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208 test1 spring 03 solution

# 208 test1 spring 03 solution - SPRING 2003 MAE 208...

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SPRING 2003 ENGINEERING DYNAMICS SILVERBERG MAE 208 SOLUTION TO TEST 1 THEORY PROBLEM (a) ( 10 points ) Draw a figure showing the planar path of a particle, it’s position vector r , the coordinates x, y, r, and θ , and the unit vectors i , j , n r , and n θ . ANSWER: (b) ( 10 points ) Show that θ θ n n = r and . r n n θ θ - = Then show that θ θ n n v v v r r + = in which . and θ θ r v r v r = = ANSWER: Differentiate with respect to time j i n θ θ sin cos + = r to get . ) cos sin ( ) (cos ) sin ( θ θ θ θ θ θ θ θ θ n j i j i n = + - = + - = r Similarly, differentiate with respect to time j i n θ θ θ cos sin + - = to get . ) sin (cos ) sin ( ) cos ( r n j i j i n θ θ θ θ θ θ θ θ θ - = + - = - + - = Next, differentiate the position vector r = r n r to get r r r r r r r v v r r r r n n n n n n v θ θ + = + = + = so . , θ θ r v r v r = = (c) ( 10 points ) Show that θ θ n n a a a r r + = in which . 2 and 2 θ θ θ θ r r a r r a r + = - = ANSWER: Differentiate v with respect to time to get ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 r

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