Centripetal Force, the planets

[StephenPrivitera]

Browsing through my physics book...
The book defines velocity as the limit(t->0) [&Delta;x/&Delta;t], which of course makes sense. An acceleration can occur in two ways: either the speed (speed= ||v||) changes or the direction of the velocity changes. Centripetal acceleration (||a|| is constant) is always the common example in which acceleration occurs without a change in speed. My book sometimes gives me the angular velocity of an object which experiences a centripetal acceleration. From here I have to convert the speed into a speed according to the above definition. This whole process seems rather arbitrary to me. I feel like I am forever committed to using rectanglar coordinates. Why can't I state the position of the object using polar coordinates: s(&theta;)=k<1,&theta;>?
Then, ds/d&theta;=<0,k>. If theta is a function of time such that &theta;(t)=mt+b, then d&theta;/dt=m and
ds/dt=[ds/d&theta;][d&theta;/dt]=<0,mk> The acceleration is then a=<0,0>. Is this result wrong by definition or wrong by principle (ie, is there a different interpretation)? (Worse yet, is the math incorrect?)
...
Next question. A body attached to a rope that causes that body to move uniformly in a circle is being acted on by a force F such that ||F||=m/r*v^2. If the rope is cut, the body will "travel along the path of the tangent" to the circle - ie, its instantaneous velocity is tangent to the curve of its path. There is a net force acting on the body because it is accelerating according to the above definition. The body pulls on the rope with the same force F. Let's say there's another body of equal mass attached to the rope half way between the first one and the center. This body is acted on by a centripetal force ||F2||=8m/r*v^2 and also an outward force by the other body ||F1||=m/r*v^2. The net force is 7m/r*v^2. On further consideration, this result shouldn't be right, because I'm making two assumptions which aren't necessarily true: (1) that the equation F=m/r*v^2 holds true for this instance in which there are two masses and (2) the force on the outer mass isn't affected by the presence of the other mass. I'm interested in discovering where this formula comes from. My book says that the formula gives us "the force required to keep a mass in uniform circular motion." I want to then apply this idea to the planets to find the force (as a function of time) required to keep them in elliptical orbits. I'm also confused by the problem that the planets exert a force on the sun (like the mass exerts a force on the rope). Does this not cause the sun to accelerate? Would I have to choose the sun as a reference frame?

I think that's enough for now.

 

[chroot]

Originally posted by StephenPrivitera
Is this result wrong by definition or wrong by principle (ie, is there a different interpretation)? (Worse yet, is the math incorrect?)

In fact, it's perfectly correct in every way. Kudos to you for going through those steps on your own. You may, in fact, choose to represent any system by any coordinate system you like -- polar coordinates on a plane are just fine. The two coordinates are then (r, &theta;). You can speak of the time derivative of rectilinear coordinates, dx/dt and so on, and refer to them as the components of velocity in the each direction. Similarly, you can speak of coordinate velocities dr/dt and d&theta;/dt, and no loss of functionality occurs.

The problem with your math is a subtle one. You are using vector notation, which makes an a priori assumption of a basis -- in rectilinear coordinates, you assume x and y. These unit basis vectors do not depend on location in space in any way. A particle can be anywhere in the space, and the unit vectors are the same.

In polar coordinates, however, it does not suffice to supply only ONE &theta; or ONE r. r points radially to the particle, and &theta; points tangentially in the direction of increasing &theta;.

The basis vectors are different at every point in the space! What you've solved is the coordinate acceleration, not the particle's acceleration through space. To do so, you have to recognize the dependence of the basis vectors at a point (r, &theta;) to that point.

The following websites will hopefully make the distinction more clear.

http://www.engin.brown.edu/courses/EN4/notes/Curvilinear3/Curvilinear3.html
http://physics.smsu.edu/faculty/broerman/fall02/phy203/polar2.htm

This body is acted on by a centripetal force ||F2||=8m/r*v^2 and also an outward force by the other body ||F1||=m/r*v^2.

I'm not sure where the factor of eight came from here, but that's okay. There is no "outward force" however, on the body -- there is tension in the rope. You could consider the inner mass to be just a lump on the rope, and analyze that small piece of the rope. You'd find that the tensions on both sides of the lump are equal and opposite and thus sum to zero. The only force acting on the lump, then, is the inward-directed centripetal force.

I want to then apply this idea to the planets to find the force (as a function of time) required to keep them in elliptical orbits. I'm also confused by the problem that the planets exert a force on the sun (like the mass exerts a force on the rope). Does this not cause the sun to accelerate? Would I have to choose the sun as a reference frame?

Both bodies travel through conic sections (usually ellipses are treated) with the center of mass at one focus. The only "axiom" you need to analytically understand this situation is Newton's law of universal gravitation, F = G(m₁ m₂)/r². The Sun does indeed move in a small ellipse, while the planet moves in a large one, but both bodies are indeed moving. All of Kepler's laws can be directly derived from Newton's law of universal gravitation.

I'm having trouble finding a suitable derivation for you to read on the web, and unfortunately I don't have the time at the moment to type one up for you -- perhaps when I get home this evening I will. In any event, any basic book on astrophysics or astrodynamics will cover this problem intricately. I recommend Carrol and Ostlie's "Introduction to Modern Astrophysics."

Let us know if you have any more questions. :)

- Warren

 

[StephenPrivitera]

I'm not sure I entirely understand anything you've told me. I understand the concept of the unit vectors (why they have to be perpendicular). In fact, that clears up a lot, and I'll be able to play around with that for a while. I'm still a bit uncertain about the coordinate versus actual acceleration, but let me looks through those sites you gave me before I ask you to elaborate.
I really am still lost on the rope with two masses issue. Apparently I don't understand tensions - this is not the first time I've been confused on a problem because I didn't understand the tensions. Let me go back through my book and see if I can figure that out. I guarantee I'll have plenty of questions later.
As for the last part, I wasn't sure whether Newton's law of gravitation was an empirical law. I thought maybe it came about through a mathematical analysis similar to the one you'd have to do to come up with the formula for centripetal force. Ironically, my approach to the problem was to use Kepler's laws to prove Newton's, because I believed that Kepler was before Newton, so Newton would have had his laws at his disposal. But if Newton's law was empirically derived, that would be a rather fruitless pursuit anyway. I'll order that book you suggested. I always thought a book with "Astrophysics" in the title would be over my head. I really don't have a strong background in physics.
As always, thank you for your help. It's always a pleasure to learn new things!

 

[chroot]

Newton was indeed a successor of Kepler. The Keplerian laws were not derived or solved -- they were simply observations Kepler made by looking at volumes of data on the positions of the planets. He didn't know what they meant.

Newton came along, studied Kepler's laws, and found that he could describe all three by one underlying rule -- his elegant law of universal gravitation. So Kepler's laws and Newton's law are mathematically equivalent representations -- you can derive Kepler from Newton or Newton from Kepler.

- Warren