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The Earth rotates around its axis once approximately every 24 hours. This rotation is responsible for the phenomenon of day and night. A complete rotation means that every point on the surface of the Earth travels through a circular path.
Understanding the Earth's rotation is crucial when calculating tangential speed, which is the linear speed of any object located on the Earth's surface.
* The Earth's equatorial circumference is about 40,070 kilometers or, in this exercise, roughly 40,110,000 meters as calculated from the given radius of 6.38 million meters.
* The rotation period is used to determine how fast a point on the Earth's surface is moving in meters per second.

• This rotation period was given as 23.9 hours, which must be converted into seconds since speed is typically expressed in meters per second.

The tangential speed varies with latitude because the distance from the axis of rotation to any point on the surface decreases as you move away from the equator to the poles. At the equator, where this distance is greatest, the tangential speed is maximized.

Latitude plays a significant role in determining the tangential speed of a point on Earth's surface. Latitude measures the angle between a point on Earth and the equator.
In this exercise, calculating the latitude where the tangential speed is one-third of that at the equator involves understanding how latitude affects radius and, consequently, speed.
* At any given latitude \( \theta \), the effective radius of rotation is \( R \cos \theta \). This adjustment accounts for the fact that as you move north or south of the equator, the circle of rotation becomes smaller.
* To solve for \( \theta \) when given a specific speed condition, we use the equation \( v_\theta = \frac{1}{3}v \), leading to \( \cos \theta = \frac{1}{3} \).
By solving \( \theta = \cos^{-1}(\frac{1}{3}) \), we determine the latitude to be approximately 70.5 degrees. This latitude is significant as it shows how rotation speed decreases as you move from the equator.

Centripetal motion governs how objects move in circles, such as points on Earth due to its rotation. A fundamental aspect of centripetal motion is that there is a constant force acting towards the center of the circle, keeping the object in motion.
In the context of Earth's rotation, this centripetal force is generated by gravity, pulling objects towards Earth’s center.
* Tangential speed and centripetal force are deeply interlinked; the required centripetal force needs to increase with tangential speed.
* The formula for centripetal force is \( F_c = \frac{mv^2}{r} \), where \( F_c \) is the centripetal force, \( m \) is mass, \( v \) is tangential speed, and \( r \) is the radius of the circular path.

• As noted in the exercise, the equator has the highest tangential speed due to the larger radius, thus requiring a larger centripetal force to maintain circular motion.
• At higher latitudes, the reduced radial distance decreases the speed and the centripetal force needed.

Understanding this balance is essential for grasping how objects move smoothly without flying off their path, thanks to Earth's gravitational pull acting as the necessary centripetal force.