Problem 103 - Precalculus : Building Concepts and Connections

Question

Earth rotates about an axis through its poles, making one revolution per day. (a) What is the exact angular speed of Earth about its axis? Express your answer in both degrees per hour and radians per hour. (b) The radius of Earth is approximately 3900 miles. What distance is traversed by a point on Earth's surface at the equator during any 8 -hour interval as a result of Earth's rotation about its axis? Express your answer in miles. (c) What is the linear speed (in miles per hour) of the point in part (b)?

Step-by-Step Solution

Answer

a) The Earth's angular speed is \( \frac{\pi}{12} \) radians per hour or 15 degrees per hour. b) A point on the Earth's surface at the equator traverses roughly \( 2600\pi \) miles in 8 hours. c) The linear speed of this point is approximately \( 325\pi \) miles per hour.

Step 1: Determine the Angular Speed

The Earth completes one revolution, or \(2\pi\) radians, in 24 hours. Hence, the angular speed \(w\) in radians can be computed as\[w = \frac{2\pi \, \text{radians}}{24 \, \text{hours}} = \frac{\pi}{12} \, \text{radians per hour}\]To find this in degrees, knowing that \(\pi\) radians equals 180 degrees, we convert as follows: \[w = \frac{180}{12} = 15 \, \text{degrees per hour}\]

Step 2: Calculate the Distance Traversed

The distance traversed by a point on Earth's surface at the equator in 8 hours is the product of the Earth's radius and the angular displacement. Here, the Earth's radius \(r\) is 3900 miles and the angular displacement for 8 hours is \(8w\). So,\[d = r * 8w = 3900 * 8 * \frac{\pi}{12} = 2600\pi \, \text{miles}\]

Step 3: Compute the Linear Speed

The linear speed \(v\) is the distance covered per unit of time. So we divide the distance \(d\) we computed by the time in hours.\[v = \frac{d}{8} = \frac{2600\pi}{8} = 325\pi \, \text{miles per hour}\]

Key Concepts

Radians per hour

Angular speed is a measure of how quickly an object moves through an angle. In the context of Earth's rotation, it helps us understand how fast the Earth spins. It’s helpful to know about radians when dealing with circular motion, because radians provide a natural way to express angles. A full circle has an angle of \(2\pi\) radians.

When calculating Earth’s angular speed in radians per hour, we recognize that one full revolution is \(2\pi\) radians. Earth completes a full rotation in 24 hours. So the angular speed \(w\) is:

• \(w = \frac{2\pi}{24} = \frac{\pi}{12}\) radians per hour

This simple calculation gives us a clear measure of how much the Earth rotates in terms of radians, making it easy to compare with other rotating objects or systems.

Linear Speed

Linear speed describes how fast a point travels over the Earth's surface, which is especially intriguing when examining points at different latitudes. At the equator, where the Earth's radius is approximately 3900 miles, the linear speed is particularly relevant.

To determine linear speed, we first calculated the distance a point travels in a given time, using the formula for arc length, which is the product of the radius and angular displacement in radians:

• Distance for 8 hours = \(3900 \times 8 \times \frac{\pi}{12} = 2600\pi\) miles

Once the distance is known, the linear speed \(v\) is simply distance divided by time:

• \(v = \frac{2600\pi}{8} = 325\pi\) miles per hour

Understanding linear speed helps us comprehend the velocity at which different parts of the Earth move, which is essential for various practical applications.

Equator

The equator is a special line around the Earth, equidistant from the poles, which divides the globe into the Northern and Southern Hemispheres. The significance of the equator in calculating speed and distance arises from its position being the furthest from the axis, making it the longest circle on the globe.

Consequently, a point on the equator travels the largest distance during Earth's rotation compared to points at other latitudes. This is why speeds measured at the equator are the maximum possible speeds on Earth's surface due to its circumference being the largest. Understanding motion at the equator offers great insights, especially in fields like meteorology and aviation, where Earth’s rotation impacts weather patterns and flight paths.