Question

A transverse wave traveling on a string has a frequency of 60 Hz and a wavelength of 0.5 m. The amplitude of the wave is 0.1 m. Determine the maximum speed, maximum acceleration, and maximum displacement of a particle on the string.

Solution

The maximum speed, maximum acceleration, and maximum displacement of a particle on a transverse wave can be determined using the formulas related to wave characteristics.

1. Maximum Speed: The maximum speed of a particle on the string can be calculated using the formula: v_max = f * λ, where v_max is the maximum speed, f is the frequency, and λ is the wavelength.

Given: f = 60 Hz, λ = 0.5 m.

Substituting the given values into the formula: v_max = 60 Hz * 0.5 m = 30 m/s.

Therefore, the maximum speed of the particle is 30 m/s.

2. Maximum Acceleration: The maximum acceleration of a particle on the string can be calculated using the formula: a_max = 4 * π^2 * f^2 * A, where a_max is the maximum acceleration, f is the frequency, and A is the amplitude.

Given: f = 60 Hz, A = 0.1 m.

Substituting the given values into the formula: a_max = 4 * π^2 * (60 Hz)^2 * 0.1 m = 452.39 m/s^2.

Therefore, the maximum acceleration of the particle is 452.39 m/s^2.

3. Maximum Displacement: The maximum displacement of a particle on the string can be calculated using the formula: d_max = A, where d_max is the maximum displacement and A is the amplitude.

Given: A = 0.1 m.

The maximum displacement is equal to the amplitude of the wave.

Therefore, the maximum displacement of the particle is 0.1 m.

In conclusion, for a transverse wave with a frequency of 60 Hz and a wavelength of 0.5 m, the maximum speed of a particle on the string is 30 m/s, the maximum acceleration is 452.39 m/s^2, and the maximum displacement is 0.1 m.