AP Physics 2 Exam Question

A wave is traveling through a medium with a wavelength of 2.5 meters and a frequency of 6 Hz. The wave is defined by the equation:

where $y$ represents the displacement of the medium particles, $x$ is the position along the wave, $t$ is the time, $k$ is the wave number, $\omega$ is the angular frequency, and $\phi$ is the phase constant.

1. Determine the wave number $k$ for this wave.
2. Calculate the angular frequency $\omega$ of the wave.
3. Find the period $T$ of the wave.
4. Calculate the velocity of the wave.
5. Determine the phase constant $\phi$ if the medium particles are initially at their maximum displacement.

Answer with Step-by-Step Explanation

1. The wave number $k$ is defined as the number of complete waves per unit distance. It is related to the wavelength by the equation $k = \frac{2 \pi}{\lambda}$, where $\lambda$ is the wavelength. Substituting the given value of $\lambda = 2.5$ meters into the equation, we have:

So, the wave number $k$ for this wave is approximately 2.513 m^(-1).

2. The angular frequency $\omega$ is related to the frequency $f$ by the equation $\omega = 2\pi f$. Substituting the given frequency $f = 6$ Hz into the equation, we have:

Therefore, the angular frequency $\omega$ of the wave is $12\pi$ rad/s.

3. The period $T$ of the wave is the time taken for one complete cycle. It is the reciprocal of the frequency, given by the equation $T = \frac{1}{f}$. Substituting the given frequency $f = 6$ Hz into the equation, we have:

So, the period $T$ of the wave is $\frac{1}{6}$ second.

4. The velocity $v$ of the wave can be found using the equation $v = \frac{\lambda}{T}$, where $\lambda$ is the wavelength and $T$ is the period. Substituting the given wavelength $\lambda = 2.5$ meters and period $T = \frac{1}{6}$ second into the equation, we have:

Thus, the velocity of the wave is 15 m/s.

5. The phase constant $\phi$ represents the initial phase of the wave. If the medium particles are initially at their maximum displacement, it means the wave is at its peak or crest. The equation $y(x, t) = 0.2 \sin{(kx - \omega t + \phi)}$ indicates that when $(kx - \omega t + \phi) = 0$, the wave reaches its maximum value.

Therefore, to find the phase constant $\phi$, we substitute the given condition $(kx - \omega t + \phi) = 0$ into the equation:

Since the sine function is zero at $\phi = n\pi$ (where $n$ is an integer), we can solve for $\phi$ using the condition:

So, if the medium particles are initially at their maximum displacement, the phase constant $\phi$ can take the values $n\pi$, where $n$ is an integer.