The speed of a transverse wave on a string is 470 m/s, and the wavelength is 0.1 m. The amplitude of the wave is 1.6 mm. How much time is required for a particle of the string to move through a total distance of 1.0 km?
To find the time required for a particle of the string to move through a total distance of 1.0 km, we need to determine the period of the wave.
The speed of a wave is given by the equation:
Speed = Frequency × Wavelength
We can rearrange this equation to solve for frequency:
Frequency = Speed / Wavelength
The period of a wave is the inverse of its frequency:
Period = 1 / Frequency
First, let's convert the amplitude from millimeters to meters:
Amplitude = 1.6 mm = 1.6 × 10^-3 m
Given:
Speed = 470 m/s
Wavelength = 0.1 m
Amplitude = 1.6 × 10^-3 m
Total Distance = 1.0 km = 1.0 × 10^3 m
Now, let's find the frequency:
Frequency = Speed / Wavelength
Frequency = 470 m/s / 0.1 m
Frequency = 4700 Hz
Next, let's find the period:
Period = 1 / Frequency
Period = 1 / 4700 Hz
Period ≈ 2.13 × 10^-4 s
To find the time required for a particle of the string to move through a total distance of 1.0 km, we need to find the number of periods in that distance. We can divide the total distance by the wavelength.
Number of Periods = Total Distance / Wavelength
Number of Periods = 1.0 × 10^3 m / 0.1 m
Number of Periods = 1.0 × 10^4
Finally, we can calculate the total time required:
Total Time = Number of Periods × Period
Total Time = 1.0 × 10^4 × 2.13 × 10^-4 s
Total Time = 2.13 s
Therefore, it takes approximately 2.13 seconds for a particle of the string to move through a total distance of 1.0 km.