Problem 33 - Applied Physics

Question

The earth rotates on its axis at an angular speed of \(1 \mathrm{rev} / 24 \mathrm{~h}\). Find the linear speed (in \(\mathrm{km} / \mathrm{h}\) ) (a) of Singapore, which is nearly on the equator. (b) of Houston, which is approximately \(30.0^{\circ}\) north latitude. (c) of Minneapolis, which is approximately \(45.0^{\circ}\) north latitude. (d) of Anchorage, which is approximately \(60.0^{\circ}\) north latitude.

Step-by-Step Solution

Answer

Singapore: 1669.14 km/h, Houston: 1444.33 km/h, Minneapolis: 1178.37 km/h, Anchorage: 832.03 km/h.

Step 1: Convert Angular Speed to Proper Units

The given angular speed is \(1 \text{ rev / 24 h}\). First, convert this to radians per hour. Since \(1 \text{ rev} = 2\pi \text{ radians}\), the angular speed \(\omega\) is:\[\omega = \frac{2\pi \text{ radians}}{24 \text{ h}} = \frac{\pi}{12} \text{ radians per hour}.\]

Step 2: Determine Radius for Each Location

For Singapore, the radius is equal to Earth's radius, approximately \(R = 6371 \text{ km}\), as it is on the equator. For other locations, the effective radius must be adjusted based on the latitude using the formula:\[R' = R \cdot \cos(\text{latitude}).\]

Step 3: Calculate Linear Speed for Singapore

For Singapore, no adjustment is needed, so the linear speed \(v\) is:\[v = R \cdot \omega = 6371 \text{ km} \cdot \frac{\pi}{12} \approx 1669.14 \text{ km/h}.\]

Step 4: Calculate Linear Speed for Houston

Houston is at \(30.0^{\circ}\) north, so:1. Find the effective radius: \[R' = 6371 \cdot \cos(30.0^{\circ}) \approx 6371 \cdot 0.8660 \approx 5517.6 \text{ km}.\]2. Calculate speed:\[v = R' \cdot \omega = 5517.6 \cdot \frac{\pi}{12} \approx 1444.33 \text{ km/h}.\]

Step 5: Calculate Linear Speed for Minneapolis

Minneapolis is at \(45.0^{\circ}\) north:1. Effective radius:\[R' = 6371 \cdot \cos(45.0^{\circ}) \approx 6371 \cdot 0.7071 \approx 4502.4 \text{ km}.\]2. Speed:\[v = 4502.4 \cdot \frac{\pi}{12} \approx 1178.37 \text{ km/h}.\]

Step 6: Calculate Linear Speed for Anchorage

Anchorage is at \(60.0^{\circ}\) north:1. Effective radius:\[R' = 6371 \cdot \cos(60.0^{\circ}) \approx 6371 \cdot 0.5 = 3185.5 \text{ km}.\]2. Speed:\[v = 3185.5 \cdot \frac{\pi}{12} \approx 832.03 \text{ km/h}.\]

Key Concepts

Linear Speed

When we talk about linear speed in the context of the Earth's rotation, we are referring to how fast a point on the Earth's surface moves in a linear path as the Earth spins. Imagine a person standing on the equator; they trace out a circle in space with the radius being the distance from the Earth's center to the equator. Every complete rotation takes 24 hours, resulting in a linear speed. The formula for linear speed is: \\[ v = R \cdot \omega \] \where \( v \) is the linear speed, \( R \) is the radius at the equator (approximately 6371 km for Earth), and \( \omega \) is the angular speed in radians per hour.
For Singapore, because it's near the equator, we don't need to adjust the radius, simply multiply Earth's radius by the angular speed. This makes linear speed calculation straightforward for locations near the equator.

Latitude Adjustments

Not all locations on Earth experience the same linear speed due to their different latitudes. As you move away from the equator towards the poles, the linear path covered decreases, which requires horizontal adjustments of the radius. This is called latitude adjustments.

• At the equator (0° latitude), the linear speed is maximized because the path is the longest.
• As you move towards the poles, say at 60° latitude, the path becomes smaller, similar to a smaller circle.
• To calculate the effective radius at any latitude \( \theta \), you use: \\[ R' = R \cdot \cos(\theta) \]

This equation helps us find the reduced radius that accounts for the position's latitude, resulting in a lower linear speed as you move away from the equator.

Earth Rotation

The Earth completes one full rotation on its axis approximately every 24 hours. This motion is predictable and forms the basis of dividing our time into the day-night cycle. The rate of rotation is called angular speed, measured typically in revolutions per hour or radians per hour.

• 1 revolution is equal to \( 2\pi \) radians.
• Angular speed for Earth is given as \( \frac{1 \text{ rev}}{24 \text{ h}} \), translated to radians by multiplying by \( 2\pi \), yielding \( \frac{\pi}{12} \) radians per hour.

This constant angular speed governs how quickly points on Earth move in a circular path, influencing phenomena like time zones and the apparent motion of stars.

Radius Calculation

The radius of the Earth is a fundamental component when calculating linear speed, particularly when factoring in latitude adjustments. At the equator, the Earth’s radius is defined as approximately 6371 km, often used as a standard measure in calculations.
The challenge arises when calculating the effective radius at any latitude besides the equator. You need to adjust the radius by the cosine of the latitude angle: \\[ R' = R \cdot \cos(\theta) \]
For instance, at 30° north latitude, used in calculating the radius for Houston, the radius effectively becomes shorter due to the cosine function. By applying this adjustment, one can determine how much smaller or shorter the circle of rotation is at different latitudes, accurately reflecting the path’s length on which a point travels.