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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

Expert verified
(a) 17.46 m/s; (b) 0.1455 m; increasing mass to 3.00 kg gives (c) 24.67 m/s and 0.2056 m respectively.

Step by step solution

01

Understanding Wave Speed on a String

The wave speed \( v \) on a string is determined by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string and \( \mu \) is the linear mass density. The tension \( T \) is the weight of the mass and is given by \( T = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately 9.8 \( m/s^2 \)).
02

Calculating Tension with 1.50 kg Mass

Calculate the tension in the rope when the mass is 1.50 kg. Using \( T = mg \), we find \( T = (1.50 \, \text{kg}) \times (9.8 \, \text{m/s}^2) = 14.7 \, \text{N} \).
03

Calculating Wave Speed with 1.50 kg Mass

Substitute the values of \( T \) (14.7 N) and \( \mu \) (0.0480 kg/m) into the wave speed formula: \( v = \sqrt{\frac{14.7}{0.0480}} \). This results in a wave speed \( v \approx 17.46 \, \text{m/s} \).
04

Calculating Wavelength for 1.50 kg Mass

The wavelength \( \lambda \) can be determined from the relationship \( v = f \lambda \), where \( f = 120 \, \text{Hz} \). Using \( v = 17.46 \, \text{m/s} \), solve for \( \lambda \): \( \lambda = \frac{v}{f} = \frac{17.46}{120} \approx 0.1455 \, \text{m} \).
05

Calculating Wave Speed with 3.00 kg Mass

Now increase the mass to 3.00 kg. Calculate the tension: \( T = (3.00 \, \text{kg}) \times (9.8 \, \text{m/s}^2) = 29.4 \, \text{N} \). Then the wave speed is \( v = \sqrt{\frac{29.4}{0.0480}} \approx 24.67 \, \text{m/s} \).
06

Calculating Wavelength for 3.00 kg Mass

Using the new wave speed \( v = 24.67 \, \text{m/s} \), calculate the new wavelength: \( \lambda = \frac{v}{f} = \frac{24.67}{120} \approx 0.2056 \, \text{m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transverse Waves
Transverse waves are a type of wave where the motion of the medium is perpendicular to the direction of the wave. In the case of a rope being vibrated by a tuning fork, the rope moves up and down, while the wave travels horizontally along the rope. These waves occur frequently in various forms, such as waves on a string or ripples on the surface of water.
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
Wave speed is how fast a wave travels through a medium. For transverse waves on a string, the wave speed can be calculated using the formula: \[ v = \sqrt{\frac{T}{\mu}} \]where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string.
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.
Understanding these factors helps in predicting and explaining the behavior of different waves in physics and in practical scenarios.
Wavelength
Wavelength refers to the distance between two consecutive crests or troughs of a wave. It represents one complete cycle of the wave. For the exercise, the wavelength was calculated using the equation:\[ \lambda = \frac{v}{f} \]where \( \lambda \) is the wavelength, \( v \) is the wave speed, and \( f \) is the frequency.
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.
When calculating, never forget that any change in wave speed or frequency impacts the wavelength directly, showing the dynamic nature of waves in physics.
Tension in Strings
Tension in strings is a critical factor influencing wave mechanics. It is the force exerted along the string when it is pulled tight. In the exercise, the tension \( T \) is calculated using:\[ T = mg \]where \( m \) is the mass hanging from the string and \( g \) is the acceleration due to gravity \((9.8 \, \text{m/s}^2)\).
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.
Understanding tension and its impact on wave speed is essential for studying mechanical waves and their applications in areas from music to engineering.

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Most popular questions from this chapter

Three pieces of string, each of length \(L\), are joined together end to end, to make a combined string of length 3\(L\). The first piece of string has mass per unit length \(\mu_1\), the second piece has mass per unit length \(\mu2 = 4\mu1\), and the third piece has mass per unit length \(\mu_3 = \mu_1/4\). (a) If the combined string is under tension F, how much time does it take a transverse wave to travel the entire length 3L? Give your answer in terms of \(L, F\), and \(\mu_1\). (b) Does your answer to part (a) depend on the order in which the three pieces are joined together? Explain.

You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 m from it, you measure its intensity to be 0.11 W/m\(^2\). An intensity of 1.0 W/m\(^2\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y(x, t) = (5.60 \, \mathrm{cm}) \mathrm{sin} [(0.0340 \mathrm{rad/cm})x] \mathrm{sin} [(150.0 \, \mathrm{rad/s})t]\), where the origin is at the left end of the string, the x-axis is along the string, and the \(y\)-axis is perpendicular to the string. (a) Draw a sketch that shows the standing-wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string. (f) What would be the equation \(y(x,t)\) for this string if it were vibrating in its eighth harmonic?

A wave on a string is described by \(y(x, t) = A \mathrm{cos}(kx - \omega t)\). (a) Graph \(y, v_y\), and \(a_y\) as functions of \(x\) for time \(t = 0\). (b) Consider the following points on the string: (i) \(x =\) 0; (ii) \(x = \pi/4k\); (iii) \(x = \pi/2k\); (iv) \(x = 3\pi/4k\); (v) \(x = \pi k\); (vi) \(x = 5\pi/4k\); (vii) \(x = 3\pi/2k\); (viii) \(x = 7\pi/4k\). For a particle at each of these points at \(t = 0\), describe in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not accelerating.

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

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