Problem 26 - College Physics

Question

The Earth rotates on its axis once a day and revolves around the Sun once a year. (a) Which is greater, the rotating angular speed or the revolving angular speed? Why? (b) Calculate both angular speeds in \(\mathrm{rad} / \mathrm{s}\).

Step-by-Step Solution

Answer

The rotational angular speed is greater. Rotation: \(7.2722 \times 10^{-5} \text{ rad/s}\); Revolution: \(1.991 \times 10^{-7} \text{ rad/s}\).

Step 1: Understanding Angular Speed

Angular speed is the rate at which an object rotates or revolves around an axis. It is measured in radians per second (\(\mathrm{rad/s}\)). In this problem, we are comparing the Earth's angular speed of rotation about its axis and its angular speed of revolution around the Sun.

Step 2: Calculate Rotational Angular Speed

To find the Earth's rotational angular speed, we consider that it completes one full rotation (\(2\pi\) radians) every 24 hours. Thus, the rotational angular speed \(\omega_r\) is:\[\omega_r = \frac{2\pi \text{ radians}}{24 \times 60 \times 60 \text{ seconds}}\]Simplifying, we get:\[\omega_r \approx 7.2722 \times 10^{-5} \text{ rad/s}\]

Step 3: Calculate Revolving Angular Speed

To find the Earth's revolving angular speed, we consider that it completes one full revolution (\(2\pi\) radians) every year. Considering a year as approximately 365.25 days (including leap years), we find \(\omega_R\):\[\omega_R = \frac{2\pi \text{ radians}}{365.25 \times 24 \times 60 \times 60 \text{ seconds}}\]Simplifying, we get:\[\omega_R \approx 1.991 \times 10^{-7} \text{ rad/s}\]

Step 4: Compare Angular Speeds

To determine which angular speed is greater, compare \(\omega_r\) and \(\omega_R\):- Rotational angular speed \(\omega_r \approx 7.2722 \times 10^{-5} \text{ rad/s}\)- Revolving angular speed \(\omega_R \approx 1.991 \times 10^{-7} \text{ rad/s}\)Clearly, \(\omega_r > \omega_R\). So, the Earth's rotational angular speed is greater than its revolving angular speed.

Step 5: Conclusion

We determined and calculated that the Earth's rotational angular speed about its axis is greater than its revolving angular speed around the Sun. The rotational angular speed is approximately \(7.2722 \times 10^{-5} \text{ rad/s}\) whereas the revolving angular speed is approximately \(1.991 \times 10^{-7} \text{ rad/s}\).

Key Concepts

Earth's Rotation

The Earth's rotation refers to its spin around its own axis. It completes one full rotation approximately every 24 hours, which results in the cycle of day and night. This constant spinning motion is responsible for the 24-hour duration of a day. The angular speed related to this rotation can be calculated in radians per second. Considering the Earth rotates once per day, which involves moving through a full circle of \(2\pi\) radians, we use the formula for angular speed:

• \(\omega_r = \frac{2\pi \text{ radians}}{24 \times 60 \times 60 \text{ seconds}}\)

After simplifying, you find the rotational angular speed of the Earth is approximately \(7.2722 \times 10^{-5} \text{ rad/s}\). This consistent rotation speed impacts many aspects of life on Earth, including the daily movement of the sun and stars in our sky.

Earth's Revolution

The Earth's revolution is the term used to describe its journey around the Sun. This movement takes about one year to complete, encompassing approximately 365.25 days, accounting for leap years. During this journey, the Earth travels in an elliptical orbit, covering a full circle of \(2\pi\) radians.To calculate the angular speed of Earth's revolution, you apply the following formula:

• \(\omega_R = \frac{2\pi \text{ radians}}{365.25 \times 24 \times 60 \times 60 \text{ seconds}}\)

This results in an angular speed of about \(1.991 \times 10^{-7} \text{ rad/s}\). This revolution is responsible for changing seasons and contributes to varying solar angles throughout the year. Though crucial for life on Earth, the angular speed of Earth's revolution is much slower than that of its rotation.

Radian per Second

A radian is the standard unit of angular measure used in many areas of mathematics. When measuring angular speed, radians per second (\(\text{rad/s}\)) is often used. 1 radian is the angle formed when the radius of a circle is wrapped along its circumference. Therefore, one complete revolution around a circle is equal to \(2\pi\) radians. In the context of Earth's movements, calculating angular speeds involves expressing these revolutions in terms of radians per second. This brings a consistent measurement unit when tackling different physics problems related to rotational or circular motion.

Physics Problem Solving

Physics often involves breaking down problems into simpler, manageable calculations. With concepts like Earth's rotation and revolution, the key steps include:

• Understand the problem: Determine what you need to find. For Earth's movement, it’s the angular speeds.
• Choose the right formula: Use the appropriate formula that relates to the scenario. For angular speed, the formula is usually distance per time, in this case, \(2\pi\) radians per rotation or revolution.
• Input correct values: Ensure you have accurate values for time periods and distances. Here, consider the length of a day and a year in seconds.
• Calculate and simplify: Perform the calculations carefully and simplify to meaningful units like rad/s.

By following these logical steps, complicated physics problems can become much easier to manage, ensuring clarity and precision in calculations and understanding results.