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PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 422 PROBLEM 12.75 For the particle of Problem 12.74, show (a) that the velocity of the particle and the central force F are proportional to the distance r from the particle to the center of force O, (b) that the radius of curvature of the path is proportional to r3. PROBLEM 12.74 A particle of mass m is projected from Point A with an initial velocity v0 perpendicular to line OA and moves under a central force F directed away from the center of force O. Knowing that the particle follows a path defined by the equation 0/ cos 2r r θ= and using Eq. (12.27), express the radial and transverse components of the velocity v of the particle as functions of θ. SOLUTION Since the particle moves under a central force, constant.h = Using Eq. (12.27), 2 0 0 0h r h r vθ= = = or 0 0 0 0 0 2 2 00 cos 2 cos 2 r v r v v rr r θθ θ= = = Differentiating the expression for r with respect to time, 0 0 0 0 03/2 3/2 0 sin 2 sin 2 sin 2 cos 2 (cos 2 ) (cos 2 )cos 2 cos 2 r vdr d r r r v d d r θ θ θθ θ θ θ θ θ θ θθ θ = = = = = Differentiating again, 22 2 2 2 0 0 0 3/2 0 sin 2 2cos 2 sin 2 2cos 2 sin 2 (cos 2 )cos 2 cos 2 vdr d r v v d d r θ θ θ θ θθ θ θ θ θ θθ θ + += = = = (a) 0 0 0 sin 2 sin 2 cos 2 r v r v r v r θ θ θ = = = 0 0 cos 2 v r v r rθ θ θ= = 2 2 2 20 0 ( ) ( ) sin 2 cos 2r v r v v v rθ θ θ= + = + 0 0 v r v r = 2 22 2 2 20 0 0 2 0 0 2 22 2 0 0 0 2 0 00 2cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 2 cos 2 cos 2 r v r v a r r r r v v v r r rr θ θθ θ θ θ θ θ θ θ += − = − += = = 2 0 2 0 :r r mv r F ma r = = 2 0 2 0 r mv r F r =
