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Ch 15: Mechanical Waves
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 15: Mechanical WavesProblem 26fg
Chapter 15, Problem 26fg

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]y(x,t)=\(\left\)(2.30\(\operatorname{mm)}\]\cos\)[\(\left\)(16.98\(\text{ }\)rad/m\(\right\))x+(742\(\text{ }\)rad/s\(\right\))t]. Being more practical, you measure the rope to have a length of 1.35 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (f) tension in the rope; (g) average power transmitted by the wave.

Verified step by step guidance
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Step 1: Identify the wave function given in the problem: y(x, t) = 2.30 mm cos[(16.98 rad/m)x + (742 rad/s)t]. This function describes a traveling wave on a rope.
Step 2: Calculate the wave speed (v) using the wave number (k) and angular frequency (ω). The wave number k is 16.98 rad/m, and the angular frequency ω is 742 rad/s. Use the formula v = ω/k to find the wave speed.
Step 3: Determine the linear mass density (μ) of the rope. The linear mass density is the mass per unit length of the rope. Use the formula μ = mass/length, where the mass is 0.00338 kg and the length is 1.35 m.
Step 4: Calculate the tension (T) in the rope using the wave speed and linear mass density. The formula for tension is T = μv², where μ is the linear mass density and v is the wave speed calculated in Step 2.
Step 5: Calculate the average power (P) transmitted by the wave. Use the formula P = (1/2)μvω²A², where A is the amplitude of the wave (2.30 mm converted to meters), μ is the linear mass density, v is the wave speed, and ω is the angular frequency.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Function

The wave function y(x, t) = 2.30mm cos[(16.98 rad/m)x + (742 rad/s)t] describes the displacement of a wave at position x and time t. It includes parameters like amplitude, wave number, and angular frequency, which are crucial for understanding wave behavior and calculating properties like speed and energy.
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Intro to Wave Functions

Tension in the Rope

Tension in the rope is a force that affects wave propagation speed. It can be calculated using the formula v = sqrt(T/μ), where v is the wave speed, T is the tension, and μ is the linear mass density (mass per unit length). Understanding tension is essential for determining how fast waves travel along the rope.
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Calculating Tension in a Pendulum with Energy Conservation

Average Power Transmitted by the Wave

The average power transmitted by a wave is the rate at which energy is transferred along the medium. It can be calculated using P_avg = (1/2)μω^2A^2v, where μ is the linear mass density, ω is the angular frequency, A is the amplitude, and v is the wave speed. This concept helps in understanding the energy dynamics of wave motion.
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Related Practice
Textbook Question

A jet plane at takeoff can produce sound of intensity 10.0 W/m2 at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 μW/m2. Assume that the plane behaves like a point source of sound. (a) What is the closest dis-tance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

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Textbook Question

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]y(x,t)=\(\left\)(2.30\(\operatorname{mm)}\]\cos\)[\(\left\)(16.98\(\text{ }\)rad/m\(\right\))x+(742\(\text{ }\)rad/s\(\right\))t]. Being more practical, you measure the rope to have a length of 1.35 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

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Textbook Question

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]y(x,t)=\(\left\)(2.30\(\operatorname{mm)}\]\cos\)[\(\left\)(16.98\(\text{ }\)rad/m\(\right\))x+(742\(\text{ }\)rad/s\(\right\))t]. Being more practical, you measure the rope to have a length of 1.35 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength.

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Textbook Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the fundamental?

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Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m2 at a distance of 4.3 m from the source. What is the intensity at a distance of 3.1 m from the source?

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Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m2 at a distance of 4.3 m from the source. How much sound energy does the source emit in one hour if its power output remains constant?

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