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Chapter 6: Problem 55

The speed, wavelength, and frequency of gravitational waves are related as \(c=\lambda \times f\). If we were to observe a gravitational wave from a distant cosmic event with a frequency of 10 hertz (Hz), what would be the wavelength of the gravitational wave?

Short Answer

Expert verified
The wavelength of the gravitational wave is 3.0 × 10^7 meters.

Step by step solution

01

Identify the Given Variables

The problem gives the frequency (f) of the gravitational wave, which is 10 Hz. It also provides the speed of the wave (c), which for gravitational waves is the speed of light: approximately 3.0 × 10^8 meters per second (m/s).
02

Use the Relationship Formula

The relationship between speed (c), wavelength (λ), and frequency (f) for gravitational waves is given by the formula: \[ c = \lambda \times f \]
03

Rearrange the Formula to Solve for Wavelength

To find the wavelength (λ), rearrange the formula to solve for λ: \[ \lambda = \frac{c}{f} \]
04

Substitute the Given Values into the Rearranged Formula

Substitute the values for c and f into the rearranged formula:\[ \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{10 \text{ Hz}} \]
05

Calculate the Wavelength

Perform the division to find the wavelength (λ): \[ \lambda = 3.0 \times 10^7 \text{ m} \]

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Key Concepts

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

wavelength
The wavelength of a wave is the distance between two consecutive peaks or troughs. This concept is essential to understand because it helps in visualizing how the wave travels through space. For gravitational waves, the wavelength tells us how 'stretched out' the wave is. It is measured in meters (m). The longer the wavelength, the lower the frequency of the wave, and vice versa. The relationship between wavelength, frequency, and speed is crucial in many physics problems, including gravitational waves.
frequency
Frequency refers to how often the peaks of a wave pass a particular point in one second. It is usually measured in hertz (Hz). In our example, we have a gravitational wave with a frequency of 10 Hz, which means 10 peaks pass a point every second. Frequency is inversely proportional to wavelength, meaning that as frequency increases, the wavelength decreases. This is key to solving many wave-related problems, as knowing one allows you to find the other when the speed is known.
speed of light
The speed of light is a constant in physics, often denoted as 'c'. It is approximately 3.0 × 10^8 meters per second (m/s). For gravitational waves, this speed is the same as that of light in a vacuum. Understanding the speed of light is crucial because it links wavelength and frequency through the formula: \(c = \lambda \times f\). This relationship helps solve problems involving how fast a wave travels, its wavelength, and its frequency. By rearranging the formula, we can find any one of the variables if we know the other two.

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Most popular questions from this chapter

You are shopping for telescopes online. You find two in your price range. One of these has an aperture of \(20 \mathrm{cm}\), and one has an aperture of \(30 \mathrm{cm} .\) Which should you choose, and why? a. The \(20 \mathrm{cm}\), because the light-gathering power will be better. b. The \(20 \mathrm{cm}\), because the image size will be larger. c. The \(30 \mathrm{cm}\), because the light-gathering power will be better. d. The \(30 \mathrm{cm},\) because the image size will be larger.

Assume that the maximum aperture of the human eye, \(D\), is approximately \(8 \mathrm{mm}\) and the average wavelength of visible light, \(\lambda,\) is \(5.5 \times 10^{-4} \mathrm{mm}\) a. Calculate the diffraction limit of the human eye in visible light. b. How does the diffraction limit compare with the actual resolution of \(1-2\) arcmin \((60-120 \text { arcsec }) ?\) c. To what do you attribute the difference?

Which of the following can be observed from Earth's surface? a. radio waves b. gamma radiation c. far UV light d. X-ray light e. visible light

Assume that you have a telescope with an aperture of 1 meter. Compare the telescope's theoretical resolution when you are observing in the near-infrared region of the spectrum \((\lambda=1,000 \mathrm{nm})\) with that when you are observing in the violet region of the spectrum \((\lambda=400 \mathrm{nm})\).

T/F: In the past, astronomers placed telescopes in highflying aircraft in an effort to rise above the water vapor in Earth's atmosphere.
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