A point on a string undergoes simple harmonic motion as a sinusoidal wave passes. When a sinusoidal wave with speed 24 m/s, wavelength 30 cm, and amplitude of 1.0 cm passes, what is the max speed of a point on the string?
To find the maximum speed of a point on the string, we can use the relation between the speed of the wave and the maximum speed of a point on the string. The maximum speed of a point on the string occurs when the displacement of the point is maximum.
The speed of a sinusoidal wave can be calculated using the equation:
v = fλ
Where:
v = speed of the wave
f = frequency of the wave
λ = wavelength of the wave
Given the speed of the wave (v = 24 m/s) and the wavelength (λ = 30 cm = 0.3 m), we can rearrange the equation to solve for the frequency:
f = v / λ
f = 24 m/s / 0.3 m
f = 80 Hz
Since the point on the string undergoes simple harmonic motion, we know that the frequency of the point is equal to the frequency of the wave. Hence, the frequency of the point on the string is also 80 Hz.
The maximum velocity of a point undergoing simple harmonic motion can be calculated using the formula:
v_max = 2πfA
Where:
v_max = maximum velocity of the point
f = frequency of the point (which is also equal to the frequency of the wave in this case)
A = amplitude of the point
Given the frequency (f = 80 Hz) and the amplitude (A = 1.0 cm = 0.01 m), we can calculate the maximum velocity of the point on the string:
v_max = 2π × 80 Hz × 0.01 m
v_max ≈ 5.026 m/s
Therefore, the maximum speed of a point on the string is approximately 5.026 m/s.