Chapter 6: Problem 12

The rays that could be used to represent the steady sound emitted by a warning siren on the top of a pole would look similar to the gravitational field lines (Section 2.8) representing the field around an object. Explain why they are alike, and point out the one important difference.

Short Answer

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Question: Compare the representation of sound rays emitted by a warning siren with the representation of gravitational field lines. Explain why they are similar in nature and point out one important difference. Answer: Sound rays and gravitational field lines are both similar in that they represent the spread of a physical phenomenon (sound waves and gravitational force) from a source (siren and massive object), and they both follow the inverse-square law. The major difference between them is that sound intensity represents a scalar quantity, while the gravitational field strength represents a vector quantity.

Step by step solution

Understanding the concept of sound rays and gravitational field lines

Sound rays represent the directions in which sound waves move away from a source, like a warning siren. Gravitational field lines, on the other hand, represent the field of gravitational force surrounding an object, showing the direction in which another object would experience a force due to the gravitational attraction between them.

Comparing sound rays and gravitational field lines

Both sound rays and gravitational field lines are similar in the sense that they represent the spread of a physical phenomenon (sound waves and gravitational force) from a source (siren and massive object). The lines/rays are used to visualize the path taken by the waves/force and how they diminish in intensity as they move away from the source.

Identifying the one important difference

The one important difference between the steady sound emitted by a warning siren and the gravitational field lines is the way they diminish as they move away from the source. Sound waves follow an inverse-square law, which means that the intensity of sound decreases with the square of the distance from the source. Thus, I \propto \frac {1}{r^2}, where I is the intensity of sound and r is the distance from the source. On the other hand, the gravitational force experienced by an object also follows the inverse-square law, where the force F between two objects with masses m1 and m2 is given by F = G\frac{m1 × m2}{r^2}, where G is the gravitational constant and r is the distance between the two objects. However, when we look at gravitational field lines, they represent the field strength (effective force) of the object on another object, not the intensity of the force. The field strength also follows the inverse-square law, but it represents a vector quantity, while sound intensity represents a scalar quantity. In summary, the major difference between sound rays and gravitational field lines is that sound intensity follows the inverse-square law and represents a scalar quantity, while the gravitational field strength also follows the inverse-square law but represents a vector quantity.

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Key Concepts

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

Understanding Sound Rays

Imagine a warning siren on top of a pole, relentlessly broadcasting its message in all directions. This broadcast is a practical example of how sound travels through the medium of air. Sound rays can be thought of as the invisible lines that illustrate the path of these sound waves emanating from the source. Just like light from a lamp, the waves spread out, and with sound, it's the vibration of air particles that carry the melody or alert across distances. That's why, if you're close to the siren, the sound is loud and clear, but as you walk away, it becomes fainter. This phenomenon is due to the fact that the energy of sound becomes spread over a larger area as you increase the distance—thus, the sound waves' intensity follows an inverse-square relationship with the distance from the source.

The concept of sound rays not only helps us understand how sound travel occurs but also aids in visualizing the way sound intensity diminishes. In an educational setting, we use the representation of sound rays to depict the direction of the waves, which also informs us about the orientation of the sound's energy flow.

Gravitational Field Lines Explained

In the realm of physics, when we venture into the domain of mass and the forces that objects exert on one another, we encounter gravitational field lines. These lines are a way to visualize how the gravitational force emanates from objects, especially those with significant mass. They are the directional paths that a smaller object would follow if it were solely under the influence of the gravitational pull of a more massive object, like a planet or a star. These field lines emanate from the center of mass of the object and extend infinitely into space, growing weaker with increasing distance.

In the educational framework, the notion of gravitational field lines becomes important as it rigorously translates the abstract concept of gravity into a comprehensible visual aid. These lines help explain not just the directionality of the gravitational force, which is towards the massive object, but also indicate the areas where that force is strongest – evident by the density of lines.

Demystifying the Inverse-Square Law

When we delve into the intricacies of wave behavior and gravitational forces, the inverse-square law stands out as a cornerstone principle. It is a mathematical expression that provides insight into how certain physical quantities decrease in intensity with the square of the distance from the source. This law applies remarkably well to both the aforementioned phenomena: sound waves and gravitational forces.

For sound rays, as it was stated earlier, the intensity of the sound diminishes according to the inverse-square law - mathematically represented by the equation \( I \propto \frac{1}{r^2} \), with \( I \) being the intensity and \( r \) the distance from the source. A similar mathematical relationship is at play for gravitational forces, where the force between two objects is calculated as \( F = G\frac{m1 \times m2}{r^2} \). Despite the similarities, this principle engages differently with the two concepts. For sound, it affects a scalar quantity (sound intensity), and for gravity, it affects a vector quantity (gravitational force). Therefore, comprehending this law is pivotal to understanding how physical influences attenuate over space and this fundamental principle has broad applications across various fields of science and engineering.

The rays that could be used to represent the steady sound emitted by a warning siren on the top of a pole would look similar to the gravitational field lines (Section 2.8) representing the field around an object. Explain why they are alike, and point out the one important difference.

Short Answer

Step by step solution

Understanding the concept of sound rays and gravitational field lines

Comparing sound rays and gravitational field lines

Identifying the one important difference

Key Concepts

Understanding Sound Rays

Gravitational Field Lines Explained

Demystifying the Inverse-Square Law

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