(a) Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the following characteristics: A = 5.00 cm, λ = 75.0 cm, f = 5.00 Hz, and y(0, t) = 0 at t = 0. (Use the following as necessary: x and t.)
A =0.0500 m
L = 0.750 m
T = 1/5 = 0.2
goes .75 m in .2 s
so v=.75/2 = .375m/s
for constant phase
(x+vt) the+vt makes it left
to make it go 2pi for every wavelength l or time T
sin [2 pi (x+vt)/L]
now the size
.05 sin[(2pi/L)(x+vt)]
more numbers in
.05 sin[(2pi/.75)(x+.375t)]
To write the expression for y as a function of x and t, we can start by considering the general equation for a sinusoidal wave:
y(x, t) = A sin(kx - ωt + φ),
Where:
- y(x, t) is the displacement of the wave at position x and time t.
- A is the amplitude of the wave.
- k is the wave number, given by 2π/λ where λ is the wavelength.
- ω is the angular frequency, given by 2πf where f is the frequency.
- φ is the phase constant.
Now, let's substitute the given values into the equation.
A = 5.00 cm (convert to meters: 0.0500 m)
λ = 75.0 cm (convert to meters: 0.750 m)
f = 5.00 Hz
y(0, t) = 0 at t = 0
First, we need to find the values of k and ω.
k = 2π/λ = 2π/0.750 = 8.37758 rad/m
ω = 2πf = 2π(5.00) = 31.4159 rad/s
Since y(0, t) = 0 at t = 0, we can determine the phase constant φ.
0 = A sin(k(0) - ω(0) + φ)
0 = A sin(φ)
sin(φ) = 0
Since sin(φ) = 0, we know that φ must be an angle where sin equals 0. This occurs at multiples of π, so we can write φ as nπ, where n is an integer.
Now, let's put everything together in the expression for y(x, t):
y(x, t) = 0.0500 sin(8.37758x - 31.4159t + nπ),
where n is an integer.
This is the expression for y as a function of x and t in SI units for the given sinusoidal wave.