Answer to Question #259441 in Mechanics | Relativity for Karthi

Question

Q1. For the shaded area shown in figure. Determine:-Co-ordinates of centroid

The removed area is semi-circle and all the given dimensions are in cm. Dimension of rectangle is 12.5*20 cm² and diameter of semi-circle is 7.5 cm.

12.5

7.5

Answer 1

Since figure is symmetric respect to vertical axis:

$x_C=20/2=10$

$y_C=S_x/A$

where S_x is static moments around x axis,

A is area of figure

$S_x=\sum S_{x_i}$

$S_{x_i}=A_i y_{C_i}$

where A_i are subarea,

y_ci is centroid coordinate of subarea

for rectangle:

$A_1=20\cdot 12.5=250$ cm²

$y_{c1}=12.5/2=6.25$

for semi-circle:

$A_2=\pi R^2/2=3.75^2\pi/2=22$ cm²

$y_{c2}=12.5-4R/(3\pi)=12.5-4\cdot 3.75/(3\pi)=10.9$

$A=A_1-A_2=250-22=228$ cm²

$S_x=A_1y_{c1}-A_2y_{c2}=250\cdot6.25-22\cdot10.9=1322.7$

$y_C=\frac{1322.7}{228}=5.8$