Question

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3. Polar Basis Vectors (Adapted from Taylor 1.43) (a) Show that the unit vector s of two-dimensional polar coordinates is equal to: and find the corresponding expression for (b) Sensemaking: Special Cases Discuss whether your answer is sensible for some special cases. Draw pictures to help with your explanations. (c) Assume that φ changes with time and show that: ds dt do dt where ψ is the time derivative of ψ. (d) Sensemaking: Visualize Geometric Relationships Find the direction of d by: (1) drawing two points near each other representing the position of an object at time t and t + dt (a short time later), (2) drawing & for each point, and (3) performing the vector subtraction. Do the same thing for o. Discuss how your answer corresponds to the algebraic answer.

Answer 1

Take a point p(r, phi) in polar coordinate or p(x, y) in Cartesian coordinate r^= p^rightarrow/ Vertical-bar p Vertical-bar = xx^+ yy^/ Vertical-bar r^rightarrow Vertical-bar = Vertical-bar x^rightarrow Vertical-bar cos phi x^+ Vertical-bar r^rightarrow Vertical-bar sin phi y^/ Vertical-bar r^rightarrow Vertical-bar so s^= r^ cos phi x^+ sin phi y^so x^= r^cos phi - phi^sin phi y^= phi cos phi + r^sin phi -x^sin phi = -r^cos phi sin phi + phi^sin^2 phi y^cos phi = r^sin phi cos phi + phi^cos^2 phi / y^cos phi - x^sin phi = 0 + phi^times (1) phi^= -x^sin phi + y^cos phi special case phi^at (o, r) must be parallel to y axis from the figure phi^= -x^sin (0) + y^cos (0) = 0 + y^(true) phi^at (a, 0) must be -x^from figure phi^= -x^sin (90 degree) + y^cos (90) = -x^ true