Explanation Given information: The radius is, r ( t ) = 〈 R cos ( 2 π N t h ) , R sin ( 2 π N t h ) , t 〉 , 0 ≤ t ≤ h . Concept used: The general form of the curve is r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 . The arc length of the curve can be defined as follows, S = ∫ 0 h ‖ r ′ ( t ) ‖ d t . Consider the given curve. r ( t ) = 〈 R cos ( 2 π N t h ) , R sin ( 2 π N t h ) , t 〉 Compare the general form of the curve. r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 Here, x ( t ) = R cos ( 2 π N t h ) y ( t ) = R sin ( 2 π N t h ) z ( t ) = t Differentiate x ( t ) with respect to t . x ′ ( t ) = R [ − sin ( 2 π N t h ) ] d d t ( 2 π N t h ) = − 2 π N h sin ( 2 π N t h ) x ( t ) 2 = ( 2 π N h ) 2 sin 2 ( 2 π N t h ) Differentiate y ( t ) with respect to t . y ′ ( t ) = R [ cos ( 2 π N t h ) ] d d t ( 2 π N t h ) = 2 π N h cos ( 2 π N t h ) y ′ ( t ) 2 = ( 2 π N h ) 2 cos 2 ( 2 π N t h ) Differentiate z ( t ) with respect to t

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 14.3, Problem 22E

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The general formula for the length of a helix of radius R and height h that makes N complete turns.

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Chapter 14.3, Problem 21E

Chapter 14.3, Problem 23E

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Chapter 14 Solutions

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Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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The general formula for the length of a helix of radius R and height h that makes N complete turns.

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To find: The general formula for the length of a helix of radius R and height h that makes N complete turns.

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Chapter 14 Solutions

To find:

The general formula for the length of a helix of radius R and height h that makes N complete turns.