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QUESTION: 1

A solid cylinder of diameter 5.0 m has a height of 6.0 m. Find the meta-centric height of the cylinder if the specific gravity of the material of cylinder 0.45 and it is floating in water with its axis vertical. State whether the equilibrium is stable or unstable.

Solution:

Explanation: BG=Centre of pontoon – Centre of immersed portion=0.3-0.45*0.3=1.65

Metacentric height=I/∀ -BG

I=π*r⁴=π*2.5⁴

∀=π*r*r*h=π*2.5*2.5*6

Metacentric height=-0.61.

QUESTION: 2

A solid cylinder of 15 cm diameter and 40 cm long, consists of two parts made of different materials. The first part at the base is 1.5 cm long and of specific gravity=6.5. The other part of the cylinder is made of the material having specific gravity 0.75. State, if the it can float vertically in water.

Solution:

Explanation: AG=(weight of base*distance of C.G from base point A) + (weight of upper part*distance of C.G from point A)/ )weight of base + weight of upper part)

= 14.52

By principle of buoyancy,

Weight of cylinder = Weight of water displaced

h=38.625

AB=19.31

BG=14.25-19.31= -4.79

GM= Metacentric height=I/∀ -BG

= 6.16

As metacentric height is positive, it will float.

QUESTION: 3

A wooden cylinder of sp.gr. = 0.6 and circular in cross-section is required to float in oil(sp.gr. = 0.90). Find the L/D ratio for the cylinder to float with its longitudinal axis vertical in oil, where L is the height of cylinder and D is its diameter.

Solution:

Explanation: By principle of buoyancy,

Weight of cylinder = Weight of water displaced

h=2L/3

AG=L/2

AB=L/3

BG=AG-AB=L/6

GM= Metacentric height=I/∀ – BG=3D2/32L-L/6

For stable equilibrium, GM should be positive

GM>0

i.e L/D<3/4.

QUESTION: 4

A cylinder(uniform density distribution) of radius 3.0 m has a height of 9.0 m. The specific gravity of the material of cylinder 0.85 and it is floating in water with its axis vertical. State whether the equilibrium is stable or unstable.

Solution:

Explanation: BG=Centre of pontoon – Centre of immersed portion=0.3-0.45*0.3=1.65

Metacentric height=I/∀ -BG

I=π*r⁴=π*3⁴

∀=π*r*r*h=π*3*3*9

Metacentric height=0.325.

QUESTION: 5

If the magnitude of dimension of a rectangular wooden block is length>breadth>height, then for it to float on the water, it should be immersed in what manner?

Solution:

Explanation: When it is immersed in such a manner where height is partially immersed, its stability is most as moment of inertia is most about that axis.

QUESTION: 6

When body is completely or partially immersed in a fluid, how much its weight be distributed for it to be in stable equilibrium.

Solution:

Explanation: When the weight distribution is around the lower part, the centre of gravity is at lower portion and hence below the centre of buoyancy which is condition for stable equilibrium.

QUESTION: 7

In unstable equilibrium what is the relation between forces?

Solution:

Explanation: Fb=W and the the centre of buoyancy is below the centre of gravity.

QUESTION: 8

The floating body is said to be in unstable equilibrium if the metacentre is below the centre of gravity.

Solution:

For a floating body

Metacentre above centre of gravity→ Stable Equilibrium

Metacentre coincides centre of gravity→ Neutral Equilibrium

Metacentre below centre of gravity→ Unstable Equilibrium

For a submerged body

Centre of buoyancy above centre of gravity→ Stable Equilibrium

Centre of buoyancy coincides centre of gravity→ Neutral Equilibrium

Centre of buoyancy below centre of gravity→ Unstable Equilibrium

QUESTION: 9

The floating body is said to be in neutraL equilibrium if the metacentre is above the centre of gravity.

Solution:

Explanation: The floating body is said to be in unstable equilibrium if the metacentre coincides with the centre of gravity.

QUESTION: 10

In stable equilibrium for completely submerged bodies what is the relation between forces?

Solution:

Explanation: Fb=W and the the centre of buoyancy is above the centre of gravity.

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