5.48 and *5.49 Determine by direct integration the centroid of the area shown.
Fig. P5.48
The centroid of shaded area in Fig. P5.48 by method of direct integration.
Answer to Problem 5.48P
Centroid is located at
Explanation of Solution
Refer the figure P5.48 and figure given below.
Write the equation for curve
Here,
In the figure given above, black dot denotes the center of mass of triangular shaped differential area element.
Write the expression for the x-coordinate of center of mass of triangular shaped differential area element.
Here,
Rewrite the above relation by substituting
Write the expression for the y-coordinate of center of mass of triangular shaped differential area element.
Here,
Rewrite the above relation by substituting
Write the expression to calculate the triangular differential area element.
Here,
Rewrite the above relation by substituting
Write the expression to calculate the total area of shaded region in P5.48.
Here,
Rewrite the above equation by substituting
Calculate
Apply the integration by parts method to solve the above integral. The required formula is given below.
Here,
Substitute
Rewrite the above equation in terms of
Calculate
Rewrite equation (I) by substituting the above result.
\
Calculate
Apply the integration by parts method to solve the above integral. The required formula is given below.
Here,
Substitute
Rewrite the above equation in terms of
Calculate
Rewrite equation (I) by substituting the above result.
Write the expression for first moment of whole area about y-axis.
Here,
Rewrite the above relation by substituting
Rewrite the above relation in terms of
Write the expression for first moment of whole area about x-axis.
Here,
Rewrite the above relation in terms of
Therefore, the centroid is located at
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Chapter 5 Solutions
Vector Mechanics for Engineers: Statics
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