Higher Engineering Mathematics

382 INTEGRAL CALCULUS

When areaPQRSis rotated about axisXXthe vol-

ume generated is that of the pulley. The centroid of

the semicircular area removed is at a distance of

4 r

3 π

from its diameter (see ‘Engineering Mathematics4th

edition’, page 471), i.e.

4(1.0)

3 π

, i.e. 0.424 cm from
PQ. Thus the distance of the centroid fromXXis
5. 0 − 0 .424, i.e. 4.576 cm.
The distance moved through in one revolution by the
centroid is 2π(4.576) cm.

Area of semicircle=

πr^2

2

=

π(1.0)^2

2

=

π

2

cm^2

By the theorem of Pappus,

volume generated=area×distance moved by

centroid=

(π

2

)

(2π)(4.576).

i.e.volume of metal removed=45.16 cm^3

Mass of metal removed=density×volume

=8000 kg m−^3 ×

45. 16

106

m^3

= 0 .3613 kg or 361.3g

volume of pulley=volume of cylindrical disc

−volume of metal removed

=π(5.0)^2 (2.0)− 45. 16

=111.9 cm^3

Mass of pulley=density×volume

=8000 kg m−^3 ×

111. 9

106

m^3

=0.8952 kgor895.2 g

Now try the following exercise.

Exercise 152 Further problems on the the-

orem of Pappus

1. A right angled isosceles triangle having a
   hypotenuse of 8 cm is revolved one revolution
   about one of its equal sides as axis. Deter-
   mine the volume of the solid generated using
   Pappus’ theorem. [189.6 cm^3 ]


2. Using (a) the theorem of Pappus, and (b) inte-
   gration, determine the position of the centroid

of a metal template in the form of a quadrant

of a circle of radius 4 cm. (The equation of a

circle, centre 0, radiusrisx^2 +y^2 =r^2 ).

⎡

⎢

⎣

On the centre line, distance

2 .40 cm from the centre,

i.e. at co-ordinates

(1.70, 1.70)

⎤

⎥

⎦

3. (a) Determine the area bounded by the curve
   y= 5 x^2 , the x-axis and the ordinates
   x=0 andx=3.

(b) If this area is revolved 360◦about (i) the

x-axis, and (ii) they-axis, find the vol-

umes of the solids of revolution produced

in each case.

(c) Determine the co-ordinates of the cen-

troid of the area using (i) integral calcu-

lus, and (ii) the theorem of Pappus.

⎡

⎢

⎣

(a) 45 square units

(b) (i) 1215πcubic units

(ii) 202.5πcubic units

(c) (2.25, 13.5)

⎤

⎥

⎦

4. A metal disc has a radius of 7.0 cm and is
   of thickness 2.5 cm. A semicircular groove of
   diameter 2.0 cm is machined centrally around
   the rim to form a pulley. Determine the vol-
   ume of metal removed using Pappus’ theorem
   and express this as a percentage of the origi-
   nal volume of the disc. Find also the mass of
   metal removed if the density of the metal is
   7800 kg m−^3.
   [64.90 cm^3 , 16.86%, 506.2 g]

For more on areas, mean and r.m.s. values, volumes

and centroids, see ‘Engineering Mathematics4th

edition’, Chapters 54 to 57.

38.7 Second moments of area of

regular sections

Thefirst moment of areaabout a fixed axis of a

lamina of areaA, perpendicular distanceyfrom the

centroid of the lamina is defined asAycubic units.

Thesecond moment of areaof the same lamina as

above is given byAy^2 , i.e. the perpendicular distance