An Introduction to Mechanics by Kleppner D., Kolenkow R.
By Kleppner D., Kolenkow R.
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Additional info for An Introduction to Mechanics
1+x ⎞ g ⎟⎟⎟⎟⎟ ⎟ L ⎟⎠ (1) NOTES 37 Next, we make use of the following identity, which will be derived in the following section: 1 3 1 1 3 = 1 − x + x2 − x + ··· 1+x 2 8 16 (2) This expansion is valid provided x < 1. Inserting this in Eq. (1) gives 1 3 1 3 ΔT = T 0 − x + x2 − x + ··· . 2 8 16 Note that in the limit that l → 0, ΔT → 0, as we expect. For x the first term on the right may be important, in which case (3) 1, only 1 l ΔT ≈ −T 0 x = − T 0 . 01, the period is increased by about 5 × 10−3 T 0 .
Consequently, the only possible change in r is a change in its direction, which is to say that r must rotate and the trajectory is a circle, This is precisely the case: r rotates about the origin. We showed earlier that a = −ω2 r. Since r · v = 0, it follows that a · v = −ω2 r · v = 0 and a = dv/dt is perpendicular to v. This means that the velocity vector has constant magnitude, so that it too changes in time only by rotation. That v indeed rotates is readily seen from the sketch, which shows v at various positions along the trajectory.
We should emphasize that although in physics one is free to choose any coordinate system one pleases, the proper choice of a coordinate system can vastly simplify a problem. The material in this section, which introduces a coordinate system that is beautifully suited for many problems, is very much in the spirit of more advanced physics. Some of this material may be new to you. Be patient if it seems strange at first. Once you have studied the examples and worked a few problems, it will seem natural.