Question:

A wave is moving along a string with a frequency of 100 Hz and a wavelength of 0.1 m. The wave has an amplitude of 0.2 m.

a) Calculate the wave speed.

b) Determine the angular frequency (ω) of the wave.

c) Find the maximum speed (v_max) of a particle in the string as it oscillates in the wave.

d) Calculate the maximum acceleration (a_max) experienced by a particle in the string.

e) Determine the time (t) it takes for a particle in the string to complete one full cycle of oscillation.

Answer:

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

v = f * λ

where:

• v is the wave speed
• f is the frequency of the wave
• λ is the wavelength of the wave

Given:

• f = 100 Hz
• λ = 0.1 m

Substituting the given values into the formula, we have:

v = 100 Hz * 0.1 m
v = 10 m/s

Therefore, the wave speed is 10 m/s.

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

ω = 2πf

where:

• ω is the angular frequency
• f is the frequency of the wave

Given:

• f = 100 Hz

Substituting the given value into the formula, we have:

ω = 2π * 100 Hz
ω = 200π rad/s

Therefore, the angular frequency (ω) of the wave is 200π rad/s.

c) The maximum speed (v_max) of a particle in the string can be calculated using the formula:

v_max = ω * A

where:

• v_max is the maximum speed
• ω is the angular frequency
• A is the amplitude of the wave

Given:

• ω = 200π rad/s
• A = 0.2 m

Substituting the given values into the formula, we have:

v_max = 200π rad/s * 0.2 m
v_max = 40π m/s

Therefore, the maximum speed (v_max) of a particle in the string is 40π m/s.

d) The maximum acceleration (a_max) experienced by a particle in the string can be calculated using the formula:

a_max = ω^2 * A

where:

• a_max is the maximum acceleration
• ω is the angular frequency
• A is the amplitude of the wave

Given:

• ω = 200π rad/s
• A = 0.2 m

Substituting the given values into the formula, we have:

a_max = (200π rad/s)^2 * 0.2 m
a_max = 80000π^2 m/s^2

Therefore, the maximum acceleration (a_max) experienced by a particle in the string is 80000π^2 m/s^2.

e) The time (t) it takes for a particle in the string to complete one full cycle of oscillation can be calculated using the formula:

t = 1 / f

where:

• t is the time period (time for one cycle)
• f is the frequency of the wave

Given:

• f = 100 Hz

Substituting the given value into the formula, we have:

t = 1 / 100 Hz
t = 0.01 s

Therefore, the time (t) it takes for a particle in the string to complete one full cycle of oscillation is 0.01 seconds.