Question

The latitude of Lubbock, Texas (known as the Hub City of the South Plains), is \(33^{\circ}\) N. What is its rotational speed, assuming the radius of the Earth at the Equator to be \(6380 \mathrm{~km} ?\) a) \(464 \mathrm{~m} / \mathrm{s}\) d) \(0.464 \mathrm{~m} / \mathrm{s}\) b) \(389 \mathrm{~m} / \mathrm{s}\) e) \(0.389 \mathrm{~m} / \mathrm{s}\) c) \(253 \mathrm{~m} / \mathrm{s}\)

Short answer

Teacher: Based on the calculations, the rotational speed of Lubbock, Texas is approximately \(464 \mathrm{~m} / \mathrm{s}\).

Step-by-step solution

Find the Earth's circumference at Lubbock, Texas

First, we need to calculate the Earth's circumference at the latitude of Lubbock, Texas. Lubbock's latitude is 33 degrees North. So, we can use the formula for the circumference of a circle at latitude L: C(L) = 2 * pi * R * cos(L) where C(L) = Circumference at latitude L R = Radius of the Earth at the equator (6380 km) L = Latitude in radians (convert degrees to radians) pi = 3.14159265359

Calculate the latitude in radians

To convert the latitude from degrees to radians, we use the following formula: Radians = Degrees * (pi / 180) Radians = 33 * (pi / 180) Radians = 0.57595865316 (approx)

Calculate the circumference of the Earth at the latitude of Lubbock

Now, we can use the formula from Step 1 and the latitude in radians from Step 2 to calculate the circumference of the Earth at the latitude of Lubbock: C(L) = 2 * pi * R * cos(L) C(L) = 2 * pi * 6380 * cos(0.57595865316) C(L) = 40170.194158 km (approx)

Calculate the Earth's rotational speed at Lubbock

We know that the Earth completes 1 rotation every 24 hours (or 86400 seconds). Therefore, the rotational speed can be calculated by dividing the circumference by the time period of 1 rotation: Rotational Speed = Circumference / Time Period Rotational Speed = 40170.194158 * 1000 / 86400 (Converting km to m and dividing by the time period) Rotational Speed = 464.57 m/s (approx) Now, we can see that the most accurate option available is: a) \(464 \mathrm{~m} / \mathrm{s}\)

Key concepts

Circumference at Latitude

Imagine wrapping a string around the Earth exactly where Lubbock, Texas, is located. The length of that string would be the Earth's circumference at Lubbock's latitude. To understand this concept, we need to envision the Earth as a 3D object that bulges at the equator due to its rotation. This bulge means that the circumference of the Earth is not the same at every latitude.

At the equator, the circumference is at its maximum because it is the outermost circle, but as we move towards the poles, each parallel circle of latitude is slightly smaller. The formula to calculate the circumference of the Earth at a specific latitude 'L' is:
\[ C(L) = 2 \pi R \cos(L) \]
where 'R' represents the Earth's radius at the equator and 'L' must be in radians for the cosine function to work correctly in the formula. Using this formula helps students to calculate the circumference at any latitude, providing a crucial step in determining the rotational speed at that point on Earth.

Conversion of Degrees to Radians

Radians are the standard unit of angular measure used in mathematics. They provide a way to express angles based on the arc length and radius of a circle. When dealing with rotational motion and trigonometric functions such as sine and cosine, we must use radians rather than degrees.

The conversion between degrees and radians is pivotal because all trigonometric calculations in calculus are based on radians. To convert degrees to radians, the following relationship is essential:
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
Therefore, to convert degrees to radians, one must simply multiply the degree measure by \( \frac{\pi}{180} \). For example, to convert Lubbock's latitude of 33 degrees into radians, the calculation would be: \[ 33^\circ \times \frac{\pi}{180} = 0.57595865316 \text{ radians} (approx). \]
By mastering this conversion, students can shift seamlessly between degrees and radians, enabling accurate calculations in various physics and engineering tasks.

Earth's Rotation Period

The Earth's rotation period is the time it takes for the planet to make one complete turn on its axis. It is the basis of our concept of a day, commonly known to be 24 hours long. However, for more precise calculations, we must take into account that a day is actually slightly less – exactly 23 hours, 56 minutes, and 4 seconds of solar time. This period is also known as a sidereal day. Nevertheless, for most practical purposes, and considering the average student's needs, we approximate the rotation period to 24 hours when calculating rotational speed.

Knowing the rotation period is crucial for computing the rotational speed at any given latitude on Earth. For instance, if we know the Earth's circumference at the latitude of Lubbock, Texas, we can divide this by the rotation period (converted into seconds) to find the linear velocity or rotational speed at that latitude: \[ Rotational Speed = \frac{Circumference}{Time Period} \]
It's crucial for students to understand the significance of Earth's rotation period since it not only affects our daily lives but is also a fundamental measure in Earth sciences and physics.