I Classical Mechanics - Motion of a particle

AI Thread Summary
The discussion focuses on demonstrating that a particle with a specific acceleration formula moves along the surface of a cone. The acceleration is defined as a constant multiplied by the cross product of the velocity vector and the position vector, normalized by the cube of the position vector's magnitude. The key to proving the motion on the cone's surface lies in showing that a constant vector exists such that its dot product with the position vector remains constant. Participants suggest that for the motion to remain confined to the cone, certain conditions must be met, particularly regarding the particle's speed and trajectory. Understanding these constraints and vector identities is crucial for grasping the motion dynamics involved.
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How to find the trajectory given the acceleration as a cross product of velocity and acceleration
Show that a point with acceleration given by:
a=c*((dr/dt)×r)/|r|3
where c is a constant, moves on the surface of a cone.

This is jut an example to illustrate my doubt. I don't know how to obtain the tracjectory given only the acceleration in this format. I realized that if i can show that there is an constat vector 'a' that satisfy a•r=constant, than the motion would be on the surface of a cone. So i tried to make use of some vectorial identity multiplying by cross product on both sides and try to use the 'BAC-CAB' rule, but that didnt lead to anywhere.

Is there any example similar to this case or anywhere i can study to have a better understanding?
 
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Do you mean ##\mathbf{r}## is confined on the surface of a cone ? If so as an obvious case bodies moving on generatrix of a cone keep constant speed. Bodies moving around circumference of bottom of a cone gets acceleration leaving the cone surface. There should be a confinement to keep them on the surface.
 
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