a clamp, while the other end is attached to a vibration generator. The generator produces a sinusoidal wave with a frequency of 120 Hz. The student measures the tension in the string to be 5 N.

a) Calculate the wavelength of the wave produced by the generator.

b) The student notices that when the string is excited, it forms a standing wave pattern with four loops. Calculate the length of the string.

c) The student changes the frequency of the generator to 150 Hz and observes that the number of loops in the standing wave pattern also changes. If the number of loops increases to eight, calculate the new length of the string.

d) Calculate the speed of the wave on the string in the new situation described in part (c).

Note: Assume the string has negligible mass and does not have any boundary interactions.

Answer

a) The speed of the wave can be calculated using the wave equation:

v = λf

where v is the wave speed, λ is the wavelength, and f is the frequency.

Given: f = 120 Hz

The wave speed can be determined using the wave equation:

v = λf λ = v/f

We can assume the wave speed is equal to the speed of light in a vacuum (since the string has negligible mass and no boundary interactions), which is approximately 3.00 x 10^8 m/s. Thus:

λ = (3.00 x 10^8 m/s)/(120 Hz) = 2.50 m

Therefore, the wavelength of the wave produced by the generator is 2.50 meters.

b) For a standing wave with four loops, there are three nodes and four antinodes. The distance between consecutive nodes or antinodes is half the wavelength. Thus, the length of the string (L) can be calculated as:

L = λ/2

Given: λ = 2.50 m

Therefore,

L = 2.50 m / 2 = 1.25 m

Thus, the length of the string when it forms a standing wave pattern with four loops is 1.25 meters.

c) When the frequency of the generator changes to 150 Hz and the number of loops increases to eight, we can use the relationship between standing wave patterns and the length of the string. The number of loops in a standing wave pattern is related to the harmonic number (n) by the equation:

n = 2(L/λ)

where n is the harmonic number, L is the length of the string, and λ is the wavelength.

Rearranging the equation, we can solve for the length of the string:

L = (nλ)/2

Given: n = 8 λ = 2.50 m

Therefore,

L = (8 x 2.50 m) / 2 = 10.00 m

Thus, the new length of the string when the frequency is changed to 150 Hz and there are eight loops in the standing wave pattern is 10.00 meters.

d) The speed of the wave on the string can be calculated by rearranging the wave equation:

v = λf

Given: λ = 2.50 m f = 150 Hz

We can substitute these values into the equation to find the wave speed:

v = (2.50 m)(150 Hz) = 375 m/s

Therefore, the speed of the wave on the string in the new situation described in part (c) is 375 m/s.