Concept explainers
Locate the centroid of the section shown, which was cut f an elliptical cylinder by an oblique plane.
Fig. P5.134
The centroid of the section.
Answer to Problem 5.134P
The centroid of the section
Explanation of Solution
Refer Figure 1 and Figure 2.
Consider an elemental section of the given section.
Write an expression to calculate the volume of the element.
Here,
From the symmetry, write an expression to calculate the distance of centroid of the section from x-axis.
Here,
From the symmetry, write an expression to calculate the distance of centroid of the section from x-axis.
Here,
Write an expression to calculate the distance of centroid of element from z-axis.
Here,
Write an expression to calculate width of the element.
Here,
Write an expression to calculate the height of the element.
Here,
Write an expression to calculate the volume of the section.
Write an expression to calculate the thickness of the section.
Differentiate the equation to calculate the thickness of the element.
Write an expression to find the distance of the centroid of the section from x axis.
Here,
Write an expression to find the distance of the centroid of the section from x axis.
Here,
Write an expression to find the distance of the centroid of the section from z axis.
Here,
Conclusion:
Substitute (V), (VI) and (IX) in equation (VII) to find
write an expression to calculate the distance of centroid of the element from x-axis.
Write an expression to calculate
Substitute equation (XIII) and (XIV) in equation (XI) to find
Write an expression to calculate
Substitute equation (XIII) and (XV) in equation (XII) to find
Thus, the centroid of the section
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Chapter 5 Solutions
VECTOR MECHANICS FOR ENGINEERS: STATICS
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