A sinusoidal wave on a string is described by the equation y = (0.169 m) sin (0.713 x - 41.9 t), where x and y are in meters and t is in seconds. If the linear mass density of the string is 10.1 g/m ...
a) ... the phase of the wave at x = 2.27 cm and t = 0.175 s. ?
b)... the speed of the wave ?
c)... the wavelength.
d) ... the power transmitted by the wave. ?
To answer these questions, let's go step by step:
a) To find the phase of the wave at x = 2.27 cm and t = 0.175 s, we need to substitute these values into the equation y = (0.169 m) sin (0.713 x - 41.9 t).
So, plugging in x = 2.27 cm (which is 0.0227 m) and t = 0.175 s, the equation becomes:
y = (0.169 m) sin (0.713 * 0.0227 - 41.9 * 0.175).
Now, you can simply calculate the value of y using a calculator.
b) The speed of the wave can be found using the equation v = λ * f, where v is the speed, λ is the wavelength, and f is the frequency.
We can find the frequency (f) by taking the derivative of the equation with respect to time, since frequency is the rate of change of phase with respect to time. So, dθ/dt = -41.9.
Now, we can use the equation v = λ * f to find the speed of the wave. The frequency is the rate at which the phase changes in radians per second, and the wavelength is the distance in meters over which one complete cycle occurs.
c) The wavelength (λ) can be found using the equation λ = 2π/k, where k is the wave number.
The wave number (k) is equal to 2π divided by the wavelength. By comparing the given equation with the general form of a sinusoidal wave equation, we can find the value of k.
d) The power transmitted by the wave can be found using the equation P = 1/2 * μ * v * A^2 * ω^2.
Here, μ is the linear mass density of the string, v is the speed of the wave, A is the amplitude of the wave, and ω is the angular frequency. The linear mass density (μ) is given as 10.1 g/m, but we need to convert it to kg/m. The angular frequency (ω) is equal to 2π times the frequency (f).
Now that we have all the required values, we can plug them into the formula and calculate the power transmitted by the wave.
To summarize:
a) Substitute the given values into the equation and calculate y.
b) Use the equation v = λ * f to find the speed of the wave.
c) Use the equation λ = 2π/k to find the wavelength.
d) Use the equation P = 1/2 * μ * v * A^2 * ω^2 to find the power transmitted by the wave.