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The motion of a bead on a sinusoidal track. Very tricky and dense, thanks in advance!

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Oct26-11, 04:42 PM
P: 1
1. The problem statement, all variables and given/known data

a If the bead starts at the origin at time t=0, how long does it take for it to reach the
end of the track (30m away in the figure)? Provide the answer in terms of v[itex]_{}[/itex]. b How does the parameter α depend on time, ie. what is α(t)?
d What is the velocity vector as a function of t (in symbolic form)?
e What is the acceleration vector as a function of t (in symbolic form)?
f Based on your results for the velocity and acceleration, what is the radius of the "kissing circle" at the top of the track? Recall, the kissing circle at a given point goes through the point, has the same tangent as the curve, and a radius that reproduces the perpendicular component of the acceleration, i.e.:
R = |→v|^2/a?
f Will Rbottom the radius of the kissing circle at the bottom of the track
be <, > or = to R[itex]_{}[/itex]?
h How fast would the rod need to move for the bead to leave the track? Provide an answer in symbolic form and a numerical value for the speed.

3. The attempt at a solution
a) I tried approximating the length of the track through simply turning each curve into a line. Then divide the total distance by v[itex]_{}[/itex]?
b)α is just the x-position of the bead, so it would increase depending on time. This answer, however, seems WAY too simple...perhaps I am interpreting the question wrong?
c) I tried taking the derivative of the given parametric equation...but am not sure how to derive a parametric equation, as it has two parts.
d)I suppose I would the derive the equation obtained from deriving the original parametric equation?
After this point, I am just hopelessly lost. I think I am having trouble applying the easy concepts I learn from the textbook to more conceptual problems. Any help would be MUCH appreciated!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
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