Calculating Angular Frequency of Earth in Local Coordinates

In summary, when considering the Earth's rotation and the use of coordinate systems and Euler angles, it is important to note that the Earth's rotation vector is simplified and not fixed in space. When using local coordinate systems, the changing orientation of the coordinate axes must be taken into account. In addition, there are different conventions for defining Euler angles, and the orientation of the coordinate axes will change with each rotation. It is crucial to carefully consider these factors for accurate calculations.
  • #1
bman!!
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First: the Earth rotates with angular frequency w, with the rotation vector in the z direction. it is then useful to define a local coordinate system i', j', k' fixed on the Earth's surface (i.e. a cartesian coordinate system for us in our point of view on the surface of the earth) then to perform calculations it is easy enough to simply take components of w in the local coordinate system.


so components of w in local coordinates are (0,wsinx, wcosx) where x is the colatitude angle (at the north pole x = 0, and at equator x= 90)

having established this:

when using euler angles (im looking at yzy convention or simply y convention) are the angles still interpreted as referenced to the original i,j,k coordinate system?, so that in problems when you express k in components of e1' and e2' (the new axes after rotation) is this similar conceptually to the above situation?
 
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  • #2


I would like to clarify a few points about the use of coordinate systems and Euler angles in this scenario.

Firstly, the Earth's rotation can indeed be described by an angular frequency w, with the rotation vector in the z direction. However, it is important to note that this is a simplified model and does not take into account the Earth's precession and nutation movements, which can affect the orientation of the rotation vector.

Secondly, while it is useful to define a local coordinate system on the Earth's surface for calculations, it is important to keep in mind that this coordinate system is not fixed in space. As the Earth rotates, this coordinate system will also rotate with it. Therefore, when using this local coordinate system, it is important to take into account the changing orientation of the coordinate axes.

Moving on to the use of Euler angles, it is important to note that there are different conventions for defining these angles. In the yzy convention, the first rotation is around the y-axis, the second around the z-axis, and the third around the y-axis again. In the y convention, the first rotation is around the y-axis, the second around the x-axis, and the third around the y-axis again.

When using Euler angles, the angles are indeed still referenced to the original coordinate system. However, it is important to note that the orientation of the coordinate axes will change with each rotation. Therefore, when expressing a vector in components of the new axes after rotation, it is important to take into account the changing orientation of the axes.

In conclusion, while the concept of using local coordinate systems and Euler angles may seem similar, it is important to keep in mind the changing orientation of the coordinate axes and the different conventions for defining Euler angles. It is always important to carefully consider the specific scenario and choose the appropriate coordinate system and angle convention for accurate calculations.
 

1. What is the angular frequency of Earth in local coordinates?

The angular frequency of Earth in local coordinates is approximately 0.0000727 radians per second. This value represents the rate at which the Earth rotates about its own axis in a counterclockwise direction when viewed from above the North Pole.

2. How is the angular frequency of Earth calculated?

The angular frequency of Earth can be calculated by dividing the Earth's angular speed (rotation rate) by the Earth's radius. The angular speed of Earth is approximately 7.2921159x10^-5 radians per second, and the Earth's radius is approximately 6,371 kilometers. Therefore, the formula for calculating the angular frequency is 7.2921159x10^-5 / 6,371 = 0.0000727 radians per second.

3. Is the angular frequency of Earth constant?

No, the angular frequency of Earth is not constant. It varies slightly due to factors such as changes in the Earth's rotational speed and variations in the Earth's orbit around the Sun. However, these variations are extremely small and do not significantly affect the overall value of the angular frequency.

4. Can the angular frequency of Earth be measured?

Yes, the angular frequency of Earth can be measured using various techniques such as satellite observations, astronomical observations, and precision instruments such as gyroscopes. These measurements can be used to monitor any changes in the Earth's rotation and orbit.

5. Why is it important to calculate the angular frequency of Earth in local coordinates?

Calculating the angular frequency of Earth in local coordinates is important for various scientific and practical purposes. It can be used to accurately determine time and location on Earth, aid in navigation, and understand the Earth's rotational and orbital dynamics. It also plays a crucial role in fields such as geodesy, astronomy, and climate studies.

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