Explanation Given: The element has a thickness of dx . Show the volume of the spherical segment in Figure 1. Use a triangular “plate” element that is parallel to the x - y plane to get the value of z. z = c ( 1 − 1 b y ) (I) z = c ( 1 − 1 a x ) (II) Here, the sides of x , y , and z planes are a , b , and c respectively. Compute the volume of the element by using the x - y plane of the thickness dz . d V = 1 2 x y d z Compute the centroid of the differential element along with the z axis by using the formula. z ˜ = z Determine the location of the centre of gravity of the homogeneous rod along with the z axis ( z ¯ ) by using the relation. z ¯ = ∫ V z ˜ d V ∫ V d V (III) Conclusion: Apply the limits from 0 to c , substitute z for z ˜ and 1 2 x y d z for dV in the Equation (III). z ¯ = ∫ 0 c z ( 1 2 x y d z ) ∫ 0 c ( 1 2 x y d z ) (IV) Rearrange the equation (I) to get the value of y . z = c − c b y c b y = c − z y = ( c − z ) b c y = ( 1 − z c ) b Rearrange the equation (II) to get the value of x . z = c ( 1 − 1 a x ) z = c − c a x x = ( c − z ) a c x = ( 1 − z c ) a Substitute ( 1 − z c ) a for x and ( 1 − z c ) b for y in the Equation (IV). z ¯ = ∫ 0 c z ( 1 2 ( 1 − z c ) a ( ( 1 − z c ) b ) d z ) ∫ 0 c ( 1 2 ( 1 − z c ) a ( ( 1 − z c ) b ) d z ) (V) Calculate the numerator part from the equation (V). ∫ 0 c z ( 1 2 ( 1 − z c ) a ( ( 1 − z c ) b ) d z ) = 1 2 a b ⋅ ∫ 0 c z ( 1 − z c ) ( 1 − z c ) d z = 1 2 a b ( ( 1 − z c ) ( 1 2 z 2 ( 1 − z c ) + z 3 6 c ) − ∫ 0 c ( − 1 c ) ( 1 2 z 2 <

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ISBN: 9780137514724

Author: HIBBELER

Publisher: PEARSON

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Center of Gravity and Centroid

The term center of gravity indicates the position or location of a point about which the whole weight of a body or system is supposed to be concentrated, which means the whole weight of the body or system works through this point. It is closely related t…

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Chapter 9, Problem 50P

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The centroid z¯ of the tetrahedron segment.

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Chapter 9, Problem 49P

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The centroid z ¯ of the tetrahedron segment.

Solution Summary: The author explains how to determine the centroid of the tetrahedron segment.

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The centroid z¯ of the tetrahedron segment.

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