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?

Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

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Young & Freedman Calc 14th Edition

Ch 15: Mechanical Waves

문제 27b

15장, 문제 27b

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?

검증된 단계별 안내

First, understand that the intensity of a sound wave is defined as the power per unit area. The formula for intensity $ I $ is $ I = \frac{P}{A} $, where $ P $ is the power and $ A $ is the area over which the power is distributed.

Since the sound waves are spreading out equally in all directions, they form a sphere around the point source. The area $ A $ of a sphere is given by $ A = 4\pi r^2 $, where $ r $ is the radius of the sphere.

Substitute the given values into the intensity formula to find the power $ P $. Use $ I = 0.026 \text{ W/m}^2 $ and $ r = 4.3 \text{ m} $ to calculate $ P = I \times A = 0.026 \times 4\pi (4.3)^2 $.

Once you have the power $ P $, calculate the total energy emitted in one hour. Energy $ E $ is the product of power and time, $ E = P \times t $. Convert one hour into seconds (3600 seconds) and use $ E = P \times 3600 $.

Ensure all units are consistent and check your calculations for any errors. The final expression for energy will give you the total sound energy emitted by the source in one hour.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sound Intensity

Sound intensity is the power per unit area carried by a sound wave, measured in watts per square meter (W/m^2). It describes how much energy passes through a given area and is crucial for determining the energy output of a sound source. In this problem, the intensity at a specific distance helps calculate the total energy emitted.

Inverse Square Law

The inverse square law states that the intensity of a wave, such as sound, decreases with the square of the distance from the source. This principle is essential for understanding how sound spreads out in space, affecting the calculation of total energy emitted by the source over a given area.

Power and Energy Relationship

Power is the rate at which energy is emitted or transferred, measured in watts (W). Energy is the total amount of work done or heat transferred, measured in joules (J). The relationship between power and energy is given by the formula: Energy = Power × Time. This concept is key to calculating the total sound energy emitted over a specified time period.

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

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Threshold of Pain. 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/m². An intensity of 1.0 W/m² is often used as the 'threshold of pain.' How much closer to the source can you move before the sound intensity reaches this threshold?

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교과서 질문

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?

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=$\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 $1.35$\text{ m}$$ and a mass of $0.00338$\operatorname{kg}$$ . You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

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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