Chapter 15: Problem 15
One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 Hz. The other end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/m. (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg?
Short Answer
Step by step solution
Understanding Wave Speed on a String
Calculating Tension with 1.50 kg Mass
Calculating Wave Speed with 1.50 kg Mass
Calculating Wavelength for 1.50 kg Mass
Calculating Wave Speed with 3.00 kg Mass
Calculating Wavelength for 3.00 kg Mass
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Transverse Waves
Understanding transverse waves is crucial as it helps us analyze how energy is transferred through these mediums without the medium itself moving far from its original position. This concept applies not just to physical ropes but also to electromagnetic waves, where electric and magnetic fields oscillate perpendicular to the direction of wave travel.
For students working through the exercise, visualizing the up-and-down movement of the rope while picturing the wave moving forward can clarify the transverse nature of these waves.
Wave Speed
This formula illustrates that the speed of the wave is influenced by both the tension and the mass density of the string.
- If the tension is higher, the wave speed increases. Think of pulling a string very tight - vibrations move faster because there's more force along the string.
- a higher mass density (meaning the string is "heavier" for its length) slows down the wave, much like pushing a denser object takes more energy.
Wavelength
The frequency in this context is how often the wave cycles occur per second, measured in hertz (Hz). A higher wave speed or a lower frequency results in a longer wavelength.
- This means with a faster or less frequent wave, the distance between peaks will increase.
- Conversely, if a wave is traveling slower or cycling rapidly, the wavelength shortens.
Tension in Strings
This tension is what allows the wave to travel through the string. Larger masses increase the tension, leading to faster waves because the stronger force makes the string more responsive to vibrational movements. Conversely, less tension slows the wave. Consider:
- A dramatically tighter string (more tension) supports faster traveling waves, like a drum, which is taut so it can produce sound efficiently.
- If the string is loose, the wave speed decreases, leading to a less "energetic" system.