Explanation Show the element parallel to the x axis as in Figure (1). Express the radius of the cardioid. r = a ( 1 − cos θ ) Using Figure 1, Write the differential area of the element. d A = 1 2 r 2 d θ (I) Here, radius of the cardioid is r and differential angle is d θ . Substitute a ( 1 − cos θ ) for r in Equation (I). d A = 1 2 [ a ( 1 − cos θ ) ] 2 d θ = 1 2 a 2 ( 1 − cos θ ) 2 d θ Compute the centroid of the differential element along x axis using the formula. r ˜ = 2 3 r cos θ Determine the location of the centre of the cardioid along x axis ( x ¯ ) using the relation from Figure (1). r ¯ = ∫ A r ˜ d A 2 ⋅ ∫ A d A (II) Conclusion: Apply the limits from 0 to π and substitute 1 2 a 2 ( 1 − cos θ ) 2 d θ fort dA , and 2 3 r cos θ for r ˜ in Equation (II). r ¯ = 2 ⋅ ∫ 0 π 2 3 r cos θ [ 1 2 a 2 ( 1 − cos θ ) 2 d θ ] 2 ⋅ ∫ 0 π 1 2 a 2 ( 1 − cos θ ) 2 d θ Substitute a ( 1 − cos θ ) for r . r ¯ = 2 ⋅ ∫ 0 π 2 3 a ( 1 − cos θ ) cos θ [ 1 2 a 2 ( 1 − cos θ ) 2 d θ ] 2 ⋅ ∫ 0 π 1 2 a 2 ( 1 − cos θ ) 2 d θ = 2 ⋅ ∫ 0 π 2 3 a 3 ( 1 − cos θ ) 3 cos θ d θ 2 ⋅ ∫ 0 π 1 2 a 2 ( 1 − cos θ )

Engineering Mechanics: Statics and Study Pack (13th Edition)

13th Edition

ISBN: 9780133027990

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.1, Problem 38P

To determine

The location r¯ of the centroid C of the cardioid.

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Chapter 9.1, Problem 37P

Chapter 9.1, Problem 39P

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Chapter 9 Solutions

Engineering Mechanics: Statics and Study Pack (13th Edition)

Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid of the shaded area. Prob....Ch. 9.1 - Locate the center of mass x of the straight rod if...Ch. 9.1 - Locate the centroid of the homogeneous solid...Ch. 9.1 - Locate the centroid z of the homogeneous solid...Ch. 9.1 - Locate the center of mass of the homogeneous rod...Ch. 9.1 - Prob. 2P Ch. 9.1 - Prob. 3P Ch. 9.1 - Prob. 4P

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The location r ¯ of the centroid C of the cardioid.

The location r¯ of the centroid C of the cardioid.

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Chapter 9 Solutions