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Progress is impossible without change; and those who cannot change their minds cannot change anything. ..."George Bernard Shaw(1865-1950, Irish dramatist, essayist and critic, Nobel Prize for Literature 1925)"

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 Author Topic: Minimum energy problem  (Read 4874 times) 0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).
Fu-Kwun Hwang
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 « Embed this message on: February 22, 2010, 02:36:55 pm » posted from:Taipei,T\'ai-pei,Taiwan

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 $\vec{g}=\frac{k}{r^2}\hat{r}$
where $\hat{r}$ 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 $\vec{g}=\frac{k}{r^2}\hat{r}$ so the potential energy is $V(r)=\frac{-k}{r}$

From consevation of energy

$\frac{1}{2}mv^2-\frac{mk}{a-b}=\frac{1}{2}mu^2-\frac{mk}{a+b}$
$v^2-u^2\ge \frac{2k}{a-b}-\frac{2k}{a+b}=\frac{4kb}{a^2-b^2}$
So the minimum velocity is $v=\sqrt{\frac{4kb}{a^2-b^2}}$

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)
Mode:
1. N out the normal force is always pointing away from the center of the circle and v is the minimum velocity to reach another end.
2. N inthe 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.

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)
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Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop. This work is licensed under a Creative Commons Attribution 2.5 Taiwan License
• Please feel free to post your ideas about how to use the simulation for better teaching and learning.
• Post questions to be asked to help students to think, to explore.
• Upload worksheets as attached files to share with more users.
Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!
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Progress is impossible without change; and those who cannot change their minds cannot change anything. ..."George Bernard Shaw(1865-1950, Irish dramatist, essayist and critic, Nobel Prize for Literature 1925)"
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