Vibrations and Waves Legacy Problem #19 Guided Solution

This problem centers around a verbal statement in a diagram of a standing wave pattern that's been established in a row. We're told that Anna is standing 7.2 meters from a tree holding a rope that's tied to the tree and begins vibrating it up and down 28 cycles in 5 seconds. That will be important to write it down. 28 cycles in 5 seconds. Then I'm told that I'm asked to calculate the amplitude, wavelength, frequency, and speed. Further information is given in the diagram, like the distance from Anna to the tree is 7.2 meters. If I study that carefully, I'll recognize that 7.2 meters is equivalent to 1.5 wavelengths. If I look at Anna's hand, it's right at a rest position. I'm going to trace over one part of the pattern, going up to a crest, down to a rest, down to a trough, up to a rest. That's a complete wave. And then there's another half wave after that. So the 7.2 meters is equal to 1.5 wavelengths. I wish to find amplitude, wavelength, frequency, and speed. So I'm going to get wavelength right away. That 1.5 wavelengths equals 7.2 meters. So divide each side by 1.5. That will give me the wavelength value for these wave patterns, and it ends up being 4.8 meters. That's wavelength. Now on the diagram, I notice this distance of 0.6 meters is given as the distance from a high point to a very low point, which makes it one half of the amplitude. So if I half the 0.6 meters, I'm going to get 0.3 meters as the amplitude. Now I have to deal with this 28 cycles in 5 seconds. That's information about period and frequency. You have to know the distinction between them. Period is the time per number of cycles, and frequency is the number of cycles per time. So if I wish to find the frequency, I do the number of cycles, 28, divided by the time of 5 seconds. 28 divided by 5 gives me 5.6. That's cycles per second, also known as a hertz. So I've now found the frequency. Determining the speed demands that we use the wave equation. The wave equation is V equals f times lambda, where V is the speed, f the frequency, and lambda the wavelength. So if I take my 4.8 meters and multiply by my 5.6 hertz, that's going to give me a speed value, and it comes out to be 26.88 meters per second. I can round that to two significant digits, such that it's 27 meters per second.