By Kleppner D., Kolenkow R.
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Extra resources for An Introduction to Mechanics
8 Uniform Circular Motion Circular motion plays an important role in physics. Here we look at the simplest and most important case—uniform circular motion, which is circular motion at constant speed. x = r cos ωt r Consider a particle moving in the x−y plane according to r = r(cos ωt ˆi + sin ωt ˆj), where r and ω are constants. Find the trajectory, the velocity, and the acceleration. y = r sin ωt ωt ˆj x ˆi y |r|= r2 cos2 ωt + r2 sin2 ωt. Using the familiar identity sin2 θ + cos2 θ = 1, r ωt | r | = r = constant.
Let the scalar c and the vectors A and B be functions of time. Then d dc dA (cA) = A + c dt dt dt dA dB d (A · B) = ·B + A· dt dt dt d dA dB (A × B) = ×B + A× . dt dt dt Consider also A2 = A · A. Then dA d 2 (A ) = 2A· , dt dt and we see again that if dA/dt is perpendicular to A, the magnitude of A is constant, d(A2 )/dt = 0. 11 Motion in Plane Polar Coordinates The rectangular, or Cartesian, coordinates we have used so far are well suited to describing motion in a straight line. For instance, if we orient the coordinate system so that one axis lies in the direction of motion, then only a single coordinate changes as the point moves.
To locate the position of a point in space, we start by setting up a coordinate system. For convenience we choose a three-dimensional Cartesian system with axes x, y, and z, as shown. In order to measure position, the axes must be marked in some convenient unit of length—meters, for instance. The position of the point of interest is given by listing the values of its three coordinates, x1 , y1 , z1 , which we can write compactly as a position vector r(x1 , y1 , z1 ) or more generally as r(x, y, z).
An Introduction to Mechanics by Kleppner D., Kolenkow R.