Relative Motion - Earth's rotation speed vs latitude

Problem Statement:

The Earth radius is RT = 6.35 106 m and its rotation period is T = 24 h. Calculate:

  1. The angular speed of the Earth with respect to a reference frame at rest O (see figure).
  2. The speed of a point on the surface of the Earth as a function of its latitude λ.
  3. If the latitude of Madrid is λ = 400, find the speed of this point on the surface of the Earth with respect to O. Give the result in SI units and in km/h.
  4. Knowing that the distance between Madrid and Rome is approximately 1300 km and assuming that both cities are at the same latitude, find the time difference of the sunset in the two cities.

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Solution:

The angular velocity of the Earth ω is constant with respect to O. Its magnitude ω is given by:

Where T is its rotation period (in s). The period is defined as the time that it takes for the Earth to complete one revolution around its axis of rotation.

And substituting T in the equation above:

The angular velocity of the Earth ω is drawn in the figure that accompanies the problem statement. For any circular motion is a vector perpendicular to the plane ot the motion. Its direction is given by the right hand screw rule:


Every point on the Surface of the Earth describes a circular motion of radius R (in red in the figure) with respect to an inertial reference frame O. In a circular motion, the relationship between angular and linear speed is given by:

The radius of each point on the surface of the Earth depends on its latitude λ, which is defined as the angular distance of a place north or south of the Earth’s equator (see figure).

In the figure below you can see the relationship between R and the latitude:

And using the trigonometric relationship:

We can determine the radius R of a point on the surface of the earth as a function of its latitude and the radius of the Earth.

Now we substitute this relation in the equation of the speed:

This speed reaches its maximum value at the equator (λ = 00) and is zero at the poles (λ = 900).


If we substitute in the equation above the latitude of Madrid (λ = 400) as well as the angular speed ant the radius of the Earth we have:

And in km/h:

As you can see, at this very moment everyone of us is moving at a great speed. We don’t realize it because the brain detects changes in velocity (acceleration and therefore forces) but not the velocity itself.


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To determine the time difference of the sunset between Madrid and Rome, we have just calculated the speed of a point on the surface of the Earth as a function of its latitude. We also know the distance between them, which we will call s. In the figure below two points at the same latitude are drawn. These points can be both cities.

If the sunset at Rome takes place at a given time, the time it will take for Madrid to “reach” the position of Rome due to the Earth rotational motion is:

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