Question

Earth rotates on an axis through its poles. The distance from the axis to a location on Earth at \(30^{\circ}\) north latitude is about 3429.5 miles Therefore, a location on Earth at \(30^{\circ}\) north latitude is spinning on a circle of radius 3429.5 miles. Compute the linear speed on the surface of Earth at \(30^{\circ}\) north latitude.

Short answer

The linear speed is approximately 897 miles per hour.

Step-by-step solution

- Understand the Problem

We need to calculate the linear speed of a location on Earth at 30° north latitude. The given radius from the axis to the location is 3429.5 miles. We'll use the formula for linear speed: \( v = \omega \times r \), where \( v \) is the linear speed, \( \omega \) is the angular speed, and \( r \) is the given radius.

- Determine Earth's Angular Speed

Earth completes one full rotation every 24 hours. This means it rotates \( 2\pi \) radians in 24 hours. Angular speed is given by: \( \omega = \frac{2\pi}{24 \, hours} = \frac{2\pi}{86400 \, seconds} \).

- Simplify Angular Speed

To simplify: \( \omega = \frac{2\pi}{86400} \) radians per second. This reduces to approximately \( \omega \approx 7.2722 \times 10^{-5} \) radians per second.

- Calculate Linear Speed

Using the formula for linear speed, \( v = \omega \times r \), substitute the values: \( v = (7.2722 \times 10^{-5}) \times 3429.5 \.\) Multiply to find: \( v \approx 0.249 \) miles per second.

- Convert to Preferred Units

To convert the speed to miles per hour, multiply by the number of seconds in an hour (3600 seconds): \( 0.249 \, miles/second \times 3600 \, seconds/hour \approx 897 \, miles/hour.\)

Key concepts

Angular Speed

Angular speed is a measure of how fast an object rotates or spins. It shows the angle of rotation per unit time. Since Earth's rotation is considered constant, angular speed remains the same at any point on Earth.
For the Earth, one full rotation equals to 360 degrees or \(2\pi\) radians. This rotation happens every 24 hours. We then convert the time into seconds to ease further calculations, noting that 24 hours equals 86400 seconds.
Using these values, angular speed (\( \omega \)) can be computed as follows: \( \omega = \frac{2\pi}{86400} \) radians per second. This value is approximately \( 7.2722 \times 10^{-5} \) radians per second.

Radians Per Second

The unit radians per second is used to measure angular speed. It’s an essential unit in rotational motion as it connects angular displacement and time.
One radian is roughly 57.3 degrees and is the angle subtended by an arc of a circle that is equal in length to the circle's radius.
Since we already know Earth's full rotation is expressed in radians (\( 2\pi \)), and the rotation time in seconds (86400 seconds), we derive the angular speed directly.
Through step-by-step calculations, we get: \( \omega = \frac{2\pi}{86400} \approx 7.2722 \times 10^{-5} \) radians per second. This is an extremely small number representing how little the Earth turns in each tiny fraction of a second.

Radius of Rotation

The radius of rotation at a specific latitude is the distance from the rotation axis to the location on Earth's surface. Here it's given as 3429.5 miles at 30° north latitude.
This radius affects the linear speed because the farther a point is from the axis, the higher its linear speed for the same angular speed.
For our calculation: \(radius = 3429.5 \) miles. Using the formula for linear speed: \( v = \omega \times r \), substituting \( \omega \approx 7.2722 \times 10^{-5} \) radians per second, we achieve \( v \approx 0.249 \) miles per second. This is the linear speed at the given latitude.

Unit Conversion

Unit conversion is the process of converting one unit of measure to another. In physics problems, this is crucial to ensure consistency and to interpret results correctly.
In our exercise, we need to convert the linear speed from miles per second to miles per hour. Since 1 hour equals 3600 seconds, we multiply: \( 0.249 \) miles per second \( \times 3600 \) seconds per hour ≈ 897 miles per hour.
The conversion confirms how speeds we compute in small units translate into more comprehensible values, such as miles per hour, which is more intuitive for understanding how fast a point is moving on the Earth's surface.