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The book defines velocity as the limit(t->0) [Δx/Δt], 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(θ)=k<1,θ>?

Then, ds/dθ=<0,k>. If theta is a function of time such that θ(t)=mt+b, then dθ/dt=m and

ds/dt=[ds/dθ][dθ/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.