# Acceleration calculation for an object with is rotating about it's COM and it's COM is revolving about another object

I am confused with Pseudo force and circular motion. Let's say a satellite is revolving in a circular orbit about a planet (Gravity is providing the centripetal acceleration). Now, an object (say A) is attached to the satellite via massless string and the object (A) is rotating about the satellite (tension in string provides the centripetal acceleration). What will be the acceleration of the object A?

the position vector to the object A given in inertial system is:

$$\begin{bmatrix} x \\ y \\ \end{bmatrix}= \left[ \begin {array}{c} r\cos \left( \omega\,t \right) +l\cos \left( \varphi(t) \right) \\ r\sin \left( \omega\,t \right) +l\sin \left( \varphi(t) \right) \end {array} \right]$$

where $$~\omega~$$ is the rotation of he satellite and $$~\dot\varphi~$$ is the angular velocity relative to the satellite.

thus the acceleration of object A is:

$$\begin{bmatrix} \ddot x \\ \ddot y \\ \end{bmatrix}= \left[ \begin {array}{c} -r\cos \left( \omega\,t \right) {\omega}^{2} -l\cos \left( \varphi \left( t \right) \right) \left( {\frac {d}{dt }}\varphi \left( t \right) \right) ^{2}-l\sin \left( \varphi \left( t \right) \right) {\frac {d^{2}}{d{t}^{2}}}\varphi \left( t \right) \\ -r\sin \left( \omega\,t \right) {\omega} ^{2}-l\sin \left( \varphi \left( t \right) \right) \left( {\frac {d }{dt}}\varphi \left( t \right) \right) ^{2}+l\cos \left( \varphi \left( t \right) \right) {\frac {d^{2}}{d{t}^{2}}}\varphi \left( t \right) \end {array} \right]$$

if you know the angular velocity $$(~\frac{d}{dt}\varphi(t)~)$$ of the object A ,you can obtain the acceleration $$~\ddot x~,\ddot y~$$ otherwise you get the $$(~\frac{d^2}{dt^2}\varphi(t)~)$$ from the equation of motion of the object A.

$$\ddot\varphi={\frac {mlr{\omega}^{2}\sin \left( -\varphi +\omega\,t \right) }{m{l}^ {2}+I_{{A}}}}$$

solving this differential equation you obtain $$~\varphi(t)~,\frac{d}{dt}\varphi(t) ~$$ and again the acceleration of A

In the frame of the satellite, the acceleration of A is just acceleration from rotation around the satellite. In the frame planet, it's the sum of the accelerations of A wrt the satellite plus the acceleration of the satellite wrt the planet.

That is, if $$\rm{\bf x}_{s,p}$$ is the vector position of the satellite in the frame of the planet, and $$\rm{\bf x}_{A,s}$$ is the vector position of A in the frame of the satellite, then, and $$\rm{\bf x}_{A,p}$$ is mass A in the frame of the planet, then, just by normal vector math, $$\rm{\bf x}_{A,p} = {\bf x}_{A,s} + {\bf x}_{s,p}$$ and then taking then taking the second time derivative of both sides $$\rm{\bf a}_{A,p} = {\bf a}_{A,s} + {\bf a}_{s,p}$$

This is more complicated if the frame of the satellite is rotating relative to the frame of the planet, but the same idea applies: the acceleration is just the vector sum.