Explanation Given information: The given force field is. F = 2 x y z i + ( x 2 z − 1 z tan − 1 y z ) j + ( x 2 y + y z 2 tan − 1 y z ) k The equation for an object moving under the force field F along the circular helix is. R ( t ) = ( sin π t ) i + ( cos π t ) j + ( 2 t + 1 ) k Calculation: Calculate the force field F along the path R ( t ) . F = 2 sin ( π t ) . cos ( π t ) . ( 2 t + 1 ) i + ( sin 2 ( π t ) ( 2 t + 1 ) − 1 2 t + 1 tan − 1 cos π t 2 t + 1 ) j + ( sin 2 ( π t ) . cos ( π t ) + cos π t ( 2 t + 1 ) 2 tan − 1 cos π t 2 t + 1 ) k Differentiate R ( t ) with respect to t . d d t R ( t ) = ( π cos π t ) i + ( − π sin π t ) j + ( 2 ) k Using above equation. F .dr = { 2 π sin ( π t ) . cos 2 ( π t ) . ( 2 t + 1 ) } d t − ( π ( sin 2 ( π t ) . ( 2 t + 1 ) − 1 2 t + 1 tan − 1 cos π t 2 t + 1 ) sin π t ) d t + 2 ( sin 2 ( π t ) . cos ( π t ) + cos π t ( 2 t + 1 ) 2 tan − 1 cos π t 2 t + 1 ) d t F .dr = { 2 π sin ( π t ) . ( 2 t + 1 ) − 3 π sin 3 ( π t ) . ( 2 t + 1 ) + sin 2 ( π t ) cos 2 ( π t ) + ( 2 cos π t ( 2 t + 1 ) 2 + π sin π t 2 t + 1 ) tan − 1 cos π t 2 t + 1 } Calculate work done by object. ∫ C F .dr = 2 π ∫ 0 1 2 sin ( π t ) . ( 2 t + 1 ) d t − 3 π ∫ 0 1 2 sin 3 ( π t ) . ( 2 t + 1 ) d t + 2 ∫ 0 1 2 sin 2 ( π t ) . cos 2 ( π t ) d t + ∫ 0 1 2 ( 2 cos π t ( 2 t + 1 ) 2 + π sin π t 2 t + 1 ) tan − 1 cos π t 2 t + 1 d t Calculate the integral ∫ 0 1 2 sin ( π t ) . ( 2 t + 1 ) d t . ∫ 0 1 2 sin ( π t ) . ( 2 t + 1 ) d t = 2 ∫ 0 1 2 t sin ( π t ) d t + ∫ 0 1 2 sin ( π t ) d t …(2) Using the values. z = π t d z = π d t Substitute the values in equation (2). ∫ 0 1 2 sin ( π t ) . ( 2 t + 1 ) d t = 2 π 2 ∫ 0 1 2 z sin z d z + 1 π ∫ 0 1 2 sin z d z = 2 π 2 ( sin z − z cos z ) | 0 π 2 + 1 π cos z | 0 π 2 = 2 π 2 + 1 π = 1 π 2 ( π + 2 ) Calculate the integral ∫ 0 1 2 sin 3 ( π t ) . ( 2 t + 1 ) d t . ∫ 0 1 2 sin 3 ( π t ) . ( 2 t + 1 ) d t = 2 ∫ 0 1 2 t sin 3 ( π t ) d t + ∫ 0 1 2 sin 3 ( π t ) d t = 2 ∫ 0 π 2 z sin 3 z d z + 1 π ∫ 0 π 2 sin 3 z d z Using the values

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 13, Problem 87SP

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To find: The work done by the objectmoving under the force field

F along the circular helix.

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Chapter 13, Problem 86SP

Chapter 13, Problem 88SP

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The work done by the objectmoving under the force field F along the circular helix.

Solution Summary: The author explains the equation for an object moving under the force field F along the circular helix.

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To find: The work done by the objectmoving under the force field F along the circular helix.

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To find: The work done by the objectmoving under the force field

F along the circular helix.