Question:

A transverse wave is traveling along a string with a frequency of 50 Hz. The wave has a wavelength of 0.2 m and an amplitude of 0.1 m.

a) Calculate the wave speed at which the wave is propagating.

b) Determine the angular frequency and the period of the wave.

c) Find the maximum velocity of a point on the string when the wave is at its maximum displacement upward.

Answer:

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

v = f * λ

where:

• v is the wave speed in meters per second (m/s),
• f is the frequency of the wave in hertz (Hz), and
• λ is the wavelength of the wave in meters (m).

Given:

• f = 50 Hz
• λ = 0.2 m

Substituting the given values into the formula:

v = 50 Hz * 0.2 m = 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 in radians per second (rad/s), and
• f is the frequency of the wave in hertz (Hz).

Given:

• f = 50 Hz

Substituting the given value into the formula:

ω = 2π * 50 Hz ≈ 314.16 rad/s

Therefore, the angular frequency is approximately 314.16 rad/s.

The period (T) of the wave can be calculated using the formula:

T = 1 / f

where:

• T is the period of the wave in seconds (s), and
• f is the frequency of the wave in hertz (Hz).

Given:

• f = 50 Hz

Substituting the given value into the formula:

T = 1 / 50 Hz = 0.02 s

Therefore, the period of the wave is 0.02 seconds.

c) The maximum velocity (v_max) of a point on the string can be calculated using the formula:

v_max = A * ω

where:

• v_max is the maximum velocity in meters per second (m/s),
• A is the amplitude of the wave in meters (m), and
• ω is the angular frequency in radians per second (rad/s).

Given:

• A = 0.1 m
• ω = 314.16 rad/s

Substituting the given values into the formula:

v_max = 0.1 m * 314.16 rad/s = 31.416 m/s

Therefore, the maximum velocity of a point on the string when the wave is at its maximum displacement upward is 31.416 m/s.