Vibrations and Waves Legacy Problem #26 Guided Solution
Problem*
A 144 cm long rope undergoes exactly 64 complete vibrational cycles in 17.6 seconds when vibrating in the third harmonic (in three equal-length sections). Determine the speed of the waves in the rope.
Audio Guided Solution
A few of the habits of effective problem solvers involve reading the problem carefully and identifying the known and unknown quantities by writing them down, expressing them in terms of variables and equations, visualizing the situation or even diagramming the situation and plotting out strategies as to how to get from the known to the unknown quantities. Here we have a rope. The length of the rope is 144 centimeters, so I write down L equal 144 centimeters. I'm told it's vibrating in the third harmonic or in three equal length sections. I take the time to actually diagram that wave pattern. I draw a line, I put two dots at the end to represent the ends of this rope, and I divide it up into three equal length sections, and then I do my best to sketch a sine wave such that there's a section with an antinode in each part of that three segment rope. I have the third harmonic here, and that 144 centimeters is going to be equal to three halves of a wavelength if it's the third harmonic. I'm also told that there's 64 vibrational cycles in 17.6 seconds. I recognize that that's information about calculating the frequency of these waves. I write down 64 cycles per 17.6 seconds. Now my unknown quantity is the speed. And the V value for the waves in this rope. So my strategy is going to center around finding the frequency, and finding the wavelength, and then multiplying using the wave equation to calculate the speed. Now the wavelength is going to come from the fact that it's a 144 centimeter long rope, and three halves of a wavelength is within it. So I'm going to write 144 centimeters equal three halves lambda, where lambda is a wavelength. Now I'm going to divide each side by 1.5, or three halves, so I can get lambda. Lambda comes out to be 96 centimeters, or 0.96 meters. Now I'm going to calculate the frequency from the fact that there's 64 cycles per 17.6 seconds. Dividing like that gives me the frequency in cycles per second. It comes out to be 3.63636 repeating hertz. Now I know what I need to know to calculate the speed. When I use the wave equation V equal F times lambda, multiplying the frequency at 3.63636 hertz by the wavelength of 0.96 meters, gets me a speed value of 3.4909 meters per second. Or if I did it in units of centimeters, it would have been 349.09 centimeters per second.
Solution
349 cm/s or 3.49 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.