AP Physics 1 Exam Question

Consider a rope fixed at one end and the other end free to move. A sinusoidal wave is generated at the fixed end with a frequency of 60 Hz and an amplitude of 0.1 m. The wave travels down the rope with a speed of 10 m/s. When the wave reaches the free end, it reflects back and interferes with the incoming wave.

a) Explain what is meant by interference in the context of waves.

b) Calculate the wavelength of the initial wave.

c) Calculate the time it takes for the reflected wave to travel back to the fixed end.

d) Explain what is meant by standing waves.

e) Determine the distances from the fixed end at which the first two nodes will occur.

f) Calculate the frequency of the standing wave that forms in the rope.

g) Determine the velocity of the reflected wave.

Solution

a) Interference refers to the meeting of two or more waves at a point in space and time, leading to changes in the resulting wave pattern. Interference can be constructive (when the waves reinforce each other) or destructive (when the waves cancel each other out).

b) The wavelength of a wave can be calculated using the formula:

wavelength = velocity / frequency

Given that the wave travels at a velocity of 10 m/s and has a frequency of 60 Hz:

wavelength = 10 m/s / 60 Hz = 0.1667 m (rounded to 4 decimal places)

Therefore, the wavelength of the initial wave is approximately 0.1667 m.

c) The time it takes for the reflected wave to travel back to the fixed end can be determined using the formula:

time = distance / velocity

The distance traveled by the wave is twice the length of the rope, as it travels from the fixed end to the free end and then back to the fixed end:

distance = 2 * length of the rope

The velocity of the wave is given as 10 m/s:

time = (2 * length of the rope) / 10 m/s

Since the length of the rope is not provided, we cannot calculate the exact time it takes for the reflected wave to travel back to the fixed end.

d) Standing waves are formed when two waves with the same frequency and amplitude traveling in opposite directions interfere with each other. The superposition of these waves results in points called nodes which remain stationary (no displacement) and points called antinodes which oscillate with the maximum displacement.

e) In a standing wave, the nodes occur at fixed distances from each other, which are integer multiples of half the wavelength. The first node is located at half a wavelength from the fixed end, and the second node is located at one and a half wavelengths from the fixed end.

Therefore, the distances from the fixed end at which the first two nodes will occur are:

First node: 0.5 * 0.1667 m = 0.08335 m (rounded to 5 decimal places) Second node: 1.5 * 0.1667 m = 0.25 m

f) The frequency of the standing wave can be determined since it is the same as the frequency of the initial wave, which is 60 Hz.

Therefore, the frequency of the standing wave is 60 Hz.

g) The velocity of the reflected wave is the same as the velocity of the initial wave, which is 10 m/s.

Therefore, the velocity of the reflected wave is 10 m/s.