Very difficult kinematics problem involving spherical coordinates

[x2carlos]

Homework Statement

A satellite is in motion over the Earth. The Earth is modeled as a sphere of radius R that rotates with constant angular velocity "$$\Omega$$" in the direction of Ez, where Ez, lies in the direction from the center of the Earth to the North Pole of the Earth at point N. The position of the satellite is known geographically in terms of its radical distance, r, from the center of the Earth, its EARTH RELATIVE longitude, "$$\theta$$", where "$$\theta$$" is the angle measured from direction Ex, where Ex lies along the line from the center of the Earth to the intersection of the Equator with the Prime Meridian, and its latitude, "$$\phi$$", where "$$\phi$$" is measured from the line that lies along the projection of the position into the equitorial plane.

Using spherical basis (Er,E"theta", E"phi") to the describe the position of the spacecraft (where Er=direction r from center of Earth to spacecraft , E"$$\theta$$"=direction of Ez x Er, and E"$$\phi$$"= Er x E"$$\theta$$"), determine the velocity and acceleration of the satellite a) as viewed by and observer fixed to the Earth b) as viewed by an observer fixed to an inertial refference frame.

Homework Equations

transport theorem

The Attempt at a Solution

I established 3 reference frames: one inertial fixed to Ex, Ey, Ez where Ey is Ez x Ex all at t=0
second one the same but fixed to earth, so it rotates with angular velocity "$$\Omega$$"
third , in the direction of r, (spherical coordinate system)

I am not sure if those are the correct ones but with those I am getting and angular velocity
("$$\theta$$dot" + $$\Omega$$)Ez - "$$\phi$$dot" E"theta"

Well I understand that it might be hard to visualize what is going on but I don't know how to upload the figure that corresponds. My main problem with spherical coordinates is that they are hard for me to visualize and in this particular problem I am having trouble determining what the angular velocity of the spacecraft is relative to an inertial reference frame in order to apply the transport theorem.

I don't know how to attach my full solution but that my main problem and i think the rest of my crazy algebra problem stem from that, I am basically wondering if there is an easier way to set the problem up to ease the algebra involved

 

[KataKoniK]

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Thank you for your post. It seems like you have a good understanding of the problem and have made some progress in setting up the reference frames and determining the angular velocity of the spacecraft.

One way to make the problem easier to solve is to use vector notation instead of spherical coordinates. This will make it easier to visualize and manipulate the vectors involved. You can define the position vector of the spacecraft as r, and the velocity and acceleration vectors as v and a, respectively. Then, you can use the transport theorem to relate the derivative of a vector in one reference frame to its derivative in another reference frame.

Also, keep in mind that the angular velocity of the spacecraft relative to an inertial reference frame is simply the angular velocity of the Earth (given by "\Omega") plus the angular velocity of the spacecraft relative to the Earth (given by "\thetadot"). This should help you in determining the angular velocity of the spacecraft in the inertial reference frame.

I hope this helps and good luck with your problem!

 

[Mene]

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I understand that kinematics problems involving spherical coordinates can be challenging. However, it is important to remember that these coordinates are often used to describe the motion of objects in space, and mastering them can greatly enhance our understanding of the physical world.

In this particular problem, it seems that you have correctly established the three reference frames. However, I recommend double checking your calculations to ensure that you have correctly determined the angular velocity of the spacecraft relative to the inertial reference frame. This may involve considering the rotation of the Earth and the motion of the spacecraft in relation to the Earth's rotation.

In terms of visualizing the problem, it may be helpful to create a diagram or a 3D model to better understand the position and motion of the satellite. You could also try breaking down the problem into smaller, more manageable parts and then combining them to find the overall solution.

In terms of the transport theorem, make sure you are applying it correctly and using the appropriate equations for spherical coordinates. It may also be helpful to consult with a classmate or your instructor for additional guidance and clarification.

Overall, mastering kinematics problems involving spherical coordinates takes practice and patience. Keep working through the problem and don't be afraid to ask for help when needed. Good luck!

 

[piercas]

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I understand that kinematics problems involving spherical coordinates can be challenging to visualize and solve. In this particular problem, it may be helpful to first establish the reference frames and clearly define the coordinate system being used. It is also important to carefully consider the given information and understand the motion of the satellite relative to the Earth.

In order to determine the velocity and acceleration of the satellite as viewed by an observer fixed to the Earth, it may be helpful to use the transport theorem. This theorem allows us to relate the velocities and accelerations in different reference frames, and can simplify the problem by reducing the number of variables involved.

Furthermore, it may be beneficial to break down the problem into smaller parts and solve them separately. For example, you could first determine the velocity and acceleration of the satellite as viewed by an observer fixed to the Earth's rotating reference frame, and then use the transport theorem to relate it to the velocity and acceleration as viewed by an observer fixed to an inertial reference frame.

Overall, the key to solving a difficult kinematics problem involving spherical coordinates is to carefully establish the reference frames, clearly define the coordinate system, and use appropriate mathematical tools such as the transport theorem to simplify the problem.