Vibrations and Waves Legacy Problem #27 Guided Solution
Problem*
In a physics lab, a rope is observed to make 240 complete vibrational cycles in 15 seconds. The length of the rope is 2.8 meters and the measurements are made for the 6th harmonic (with six equal length sections). Determine the speed of the waves in the rope.
Audio Guided Solution
In this problem, a rope is vibrating with a standing wave pattern, the pattern being the pattern for the sixth harmonic. The rope itself is 2.8 meters long, and it makes 240 complete vibrational cycles in 15 seconds. I have to calculate the speed of these waves. Thinking through what I know about physics, one of the things that I could say is that the speed of the waves is the frequency times the wavelength. So in order to solve this problem, one of the obvious strategies would be to somehow find the frequency and the wavelength, and then to multiply to get the speed. Now in the first sentence, we read of 240 complete cycles in 15 seconds, and that's frequency type information. Frequency is the number of cycles per second, so if you have 240 in 15 seconds, you could divide the 240 by 15 seconds, you'll get the frequency in units of cycles per second, or hertz. That comes out to be 16 hertz. Now in the second sentence, we read that the length of the rope is 2.8 meters long. That's not the wavelength, that's how long the rope is. The rope is divided into 6 equal length sections, with each one having a vibrating antinode. So that tells me that the number of wavelengths within the 2.8 meter length of rope is 3 wavelengths. I got that by sketching myself, or at least visualizing for myself, the standing wave pattern. If you're having difficulties, you should probably sketch it yourself, or look one up that is already sketched. Draw a rope horizontal line on your sheet of paper. Put dots at each end to represent the ends of the rope. Divide it up into 6 equal length sections, and then begin sketching a standing wave pattern, a sine wave pattern, such that there's an antinode in the middle of each section. What you'll find is that you've drawn 3 wavelengths. So 2.8 meters equals 3 wavelengths. If I divide by 3, I'll get the wavelength. Dividing 2.8 by 3 gives me a wavelength value of 0.933 repeating meters. Now I know the frequency, and I know the wavelength. And the wave equation states that v equals f times lambda. So multiplying this f of 16 hertz by this wavelength of 0.93333 meters gives me a speed of 14.933 meters per second. And I can round that to 2 significant digits.
Solution
15 m/s
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = 12.8 \unit{\meter\per\second}\), \(\descriptive{λ}{λ,wave length} = 4.52 \unit{m}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Vibrations and Waves at The Physics Classroom Tutorial.