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. ?
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).
First, let's convert x into meters:
x = 2.27 cm = 2.27 * 10^(-2) m
Now, substitute the values into the equation:
y = (0.169 m) sin(0.713 * 2.27 * 10^(-2) - 41.9 * 0.175)
Using a calculator, we can evaluate the expression to get the phase of the wave at x = 2.27 cm and t = 0.175 s.
b) The speed of the wave can be calculated using the formula v = λf, where v is the speed, λ is the wavelength, and f is the frequency.
In the given equation, the coefficient of t represents the angular frequency (ω), which is related to the frequency (f) by the equation ω = 2πf. We can find the angular frequency by comparing the given equation to the standard form y = a sin(bx - ωt). In this case, ω = 41.9.
Now, we can find the frequency (f) using:
f = ω / (2π) = 41.9 / (2π)
The speed (v) of the wave can be found by multiplying the frequency (f) with the wavelength (λ):
v = fλ
c) The wavelength (λ) can be found using the formula λ = 2π / k, where k is the wave number.
In the given equation, the coefficient of x represents the wave number (k). In this case, k = 0.713.
Substituting the value of k into the formula, we get:
λ = 2π / 0.713
d) The power transmitted by the wave can be found using the formula P = (1/2)ρAv^2, where P is the power, ρ is the linear mass density, A is the amplitude, and v is the speed of the wave.
In this case, the linear mass density (ρ) is given as 10.1 g/m, which needs to be converted to kg/m:
ρ = 10.1 g/m = 10.1 * 10^(-3) kg/m
The amplitude (A) can be found from the given equation: y = (0.169 m) sin(0.713 x - 41.9 t). In this case, the amplitude (A) is 0.169 m.
We have already found the speed (v) in part b.
Now substituting the values into the formula, we get:
P = (1/2) * (10.1 * 10^(-3) kg/m) * (0.169 m)^2 * v^2
Using a calculator, we can evaluate the expression to find the power transmitted by the wave.
To find the answers to these questions, we need to analyze the given sinusoidal wave equation and apply some relevant equations and principles.
a) To find the phase of the wave at x = 2.27 cm and t = 0.175 s, we can plug these values into the equation:
y = (0.169 m) sin(0.713 x - 41.9 t)
At x = 2.27 cm = 0.0227 m and t = 0.175 s:
y = (0.169 m) sin(0.713 * 0.0227 - 41.9 * 0.175)
You can calculate the value of y using a calculator.
b) To find the speed of the wave, we can use the equation:
v = λ * f
where v is the speed of the wave, λ is the wavelength, and f is the frequency. In this case, frequency can be determined from the equation:
f = ω / (2π)
where ω is the angular frequency, given by:
ω = 0.713 rad/m * (2π rad/cycle)
Substituting this back into the equation for v, we get:
v = λ * 0.713 rad/m * (2π rad/cycle) / (2π)
The wavelength λ can be found using the equation:
λ = 2π / k
where k is the wave number, given by:
k = 0.713 rad/m
Substituting this back into the equation for v, we get:
v = (2π / 0.713) * 0.713 rad/m * (2π rad/cycle) / (2π)
You can simplify this equation to find the value of v.
c) To find the wavelength, we can use the equation mentioned earlier:
λ = 2π / k
Substituting the value of k into the equation, we get:
λ = 2π / 0.713
You can calculate the value of λ using a calculator.
d) To find the power transmitted by the wave, we can use the equation:
P = 0.5 * ρ * v * A²
where P is the power, ρ is the linear mass density of the string, v is the speed of the wave, and A is the amplitude of the wave.
Plugging in the values provided:
P = 0.5 * (10.1 g/m) * (speed of wave) * (amplitude)²
You can calculate the value of P using a calculator.
Please note that in some parts of the calculations, the units were converted to ensure consistent units throughout the equations.