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Chapter 8: Problem 51

Calculate the angular speed of the Moon as it orbits Earth (recall, the Moon completes one orbit about Earth in \(27.4\) days and the Earth-Moon distance is \(3.84 \times 10^{8} \mathrm{~m}\) ). SSM

Short Answer

Expert verified
The angular speed of the Moon as it orbits Earth is approximately \(2.66 \times 10^{-6} \, s^{-1}\).

Step by step solution

01

Convert Time Units

First, it's important to realize that we cannot use days directly in our formula. Instead, we need to convert the time to seconds since \(s^{-1}\) is the SI unit of angular speed. 1 day equals to 86400 seconds, so 27.4 days equals to \(27.4 \times 86400 = 2,366,560 \) seconds.
02

Plug Values Into Formula

Use the formula for angular speed \(\omega = \frac{2\pi}{T}\). Here, T is the period of rotation. Plug in the values into the formula, \(\omega = \frac{2\pi}{2,366,560 \, s}\).
03

Do the Calculation

Now, perform the division to find the angular speed. A calculator gives \(\omega \approx 2.66 \times 10^{-6} \, s^{-1}\).

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Key Concepts

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

moon orbit
The Moon orbits the Earth in a path that is almost circular. It completes one full orbit around the Earth approximately every 27.4 days. This movement is essential for our understanding of many celestial calculations, such as determining the angular speed of the Moon. The orbit is elliptical, but for the simplicity of calculations, it can be approximated as a circular path when calculating angular speed.
The orbit of the Moon affects various phenomena on Earth, like tides, and plays a significant role in our calendar systems. Moreover, understanding its orbit is important for planning space missions to and from the Moon.
SI units
SI units, also known as the International System of Units, are the standard units used in scientific calculations worldwide. When computing quantities such as angular speed, it becomes crucial to convert all measurements into SI units. This ensures consistency and comprehensibility in the data. For angular speed, the SI unit used is radians per second \(s^{-1}\).
Using SI units helps scientists and engineers to communicate measurements and results without confusion. It's why in science education, emphasis is placed on learning these units thoroughly across different domains.
period of rotation
The period of rotation refers to the time it takes for an object to complete one full rotation or orbit around another object. In the context of the Moon orbiting Earth, this is the 27.4 days that the Moon takes to go around Earth. Understanding this concept is fundamental for calculating angular speed.
The formula for angular speed is \( \omega = \frac{2\pi}{T} \), where \( \omega \) is the angular speed and \( T \) is the period of rotation in seconds. By knowing how long it takes for the Moon to complete its orbit, we can use this information to calculate its angular speed.
unit conversion
Unit conversion is an essential step when performing calculations that involve different units like days and seconds. Since angular speed must be in radians per second, converting the Moon's orbital period from days to seconds is necessary.
To convert days into seconds, multiply the number of days by the number of seconds in a day: \( 27.4 \times 86400 = 2,366,560 \) seconds. This ensures that the period of rotation is consistent with the SI unit of time, facilitating accurate calculations of angular speed.
  • Important for consistency in scientific measurements
  • Essential when using formulas with specific unit requirements

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Most popular questions from this chapter

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