(^754) „„„„„ A Textbook of Engineering Mechanics
We know that distance of centre of buoyancy from the bottom of the buoy,
OB=
1
(0.5 20) 0.25 10 cm
2
ll+= +
and volume of water displaced,
V= (20) (0.5^2 20) 100 (0.5 20)
4
ll
π
+=π +
∴ BM=
2500 25 50
100 (0.5 20) 0.5 20 40
I
Vlll
π

π+ + +
Now OM=OB + BM = (0.25l + 10) +
50
l+ 40
For stable equilibrium, the metacentre (M) should be above centre of gravity (G) or may
coincide with G.
i.e., OMñOG
50
(0.25 10)
40
l
l
++

• ñ
  (^25100)
  280
  ll
  l
  ++


• 
  

  ( 40) (0.25 10) 50
  40
  ll
  l
  +++

  
  


• ñ
  (^25100)
  280
  ll
  l
  ++


• 
  

  2 (0.25^21010400 50)
  2( 40)
  lll
  l
  +++ +

  
  


• 
  

  ñ
  (^25100)
  280
  ll
  l
  ++

  
  


• 
  

  ...(Multiplying and dividing the L.H.S. of the equation by 2)
  0.5l^2 + 40l + 800 + 100ñl^2 + 5l + 100
  or l^2 – 70l – 1600ñ 0
  ...[Multiplying both sides by (2l + 80)]
  Solving this quadratic equation for l,
  lñ
  70 (70)^2 4 1600
  cm
  2
  ++ +×
  ñ88.15 cm
  ∴ Maximum permissible length of the cylinder including the metal portion
  = 88.15 + 2.5 = 90.65 cm Ans.
  36.13.CONICAL BUOYS FLOATING IN A LIQUID
  A conical buoy, as the name indicates, is a buoy which is shaped
  like a cone or a solid body that tapers uniformly from a circular base to a
  point. Now consider a conical buoy floating in same liquid as shown in
  Fig. 36.10.
  Let D= Diameter of the cone,
  d= Diameter of the cone at the liquid
  level,
  2 α= Apex angle of the cone,
  L= Length of the cone,
  l= Length of the cone immersed in
  liquid. Fig. 36.10. Conical buoy