(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.)

(b) Write the expression for y as a function of x and t for the wave in part (a) assuming y(x, 0) = 0 at the point x = 50.0 cm. (Use the following as necessary: x and t.)

for the second part, out in a phase angle

.05 sin[(2pi/.75)(x+.375t) -PHI]

(2pi/.75)(0.50) -PHI = 0
PHI = 2 pi (.5/.75)

To write the expression for y as a function of x and t, we can use the general form of a sinusoidal wave:

y(x, t) = A * sin(kx - ωt + φ)

where:
- 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

(a) Given:
A = 5.00 cm
λ = 75.0 cm
f = 5.00 Hz

First, we need to convert these values to SI units:
A = 5.00 cm = 0.05 m
λ = 75.0 cm = 0.75 m
f = 5.00 Hz

Next, let's calculate the wave number and angular frequency:
k = 2π/λ = 2π/0.75 = 8.3776 rad/m
ω = 2πf = 2π*5.00 = 31.4159 rad/s

Now, let's substitute these values into the equation:
y(x, t) = 0.05 * sin(8.3776x - 31.4159t + φ)

We need to find the phase constant φ. Given that y(0, t) = 0 at t = 0, we can substitute these values:
0 = 0.05 * sin(-31.4159*0 + φ)
0 = 0.05 * sin(φ)

Since sin(φ) = 0 gives us the desired condition, we can choose φ = 0.

Therefore, the expression for y as a function of x and t is:
y(x, t) = 0.05 * sin(8.3776x - 31.4159t)

(b) Given:
y(x, 0) = 0 at x = 50.0 cm

We need to find the phase constant φ. Substituting y(x, 0) = 0 and x = 50.0 cm into the equation:
0 = 0.05 * sin(8.3776*0.50 - 31.4159*0 + φ)
0 = 0.05 * sin(4.1888 - 0 + φ)
0 = 0.05 * sin(4.1888 + φ)

To satisfy sin(4.1888 + φ) = 0, we can choose φ = -4.1888.

Therefore, the expression for y as a function of x and t is:
y(x, t) = 0.05 * sin(8.3776x - 31.4159t - 4.1888)