Question

Find the angular speed of the Earth as it orbits about the Sun. Give your answer in radians per second (rad/s).

Short answer

The angular speed of the Earth as it orbits the Sun is approximately \\(1.99 \times 10^{-7}\\) rad/s.

Step-by-step solution

Identify Known Values

We know that the Earth completes one orbit around the Sun in one year. The orbit is approximately circular, so we can use this to determine the angular speed.

Calculate Total Angle in Radians

For a circular path, the Earth completes one full revolution around the Sun, which is equivalent to an angle of \(2\pi\) radians per year.

Convert Time from Years to Seconds

First, convert one year into seconds. There are 365.25 days on average in a year (accounting for leap years), and each day consists of 24 hours, each hour has 3600 seconds. Therefore, the number of seconds in a year is \(365.25 \times 24 \times 3600\).

Calculate Angular Speed

Angular speed, \(\omega\), is calculated using the formula \(\omega = \frac{\theta}{t}\). Here, \(\theta\) is \(2\pi\) radians, and \(t\) is the number of seconds in a year. Thus, \(\omega = \frac{2\pi}{365.25 \times 24 \times 3600}\) radians per second.

Simplify the Expression

Perform the calculation: \(\omega = \frac{2\pi}{31,557,600}\) radians per second, which simplifies to approximately \(1.99 \times 10^{-7}\) rad/s.

Key concepts

circular orbit

A circular orbit is a path that a planet, like Earth, takes around a star, such as the Sun, where the distance from the planet to the star remains constant. In simpler terms, if you think of the Earth's path around the Sun as a circle, the orbit is circular. This is an important assumption that makes complex calculations more manageable, allowing us to use geometric principles to find parameters like angular speed. For Earth, although its orbit isn't a perfect circle, it's very close, so we often approximate it as circular for calculations.

radians per second

Radians per second, is a unit of angular speed. Angular speed measures how quickly an object travels around a circular path in terms of the angle it moves through. This can be thought of as how fast something rotates or spins. When we say Earth's angular speed is in radians per second, we're talking about the angle the Earth sweeps out about the Sun per second of time. Since a full circle has \(2\pi\) radians, if an object makes a revolution over a known time, we can express that time in radians per second.

leap year conversion

Leap year conversion takes into account the way we count years. A solar year, the time it takes Earth to complete one full orbit around the Sun, is about \(365.25\) days long.To keep our calendar years aligned with this solar year, we add an extra day every four years—this is a leap year. When calculating time-related quantities, such as the number of seconds in a year, leap year conversion ensures we account for this extra quarter-day each year, allowing for accurate long-term calculations.

year to second conversion

Converting years to seconds involves a bit of arithmetic. To do this conversion precisely, you multiply the number of days in a year by the number of hours in a day, the number of minutes in an hour, and the number of seconds in a minute.

• Days in a year (on average): 365.25 (accounting for leap years).
• Hours in a day: 24.
• Minutes in an hour: 60.
• Seconds in a minute: 60.

Putting this together, the calculation is:\[ 365.25 \times 24 \times 60 \times 60 = 31,557,600 \text{ seconds} \]This number is essential for many scientific calculations involving annual cycles.