(^756) „„„„„ A Textbook of Engineering Mechanics
and distance of c.g. from the apex,
OG= 0.75 L
For stable equilibrium, the metacentric (M) should be above G or may coincide with c.g.
i.e., BGñBM
OG – OBñBM
0.75 L – 0.75 L (0.8)1/3ñ0.75 l tan^2 α
L [1 – (0.8)1/3]ñL (0.8)1/3 tan^2 α
∴ tan^2 αú
1/3
1/ 3
[1 – ( 0. 8) ]
(0.8)
ú0.08
∴ tan αú0.2828
or αú15.8°
∴ Least apex angle, 2 α= 31.6° Ans.
and moment of inertia of the circular section about the liquid level
I=
(^44) (2 tan )
64 64
dl
ππ
=× +α =^44 tan
4
l
π
α
We know that BM=
44
2
32
tan
(^4) 0.75 tan
1
tan
3
I l
l
V l
π
α
==+α
πα
Example 36.11. A conical buoy 1 metre long, and of base diameter 1.2 metre, floats in
water with its apex downwards. Determine the minimum weight of the buoy, for stable equilibrium.
Take weight of water as 9·8 kN/m^3.
Solution. Given: Length of the conical buoy (L) = 1 m and diameter of base of the conical
buoy (D) = 1.2 m.
Let l= Length of the cone immersed in water,
∴ Volume of water displaced
V=^23
1
(0.6 ) m
3
π×ll
= 0.377 l^3 m^3
and moment of inertia of circular section,
I=
(1.2 )^44 0.1018
64
ll
π

∴ BM=
4
3
0.1018
0.27 m
0.377
Il
l
V l

We know that distance of centre of buoyancy from the apex,
OB= 0.75 l
and distance of c.g. from the apex,
OG= 0.75 × 1 = 0.75 m
Fig. 36.12.