Lesson 14 centroid of volume
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Transcript of Lesson 14 centroid of volume
TOPIC
APPLICATIONSCENTROIDS OF SOLIDS OF REVOLUTION
CENTER OF GRAVITY OF A SOLID OF REVOLUTION
The coordinates of the centre of gravity of a solid of revolution are obtained by taking the moment of an elementary disc about the coordinate axis and then summing over all such discs. Each sum is then approximately equal to the moment of the total volume taken as acting at the centre of gravity. Again, as the disc thickness approaches zero the sums become integrals.
THE MOMENT OF A SOLID of volume V, generated by revolving a plane area about a horizontal or vertical axis, with respect to the plane through the origin and perpendicular to the axis may be found as follows:
1.Sketch the region, showing a representative strip and the approximating rectangle.
2.Form the product of the volume, disc or shell generated by revolving the rectangle about the axis and the distance of the centroid of the rectangle from the plane, and sum for all the rectangles.
3.Assume the number of rectangles to be indefinitely increased and apply the fundamental theorem.
When the area is revolved about the x-axis, the centroid is on that axis. Which means that, for solids generated by revolving the plane area about an axis, its centroid is on that axis, thus, giving one coordinate.
)z ,y ,x(C
If Myz is the moment of the solid to the y-z plane through the origin and
perpendicular to the x-axis, thenxvMyz
xzMSimilarly if is the moment of the solid with respect to the x-z plane through the origin, and perpendicular to the y-axis, then
is zero, considering that solids generated by revolving a plane area only about a horizontal or vertical axis, where which is the moment about an x-y plane, is always zero. ( ).
zxyM
0xyM
yvMxz
1. Determine the centroid of the solid generated by revolving the area bounded by the curve y = x2 , y = 9, and x = 0, about the y-axis.
Since the axis of revolution is the y-axis, the centroid of the solid is on that axis, giving
.
.0x
h= 9-y
y = x2
y=y
y = 9
( 3 , 9 )
( x , y )
),,( zyxC
)0,6,0(C
EXAMPLE
,
dx)y(xdv 92
3x
0xdxy9x22V
2xy cesin
units cu. 2
81
034
13
2
92
4
x
2
x92
dxxx92
xdx)x9(2V
42
3
0
42
3
03
3
02
ydVdMxz 2
9 yy
ydxyxdMxz 92
dxy
yxdMxz 2
992
243
036
13
2
81
6
x
2
x81
dxxx81
xdxx81
xdxy81M
62
3
0
62
3
05
3
04
3
02
xz
Moment about the x-z plane:
yVMxz
units 6y
2/81
243V
My xz
Centroid:
Centroid C (0, 6, 0)
xy 42 42 xy .4x
2. Find the coordinates of the centroid of the solid generated by revolving the area within and the line about
EXAMPLE
)0,8
5,4(Canswer
units .cuV5
216
27xzM
EXERCISES:
Determine the coordinates of the centroids of the solids generated by revolving:
1. the first quadrant region bounded by the curve y = 4 – x2 about the y –axis.
2. the third quadrant region bounded by the curve y = x3 and y = x about y – 1 = 0.
3. the region bounded by the curve y2 = 4x , the lines x = 0 and y = 4, about x + 1 = 0.
4. the region bounded by the curves y = x2 and y2 = – x about the line y+1 = 0.
5. the region bounded by the curves y = 6x – x2 and y = x2 – 2x about x – 4 = 0.