Vibrations and Waves Legacy Problem #28 Guided Solution
Problem*
Winston and Michal hold opposite ends of a stretched rope. Winston introduces a 68-cm tall upward-displaced pulse on his end while Michal simultaneously introduces a 42-cm tall upward-displaced pulse on his end. The two pulses meet in the middle of the rope.
- What is the resultant displacement of the rope when they completely overlap?
- What would be the resultant displacement of the rope if Michal's pulse was displaced downward?
Audio Guided Solution
This question pertains to the concept of interference of waves, and if you're having difficulties picturing the situation, you might want to click on the link back to the Physics Classroom tutorial page on wave interference. One of the big principles discussed on that page is the principle of superposition. The idea is that when waves meet, that the resulting displacement of the medium in any location is simply the algebraic sum of the displacements of the individual waves. So here Winston's introducing a wave in one end and Michael in the other. What we know is that Winston's pulse is displaced 68 centimeters upwards, and Michael's pulse is displaced 42 centimeters upward. When these two pulses meet in the middle of the rope and completely overlap, the medium will be displaced upward in such a manner that the resultant displacement will simply be the sum of the two individual displacements. So the resultant displacement would be 68 centimeters plus 42 centimeters. That gives us 110 centimeters upwards. Now in part B of the problem, the question is asked, what if Michael's pulse was displaced downwards, such that Winston's 680 centimeter tall upwards displaced pulse meets up with a 42 centimeter tall downward displaced pulse? In this situation, we would have some destructive interference occurring, and the result would be that the 68 centimeters up plus the 42 centimeters down would give us a net resulting displacement of 26 centimeters up. If you're still having trouble picturing this, if you turn back to the wave interference page in the physics classroom tutorial, you'll see some wonderful graphics and animations.
Solution
- 110 cm, up
- 26 cm, up
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.