Find the linear speed v for each of the following.


a point on the equator moving due to Earth's rotation, if the radius is 3960 mi

Verified step by step guidance

1. Identify the formula for linear speed: \( v = r \cdot \omega \), where \( r \) is the radius and \( \omega \) is the angular speed.
2. Determine the angular speed \( \omega \) of the Earth. Since the Earth makes one full rotation (\( 2\pi \) radians) in 24 hours, \( \omega = \frac{2\pi}{24} \) radians per hour.
3. Substitute the given radius \( r = 3960 \) miles and the calculated \( \omega \) into the linear speed formula: \( v = 3960 \cdot \frac{2\pi}{24} \).
4. Simplify the expression by calculating \( \frac{2\pi}{24} \) to find the angular speed in radians per hour.
5. Multiply the simplified angular speed by the radius to find the linear speed \( v \).

Key Concepts

Linear Speed

Linear speed refers to the distance traveled per unit of time. In the context of circular motion, it can be calculated using the formula v = d/t, where d is the distance traveled along the circular path and t is the time taken. For a point on the equator, this speed is determined by the circumference of the Earth and the time it takes for one complete rotation.

Circumference of a Circle

The circumference of a circle is the total distance around it, calculated using the formula C = 2πr, where r is the radius. For the Earth, with a radius of 3960 miles, the circumference provides the distance a point on the equator travels in one complete rotation. This value is essential for determining the linear speed of a point on the equator.

Earth's Rotation Period

The Earth's rotation period is the time it takes for the Earth to complete one full rotation on its axis, approximately 24 hours. This period is crucial for calculating linear speed, as it serves as the time variable in the speed formula. Understanding this concept allows us to relate the distance traveled (circumference) to the time taken, yielding the linear speed of a point on the equator.