Acceleration of an offset rotating point on a sphere

1. Apr 19, 2012

S1mWa1k

1. The problem statement, all variables and given/known data

I'm trying to sort out some equations for an academic paper I'm writing. I need to work out the acceleration of a point that rotates around another point that is moving on a sphere.

In the attached figure the black dot lies on the surface of a sphere, with a fixed radius. The red dot is connected to the black dot. The line joining the red dot and the black dot is orthogonal to the line joining the black dot and the centre of the sphere. The red dot rotates about the line joining the black dot and the centre of the sphere. I need to calculate the acceleration of the red dot in x,y,z-cartesian coordinates.

The relevant parameters and variables are:

$$r \text{- the distance between the black point and the centre of the sphere (fixed parameter)}$$
$$c \text{- the distance between the red point and the black point (fixed parameter)}$$
$$\theta \dot{\theta} \ddot{\theta} \text{- azimuth angle, velocity and acceleration}$$
$$\phi \dot{\phi} \ddot{\phi} \text{- polar angle, velocity and acceleration}$$
$$\alpha\dot{\alpha} \ddot{\alpha} \text{- the angle, velocity and acceleration of the red point relative to the black point. If } \phi=\pi/2 \text{ then } \alpha=0 \text{ when the red dot lies on the } xy \text{plane}$$

2. Relevant equations

Using the equations from Wolfram Mathworld (http://mathworld.wolfram.com/SphericalCoordinates.html) I can get the accelerations for a point on a sphere when the radius is constant. This gives:

$$\ddot{x}=r \text{Cos}\theta \text{Cos}\phi \ddot{\phi }-2 r \text{Cos}\phi \dot{\theta } \dot{\phi } \text{Sin}\theta-r \text{Cos}\theta \left(\dot{\theta }^2+\dot{\phi }^2\right) \text{Sin}\phi -r \ddot{\theta } \text{Sin}\theta \text{Sin}\phi\\ \\ \ddot{y}=2 r \text{Cos}\theta \text{Cos}\phi \dot{\theta } \dot{\phi}+r \text{Cos}\theta \ddot{\theta } \text{Sin}\phi+r \text{Cos}\phi \ddot{\phi } \text{Sin}\phi-r \left(\dot{\theta }^2+\dot{\phi }^2\right) \text{Sin}\theta \text{Sin}\phi \\ \\ \ddot{z}=-r \text{Cos}\phi \dot{\phi }^2-r \ddot{\phi } \text{Sin}\phi\\\$$

I also have equations for calculating the acceleration when
$$\phi=pi/2$$
which greatly simplifies the problem:

$$\ddot{x}=\text{Cos}\theta \left(c \text{Cos}\alpha -r \dot{\theta }^2-2 c \dot{\alpha } \dot{\theta } \text{Sin}\alpha\right)-\text{Sin}\theta\left(c \text{Cos}\alpha \dot{\alpha }^2+c \text{Cos} \alpha \dot{\theta}^2+r \ddot{\theta}+c \ddot{\alpha } \text{Sin}\alpha\right) \\ \ddot{y}=\text{Sin}\theta \left(c \text{Cos}\alpha -r \dot{\theta }^2-2 c \dot{\alpha } \dot{\theta } \text{Sin}\alpha\right)+\text{Cos}\theta\left(c \text{Cos}\alpha \dot{\alpha }^2+c \text{Cos} \alpha \dot{\theta}^2+r \ddot{\theta}+c \ddot{\alpha } \text{Sin}\alpha\right) \\ \ddot{z}=-c \ddot{\alpha } \text{Cos}\alpha+c \dot{\alpha }^2 \text{Sin}\alpha$$

This effectively restricts the black point to lying on the xy plane, so that only movement in z is due to the changes in α.

3. The attempt at a solution

I've been through these equations, particularly the second set, and I think I understand where the various terms come from. I realise I need to combine the two sets of equations, but I can't work out how to get it done, due to the multitude of cross terms between
phi and theta.

Any help would be hugely appreciated

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• spherical_coords.png
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Last edited: Apr 19, 2012