Assume a partciel A is moving in a confined circular orbit (with radius a), another particle B is located away from the center (at r=b).
And there is a gravitation field between particles
where is the unit vector between those two particles.
What is the minimum initial velocity for particle a to circular the orbit?
Because the field is so the potential energy is
From consevation of energy
So the minimum velocity is
What if we want the particle always touch the inner surface at r=a (i.e. the Normal force provided by the circular orbit is always pointing into the center of the circle)
Do you know how to solve it?
The following is the simulation for you to play with.
You can drag particle B ( to change b)
1. [b]N out[/b] the normal force is always pointing away from the center of the circle and v is the minimum velocity to reach another end.
2. [b]N in[/b]the normal force is always pointing into the center of the circle and v is the minimum velocity
3. You can change cst to change the velocity ratio.
The red arrow is the velocity of the particle a (You can drag the arrow to change velocity)
The blue arrow is the gravitation force between particle A and B.
The magenta arrow is the required Centripetal force.
Another black arrow is the normal force supplied by the circular orbit.
The kinetic energy/potential energy and total energy as function of angle are drawn as red/blue and green curves.