Explanation Given information: A 2 meters connecting rod OA is rotating counter clockwise in a plane about O at the rate of 3 rev/sec . The length of rod AB is attached to length OA at point A. The length of rod AB is 5 meters. And the graph is, Cosine rule: Let a, b, and c are the sides on a triangle. C is the angle opposite side c. Then, From the above figure, c 2 = a 2 + b 2 − 2 a b cos ( C ) Let the length of OB is x. Then, Using the cosine rule 5 2 = 2 2 + x 2 − 2 ( 2 ) x cos ( θ ) So, 25 = 4 + x 2 − 4 x cos ( θ ) 25 − 4 = x 2 − 4 x cos ( θ ) [ subtract 4 from both sides ] 21 = x 2 − 4 x cos ( θ ) 25 − 21 = x 2 − 4 x cos ( θ ) − 21 [ subtract 21 from both sides ] x 2 − 4 x cos ( θ ) − 21 = 0 For velocity, differentiate the above equation with respect to time t. d d t ( x 2 − 4 x cos ( θ ) − 21 ) = 0 d d t ( x 2 ) − 4 d d t ( x cos ( θ ) ) − d d t ( 21 ) = 0 ( 2 x ) d x d t − 4 d x d t ( cos ( θ ) ) + 4 x sin ( θ ) d θ d t − ( 0 ) = 0 [ from product rule ] 2 d x d t ( x − 2 ( cos ( θ ) ) ) + 4 x sin ( θ ) d θ d t = 0 2 d x d t ( x − 2 ( cos ( θ ) ) ) = − 4 x sin ( θ ) d θ d t [ subtract 4 x sin ( θ ) d θ d t from both sides ] Further simplify the above equation. 2 d x d t ( x − 2 ( cos ( θ ) ) ) = − 4 x sin ( θ ) d θ d t d x d t = − 4 x sin ( θ ) 2 ( x − 2 ( cos ( θ ) ) ) d θ d t [ D i v i d e b y ( 2 x − 4 ( cos ( θ ) ) ) ] d x d t = − 2 x sin ( θ ) ( x − 2 ( cos ( θ ) ) ) d θ d t [ Cancel out 2 ] d x d t = 2 x sin ( θ ) ( − x + 2 ( cos ( θ ) ) ) d θ d t [ multiply and divide

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3, Problem 95SP

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To find the velocity and the acceleration of the motion of the point B along the x-axis.

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Chapter 3, Problem 94SP

Chapter 3, Problem 96SP

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Chapter 3 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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To find the velocity and the acceleration of the motion of the point B along the x-axis.

Solution Summary: The author analyzes the velocity and acceleration of the point B along the x-axis. The length of rod AB is attached to length OA at point A.

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To find the velocity and the acceleration of the motion of the point B along the x-axis.

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Chapter 3 Solutions