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- 01-03-2016 20:54
(Original post by atsruser)
- 01-03-2016 21:32
The question relies on using the so-called "intrinsic coordinates" of a curve. The idea is that we know how to handle the motion of a particle constrained to circular motion, and we extend this to more general curvilinear motion.
The physics is as follows: a particle constrained to a smooth curve (say a bead on a wire) knows only about its local environment - it can feel only the forces exerted on it by the wire and external forces at its current location, P.
At any instant in time, the particle has a velocity tangent to the wire, and feels an component of external force accelerating it in the direction of tangent. Also, it feels a force exerted by the wire normal to the tangent at that point. Locally the wire is curved, and over an infinitesimal distance before and after P, that curvature is constant. So at P, we can model the wire as a circle of radius , chosen to give the same curvature as that of the wire at P. So when the particle is at P, we can write down normal and tangential equations of motion, if we know the curvature, by treating the particle as being on a circle, instantaneously.
Of course, an instant later the bead is at another point and the tangent direction has changed, and so has the radius of curvature , so we need to be able to write down the equations of motion in a general form, taking into account the fact that the tangent and are functions of the position, and that from the POV of the bead, it's now on a circle with a different curvature.
To do so, we measure the position of the particle with coords where is the arc length along the curve measured from some origin, and is the angle that the tangent at P makes with the x-axis. These coords are used because it turns out that there are nice formulae relating them to the radius of curvature of the curve at any point - in fact, you can show that , and since, if we have the equation of the constraining curve in the form (as we usually do), then also , so with a bit of work we can use the cartesian eqn of the curve to calculate at any point . [Fun exercise: work out the details yourself].
Having done all that, we can write down the radial and tangential equations of motion by noting that , the tangential velocity, and is the tangential acceleration so we have, by Newton II:
by using the results from circular motion, generalised to our situation where we don't have a constant radius of curvature , but a variable one, namely .
- Thread Starter
- 02-03-2016 12:07