Centripetal Force and Orbitals Help
As far as I understand (this is only what I have learned from studying OCR A2 physics, and a little outside reading, I'm not very clued up):
Centripetal does not affect velocity, so if velocity of an orbit changes, the radius of its orbit must change.
Centripetal F = gravitational F for an orbit in uniform circular motion.
Hence GMm/r^2 = mv^2/r
And rearranging, v^2=GM/r.
From the first point, I understand that if an orbital changes its velocity it will slip into another orbit as radius must change (F=mv^2/r), but I think I'm not understanding something somewhere along the line because when you take into account the factors of F to get v^2=GM/r, is seems that r is inversely proportional to v^2 and so a small velocity means a larger radius, but instinct tells me that if an orbital were to slow down it would go into a lower orbit?
Centripetal does not affect velocity, so if velocity of an orbit changes, the radius of its orbit must change.
Centripetal F = gravitational F for an orbit in uniform circular motion.
Hence GMm/r^2 = mv^2/r
And rearranging, v^2=GM/r.
From the first point, I understand that if an orbital changes its velocity it will slip into another orbit as radius must change (F=mv^2/r), but I think I'm not understanding something somewhere along the line because when you take into account the factors of F to get v^2=GM/r, is seems that r is inversely proportional to v^2 and so a small velocity means a larger radius, but instinct tells me that if an orbital were to slow down it would go into a lower orbit?
Thought:
If the orbit is changing it's no longer in uniform circular motion so the formulas don't apply any more. If speed slows down, then v^2/r is smaller than the force of gravity, so it falls towards earth? (speed not enough to accelerate around earth, so instead starts to accelerate towards earth?)
If the orbit is changing it's no longer in uniform circular motion so the formulas don't apply any more. If speed slows down, then v^2/r is smaller than the force of gravity, so it falls towards earth? (speed not enough to accelerate around earth, so instead starts to accelerate towards earth?)
The equations you have at A2 apply to a mass in a steady orbit. In this case the only force acting is that of the gravitational force between the two masses. The orbit is generally an ellipse, but for the sake of simplicity of the maths, you look at circular orbits.
In such a stable orbit, the relationship between r and v is as you say. If r is large (planet far from Sun, for example) the orbital speed is low. If the planet is near the Sun (Mercury for example) the orbital speed is large. Take a look at any table of planetary data to check this out.
When the orbit is an ellipse, the planet gets nearer to and further from the sun, yet stays in a stable orbit. When the planet is nearer it is moving faster, and when further it moves more slowly. (Kepler's Laws)
So I think you are on the right track with your thoughts.
In such a stable orbit, the relationship between r and v is as you say. If r is large (planet far from Sun, for example) the orbital speed is low. If the planet is near the Sun (Mercury for example) the orbital speed is large. Take a look at any table of planetary data to check this out.
When the orbit is an ellipse, the planet gets nearer to and further from the sun, yet stays in a stable orbit. When the planet is nearer it is moving faster, and when further it moves more slowly. (Kepler's Laws)
So I think you are on the right track with your thoughts.
Original post by hhattiecc
...
Think about it this way. If Earth is orbiting the Sun at a certain angular velocity and you moved it to a greater radius, how could you expect to have the same or an even greater angular velocity?
Earth would move through a greater distance in the same amount of time, implying it has suddenly increased its speed --> violates conservation of energy
Original post by hhattiecc
As far as I understand (this is only what I have learned from studying OCR A2 physics, and a little outside reading, I'm not very clued up):
Centripetal does not affect velocity, so if velocity of an orbit changes, the radius of its orbit must change.
Centripetal F = gravitational F for an orbit in uniform circular motion.
Hence GMm/r^2 = mv^2/r
And rearranging, v^2=GM/r.
From the first point, I understand that if an orbital changes its velocity it will slip into another orbit as radius must change (F=mv^2/r), but I think I'm not understanding something somewhere along the line because when you take into account the factors of F to get v^2=GM/r, is seems that r is inversely proportional to v^2 and so a small velocity means a larger radius, but instinct tells me that if an orbital were to slow down it would go into a lower orbit?
Centripetal does not affect velocity, so if velocity of an orbit changes, the radius of its orbit must change.
Centripetal F = gravitational F for an orbit in uniform circular motion.
Hence GMm/r^2 = mv^2/r
And rearranging, v^2=GM/r.
From the first point, I understand that if an orbital changes its velocity it will slip into another orbit as radius must change (F=mv^2/r), but I think I'm not understanding something somewhere along the line because when you take into account the factors of F to get v^2=GM/r, is seems that r is inversely proportional to v^2 and so a small velocity means a larger radius, but instinct tells me that if an orbital were to slow down it would go into a lower orbit?
I think the concept of angular momentum might help, as it's always conserved. If you are travelling with a certain angular momentum, then going into a lower radius orbit means you will need to speed up to compensate.
If something is actually causing you to slow down then I guess it would be applying a force and so quite a few laws wouldn't work any more. But it should hold true that a lower orbit is faster than a higher one (think of it as being more strongly attracted by gravity).
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