Time derivative with three variables

[x2carlos]

Homework Statement

Im am attempting to solve very difficult kinematics for a problem in my Dynamics course, and after what I got for the velocity of the particle, I come accros the problem that i can't diferentiate one part.

Essentially I have to get the time derivative of r($$\dot{\theta}$$+$$\Omega$$)cos$$\phi$$
where r, $$\phi$$, AND $$\theta$$ are variables and $$\Omega$$ is the only constant

Homework Equations

The Attempt at a Solution

I believe if I can figure this out I can go on with the problem, I may have well made a mistake in my assumptions early on which would make me not run into this problem, but I can't figure out another way to do it. Is it even possible?
product rule twice was the only thing I could see plausible but even then I wouldn't know how to implement it.

Thanks

 

[gabbagabbahey]

The product rule for three variables A,B and C is just:

$$\frac{d( ABC)}{d t}=\left( \frac{d A}{d t} \right) BC+\left( \frac{d B}{d t} \right) AC+\left( \frac{d C}{d t} \right) AB$$

...In this case you would just use the above rule with $A=r$, $B=\theta+\Omega$ and$C=\cos \phi$

That being said, $r(\theta+\Omega)\cos \phi$ is a VERY unusual form to encounter in a physics problem, so I suspect you may have an error...if you post the details of the original problem, I can check.

 

[x2carlos]

thanks a lot, I think I will be able to solve it knowing that rule. In any case I already posted the original problem word for word but haven't gotten any answers, I suspect is partly because it is difficult to visualize the problem without the figure that comes with it, but I am unable to upload it.
The whole original problem is


1. 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"phi"), 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.


2. Homework Equations

transport theorem


3. The Attempt at a Solution

I established 3 reference frames: one inertial fixed to Ex, Ey, Ez
second one,
Ur=direction of OA where A is the point of the projection of the satellite on the Ey,Ex plane.
Uz=Ez
U"theta"= Uz x Ur

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
($$\dot{\theta}$$+$$\Omega$$)Uz- $$\dot{\phi}$$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

I realized that if for some reason I were to assume r, theta, or phi were a constant the algebra with become extremely easy and I can solve it with no problem, but I can't find a reason why I would assume such a thing

Thanks