Question:

A rope is fixed at both ends and is set into motion to create transverse waves. The frequency of the waves is 12 Hz, and the wavelength is 2.5 meters. The amplitude of the waves is 0.5 meters.

a) Calculate the wave speed of the transverse waves.

b) Calculate the angular frequency (ω) of the waves.

c) Determine the maximum velocity and maximum acceleration of a particle of the rope when it is at its maximum displacement.

d) Write the equation of the wave in terms of time (t), amplitude (A), angular frequency (ω), and wave number (k).

Answer:

a) The wave speed (v) can be calculated using the formula:

v = fλ

where f is the frequency and λ is the wavelength.

Given: f = 12 Hz λ = 2.5 m

Using the formula:

v = 12 Hz x 2.5 m

v = 30 m/s

Therefore, the wave speed is 30 m/s.

b) The angular frequency (ω) for a wave can be calculated using the formula:

ω = 2πf

where f is the frequency.

Given: f = 12 Hz

Using the formula:

ω = 2π(12 Hz)

ω = 24π rad/s

Therefore, the angular frequency is 24π rad/s.

c) At maximum displacement, the particle of the rope reaches its maximum velocity and maximum acceleration. The maximum velocity (vmax) and maximum acceleration (amax) can be calculated using the formulas:

vmax = Aω amax = Aω^2

Given: A = 0.5 m ω = 24π rad/s

Using the formulas:

vmax = (0.5 m)(24π rad/s) vmax ≈ 37.70 m/s

amax = (0.5 m)(24π rad/s)^2 amax ≈ 566.96 m/s^2

Therefore, the maximum velocity is approximately 37.70 m/s, and the maximum acceleration is approximately 566.96 m/s^2.

d) The equation of a transverse wave in terms of time (t), amplitude (A), angular frequency (ω), and wave number (k) is given by:

y(x, t) = A*sin(kx - ωt)

where x is the position along the wave, t is the time, and k is the wave number.

Since the question does not provide the wave number, we will assume k = 2π/λ (in rad/m) based on the given wavelength.

Given: A = 0.5 m ω = 24π rad/s λ = 2.5 m

Using the formula:

k = 2π/λ k = 2π/2.5 k = 0.8π rad/m

The equation of the wave becomes:

y(x, t) = 0.5*sin(0.8πx - 24πt)

Therefore, the equation of the wave in terms of time (t), amplitude (A), angular frequency (ω), and wave number (k) is y(x, t) = 0.5*sin(0.8πx - 24πt).