AP Physics 2 Exam Question:

A wave travelling along a string is described by the equation y(x, t) = 0.1cos(2πx - 4πt), where y is the displacement of the string from its equilibrium position, x is the position along the string in meters, and t is the time in seconds.

a) Calculate the wavelength (λ) and frequency (f) of this wave. Assume the wave is propagating in the positive x-direction.

b) Determine the amplitude (A), phase constant (φ), and speed (v) of this wave.

c) Sketch the wave at t = 0.5 seconds, showing four complete cycles of the wave.

Answer:

a) The equation representing the wave is given by y(x, t) = 0.1cos(2πx - 4πt). Comparing this with the general wave equation y(x, t) = Acos(kx - ωt + φ), we can deduce the following:

The coefficient of x is 2π, which corresponds to the wave number k. Therefore, k = 2π.

The coefficient of t is -4π, which corresponds to the angular frequency ω. Therefore, ω = 4π.

The wave equation can be written as y(x, t) = 0.1cos(2π(x - 2t)) by combining the terms.

Comparing this equation with y(x, t) = Acos(kx - ωt + φ), we can see that k = 2π and ω = 4π.

The relationship between angular frequency ω and frequency f is given by ω = 2πf. Thus, we can solve for f:

ω = 4π 2πf = 4π f = 2 Hz

The wavelength (λ) of a wave is related to the wave number (k) by the equation λ = 2π/k. Substituting the value of k, we get:

λ = 2π/(2π) λ = 1 meter

Therefore, the wavelength of this wave is 1 meter, and the frequency is 2 Hz.

b) From the given equation, the amplitude (A) is equal to 0.1.

The phase constant (φ) is determined by comparing the equation with the general wave equation. In this case, φ = 0.

The speed (v) of a wave is given by the equation v = fλ. Substituting the values of f and λ obtained previously:

v = 2 Hz * 1 meter v = 2 m/s

Therefore, the amplitude (A) is 0.1, the phase constant (φ) is 0, and the speed (v) is 2 m/s.

c) To sketch the wave at t = 0.5 seconds, we can substitute the value of t into the equation:

y(x, t = 0.5) = 0.1cos(2πx - 4π(0.5)) y(x, t = 0.5) = 0.1cos(2πx - 2π) y(x, t = 0.5) = 0.1cos(2π(x - 1))

To sketch the wave, we can use the provided equation y(x, t = 0.5) = 0.1cos(2π(x - 1)) and plot points for various values of x. Let's consider x ranging from -1 to 3 to ensure we capture four complete cycles of the wave.

Using these points, we can plot the wave on a graph, where x is the horizontal axis and y(x, t = 0.5) is the vertical axis. The resulting graph should show four complete cycles of the wave, with alternating peaks and troughs at x = -1, 0, 1, 2, and 3.

This sketch represents the wave at t = 0.5 seconds, showing four complete cycles.