(^96) „„„„„ A Textbook of Engineering Mechanics
Example 6.18. A solid hemisphere of 20 mm radius supports a solid cone of the same base
and 60 mm height as shown in Fig. 6.30. Locate the centre of gravity of the composite section.
Fig. 6.30.
If the upper portion of the cone is removed by a certain section, the centre of gravity lowers
down by 5 mm. Find the depth of the section plane (h) below the apex.
Solution. As the body is symmetrical about Y-Y axis, therefore its centre of gravity will lie on
this axis.
Let apex of the cone (O) be the axis of reference.
Centre of gravity of the composite section
(i) Right circular cone
22 3
1 (20) 60 25 133 mm
33
vrh
ππ
=××=× =
and 1
3
60 45 mm
4
y =×=
(ii) Hemisphere
23 3
2
22
(20) 16 755 mm
33
vr
ππ
=×=× =
and (^2)
320
60 67.5 mm
8
y
×
=+ =
We know that distance between centre of gravity of the body and apex of the cone,
11 2 2
12
(25 133 45) (16 755 67.5)
25 133 16 755
vy v y
y
vv

• 
  

  ×+ ×

  
  

  ++
  mm
  2 261 950
  41 888
  = = 54 mm Ans.
  Depth of the section plane below the apex
  We know that the radius of the cut out cone,
  3
  h
  r= ...
  20 60
  ⎛⎞rh
  ⎜⎟=
  ⎝⎠
  Q
  ∴ Volume of the cut out cone,
  2
  223
  3 0.1164 mm
  333
  h
  vrh h h
  ππ⎛⎞
  =××=⎜⎟×=
  ⎝⎠