AP Physics 2 Exam Question:

A string with length 2.0 meters is stretched between two fixed points, labeled A and B. The string is oscillating at a frequency of 200 Hz and forms a standing wave pattern. The amplitude of the wave is measured to be 0.1 meters.

a) Calculate the speed of the wave on the string.

b) Determine the wavelength of the standing wave.

c) How many antinodes are present in this standing wave pattern?

Answer:

a) To calculate the speed of the wave on the string, we can use the formula:

$v = f \cdot \lambda$

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

Given that f = 200 Hz, we need to find λ.

To find the wavelength, we can use the relationship between wavelength and the length of the string when forming standing waves. The fundamental frequency (first harmonic) for a string fixed at both ends is given by:

$f_1 = \frac{v}{\lambda} = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

Where L is the length of the string, T is the tension applied to the string, and μ is the linear mass density of the string. However, for the fundamental frequency, the wave on the string forms exactly one-half of a wavelength, so for the second harmonic, it will form a full wavelength.

Therefore, for the second harmonic, we have:

$\lambda = 2L = 2 \times 2.0 \, \text{m} = 4.0 \, \text{m}$

Substituting this into the wave speed formula:

$v = f \cdot \lambda = 200 \, \text{Hz} \times 4.0 \, \text{m} = 800 \, \text{m/s}$

Therefore, the speed of the wave on the string is 800 m/s.

b) The wavelength of the standing wave is already calculated in part (a) to be 4.0 meters.

c) To determine the number of antinodes in the standing wave pattern, we need to recall the pattern of nodes and antinodes in standing waves on a string.

In a standing wave, there is always a node at each end of the string (points A and B), and there is an antinode at the midpoint of the string. Additionally, there are other antinodes spread throughout the string depending on the harmonic.

From the information given, we can determine that the standing wave pattern is the second harmonic (first overtone) because it forms exactly one full wavelength.

Since there is a node at each end, and one antinode at the midpoint, the total number of antinodes in this standing wave pattern is 1.

Therefore, there is 1 antinode present in this standing wave pattern.