Linear Speed of Earth at latitude 40° N

[Reefy]

Homework Statement

a) What is the angular speed ω about the polar axis of a point on Earth's surface at latitude 40° N? (Earth rotates about that axis.) I solved this part

b) What is the linear speed v of the point?

Homework Equations

v = ωr where r = radius of Earth at latitude 40°N

The Attempt at a Solution

I found out that r = Rcos(40°) (where R = radius of Earth) but I'm failing to understand why. How am I supposed to know to use cos(40°)? Is it a component of some sort? Where does it come from?

If the problem said longitude 40° E or W, would I then use sin(40°)? Also would I have to know to put a negative sign depending on N, S, E, W?

 

[voko]

Think about a circle of latitude. It is also known as a parallel. The circle of latitude 0, the equator, has the radius equal to that of the Earth. As the latitudes go up (Northbound or Southbound), their circles become smaller, eventually becoming just points at the poles.

Now, the question is, what is the radius of the parallel at a latitude between the equator and the pole? The truth is, it is complicated. But frequently we can assume the Earth is a perfect sphere.

Refer to the attached diagram. It depicts the intersection of the spherical Earth with a plane perpendicular to the plane of the equator. NS is the polar axis. The equator is represented by EW, the circle of latitude is by E'W'. Obviously, CE = CW = the radius of the Earth. C'E' = C'W' = the radius of the circle of latitude.

 

[Reefy]

Wow, great explanation! Thanks so much. I wanted to ask if someone could draw a picture describing what's going on but thought that might be too much. Yet, you did it on your own.

 

[voko]

Well, there was a certain Mr. Lagrange, who wrote a book - a very important book! - on mechanics without a single drawing. It was very hard to read. His students did not understand him, either, perhaps because he had the same trait while lecturing. That shows that no amount of mathematical talent and eloquence can't substitute a good drawing. When I realized just how lengthy and bizarre a text-only explanation would be, spending a few minutes on drawing seemed like a major win :)

 

[Nickie]

As a scientist, it is important to understand the underlying principles and concepts behind the equations and solutions rather than just blindly plugging in numbers. In this case, the use of cos(40°) comes from the fact that the Earth is not a perfect sphere, but rather an oblate spheroid with a slightly flattened shape at the poles. This means that the radius of the Earth is not constant, but slightly larger at the equator and smaller at the poles.

To find the linear speed v, we need to know the distance traveled in a given amount of time. In this case, we are looking at the point on Earth's surface at latitude 40°N, which is a distance r from the polar axis. Since the Earth rotates at a constant angular speed ω, the distance traveled in a given time is equal to the circumference of the circle described by the point on Earth's surface. However, since the Earth is not a perfect sphere, the circumference of the circle will be different at different latitudes.

To account for this, we use the formula for the circumference of a circle, 2πr, but instead of using the equatorial radius R, we use the radius at latitude 40°N, which is Rcos(40°). This is because the cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle, and in this case, it represents the ratio of the radius at 40°N to the equatorial radius.

If the problem had asked for longitude, we would use the sine function instead, as it represents the ratio of the opposite side to the hypotenuse in a right triangle. And yes, if the point was at a latitude or longitude south or west of the polar axis, we would use a negative sign to indicate the direction of rotation.

In summary, as a scientist, it is important to understand the concepts and principles behind the equations and solutions, rather than just blindly plugging in numbers. This will not only help you to better understand and explain your results, but also to apply them to different situations and make accurate predictions.