International Edition---engineering Mechanics: Statics, 4th Edition
International Edition---engineering Mechanics: Statics, 4th Edition
4th Edition
ISBN: 9781305501607
Author: Andrew Pytel And Jaan Kiusalaas
Publisher: CENGAGE L
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Chapter 8, Problem 8.50P
To determine

Centroid of the curved surface

Expert Solution & Answer
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Answer to Problem 8.50P

The centroid (x¯,y¯,z¯) of curved surface is [( 2 π)2R,2πR,h3].

Explanation of Solution

Given information:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 8, Problem 8.50P , additional homework tip  1

The centroid of the curved surface is defined as:

  x¯=Q yzA= A xdA A dAy¯=Q xzA= A ydA A dAz¯=Q xyA= A zdA A dA

Calculation:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 8, Problem 8.50P , additional homework tip  2

The differential area dA of element shown

  dA=zRdθ=hRπθdθ

The area A of the element

  A=0πdA=0πhRπθdθ=π2Rh

The centroidal coordinates (x el¯,y el¯,z el¯) of element

  x el¯=Rcosθy el¯=Rsinθz el¯=z2=h2πθ

The first moment Qxy of element

  Qxy=A z el ¯dA=h2R2π20πθ2dθ=π6h2R

The first moment Qyz of element

  Qyz=A x el ¯dA=hR2π0πθcosθdθ=hR2π[cosθ+θsinθ]0π=2hR2π

The first moment Qyz of element

  Qxz=A y el ¯dA=hR2π0πθsinθdθ=hR2π[sinθθcosθ]0π=hR2

The centroid (x¯,y¯,z¯) of curved surface

  x¯=Q yzA=( 2h R 2 π )( π 2 Rh)=( 2 π)2Ry¯=Q xzA=hR2( π 2 Rh)=2πR

  z¯=QxyA=π6h2R(π2Rh)=h3

Conclusion:

The centroid (x¯,y¯,z¯) of curved surface is [( 2 π)2R,2πR,h3].

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Chapter 8 Solutions

International Edition---engineering Mechanics: Statics, 4th Edition

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