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Physics 1 Problem Solutions 49

Physics 1 Problem Solutions 49 - Chapt 4 Two and...

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Chapt. 4 Two- and Three-Dimensional Kinematics 45 a) The displacement vector vector r in polar coordinates is vector r = r ˆ r, where r is the magnitude of vector r and ˆ r is unit vector pointing in the direction of vector r . Differentiate vector r with respect to time t to find vectorv using the product rule, but don’t for the moment try to determine what d ˆ r/dt is. b) There is a non-rigorous, but valid and convincing way of evaluating d ˆ r/dt . Draw a diagram with ˆ r and ˆ r + Δˆ r with angle Δ θ between them. The Δˆ r and Δ θ are the changes in, respectively, radial unit vector ˆ r and angular of position θ of the object in time Δ t . With Δ θ in radians show that Δˆ r Δ θ ˆ θ, where recall ˆ θ is always π/ 2 = 90 clockwise from ˆ r . Now show d ˆ r dt = ω ˆ θ, where ω = dθ/dt is the angular velocity. The angular velocity is usually in units of radians per unit time. c) Now write vectorv substituting for d ˆ r/dt .
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