Problem 7.29 Determine the coordinates of the
centroids.
Solution: Break into a rectangle, a triangle and a circular hole
xD
y
yD
2 in
8 in
5[108] C 12
108 C
1
1
2 86 4[108] C 13 8
108 C
2 86
1
2 86
4[22 ]
22
3[22 ]
1
2
2 86 2
D 6.97 in
D 3.79 in
x D 6.97 in
y D 3.79 in
3 in
x
4 in
6 in
10 in
Problem 7.30 Determine the coordinates of the
centroids.
Solution: The strategy is to find the centroid for the half circle
area, and use the result in the composite algorithm. The area: The
element of area is a vertical
strip y high and dx wide. From the equation
p
of the circle, y D š R2 x 2 . The
p height of the strip will be twice
the positive value, so that dA D 2 R2 x 2 dx, from which
dA D 2
AD
A
R
R2 x 2 1/2 dx
y
10 in
x
20 in
0
p
R
R2
x R2 x 2
R2
1 x
D
D2
C
sin
2
2
R
2
0
The x-coordinate:
x dA D 2
A
R
x
R2 x 2 dx
0
R
2R3
R2 x 2 3/2
D
D2 .
3
3
0
Divide by A: x D
4R
3
The y-coordinate: From symmetry, the y-coordinate is zero.
420
D 8.488 in. For
3
the inner half circle x2 D 4.244 in. The areas are
The composite: For a complete half circle x1 D
A1 D 628.32 in2
and A2 D 157.08 in2 .
c 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
525
Problem 7.38 If the cross-sectional area of the beam
shown in Problem 7.37 is 8400 mm2 and the y coordinate of the centroid of the area is y D 90 mm, what are
the dimensions b and h?
Solution: From the solution to Problem 7.37
A1 D 120 b, A2 D 200 h
and y D
y1 A1 C y2 A2
A1 C A2
h
60120 b C 120 C
200 h
2
yD
120 b C 200 h
where
y1 D 60 mm
y D 90 mm
A1 C A2 D 8400 mm2
Also,
y2 D 120 C h/2
Solving these equations simultaneously we get
h D 18.2 mm
b D 39.7 mm
200 mm
h
A2
A1
120 mm
b
c 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
531