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An important topic related to arc length is curvature. The concept of curvature provides a way to measure how sharply a smooth curve turns. A circle has constant curvature. The smaller the radius of the circle, the greater the curvature. Think of driving down a road. Suppose the road lies on an arc of a large circle.
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Dec 29, 2020 · Points corresponding to s = 0 through s = 6 are plotted. The arc length of the graph between each adjacent pair of points is 1. We can view this parameter s as distance; that is, the arc length of the graph from s = 0 to s = 3 is 3, the arc length from s = 2 to s = 6 is 4, etc.
Recall that if C is a smooth curve defined by the vector function ~r(t), and ~r′(t) 6= ~0, then the unit tangent vector is given by T~(t) = ~r(t)/||~r′(t)|| which indicates the direction of the curve. Since T~(t) pro-vides the direction of~r(t), the rate of change of T with respect to s, the
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The tangential component represents the time rate of change in the magnitude of the velocity. The particle moves along a curve at constant speed. . at = v = 0 => a = an = v2/r. The normal component represents the time rate of change in the. direction of the velocity.
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Let s(t) be an other parametrization, then by the chain rule d/dtT (s(t)) =. T ′(s(t))s′(t) and d/dtr(s(t)) = r′(s(t))s′(t). We see that the s′ cancels in T ′/r′. Especially, if the curve is parametrized by arc length, meaning that the velocity vector r′(t) has length 1, then κ(t) = |T ′(t)|.
Feb 24, 2024 · Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: →Fnet = m→a.
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Example 2.10 Curvature at the vertex of a parabola: Let y = ax2 for a>0 define a parabola. Find the best instantaneous circle approximation at the vertex (0;0) and use it to calculate the radius of curvature and the curvature at the vertex. By symmetry, we can suppose the circle to have center along the y-axis. Since the
