Mechanical Engineering Principles

86 MECHANICAL ENGINEERING PRINCIPLES

7.5 Worked problems on centroids of

simple shapes

Problem 1. Show, by integration, that the

centroid of a rectangle lies at the intersection

of the diagonal.

Let a rectangle be formed by the liney=b,the
x-axis and ordinatesx=0andx=Las shown in
Figure 7.4. Let the coordinates of the centroidCof
this area be (x,y).

y

y= b

b

C

0 Lx

x

y

Figure 7.4

By integration,

x=

∫L

0

xydx

∫L

0

ydx

=

∫L

0

(x)(b)dx

∫L

0

bdx

=

[

b

x^2

2

]L

0

[bx]L 0

=

bL^2

2

bL

=

L

2

and y=

1

2

∫L

0

y^2 dx

∫L

0

ydx

=

1

2

∫L

0

b^2 dx

bL

=

1

2

[

b^2 x

]L

0

bL

=

b^2 L

2

bL

=

b

2

i.e.the centroid lies at

(

L

2

,

b

2

)

which is at the

intersection of the diagonals.

Problem 2. Find the position of the centroid

of the area bounded by the curvey= 3 x^2 ,

thex-axis and the ordinatesx=0andx=2.

If (x,y) are the co-ordinates of the centroid of the

given area then:

x=

∫ 2

0

xydx

∫ 2

0

ydx

=

∫ 2

0

x( 3 x^2 )dx

∫ 2

0

3 x^2 dx

=

∫ 2

0

3 x^3 dx

∫ 2

0

3 x^2 dx

=

[

3 x^4

4

] 2

0

[

x^3

] 2

0

=

12

8

= 1. 5

y=

1

2

∫ 2

0

y^2 dx

∫ 2

0

ydx

=

1

2

∫ 2

0

( 3 x^2 )^2 dx

8

=

1

2

∫ 2

0

9 x^4 dx

8

=

9

2

[

x^5

5

] 2

0

8

=

9

2

(

32

5

)

8

=

18

5

= 3. 6

Hence the centroid lies at (1.5, 3.6)

Problem 3. Determine by integration the

position of the centroid of the area enclosed

by the liney= 4 x,thex-axis and ordinates

x=0andx=3.

Let the coordinates of the area be (x,y) as shown

in Figure 7.5.

Then x=

∫ 3

0

xydx

∫ 3

0

ydx

=

∫ 3

0

(x)( 4 x)dx

∫ 3

0

4 xdx

=

∫ 3

0

4 x^2 dx

∫ 3

0

4 xdx

=

[

4 x^3

3

] 3

0

[

2 x^2

] 3

0

=

36

18

= 2

y=

1

2

∫ 3

0

y^2 dx

∫ 3

0

ydx

=

1

2

∫ 3

0

( 4 x)^2 dx

18