Hydrostatics Notes

Hydrostatics Fluid

Hydrostatics Basics and Definitions

Fluid

Liquids and gases do not have any definite shape of their own and thus can change their shapes easily. As these substances are not rigid, they do not have the capability to restore their original shape.

For this reason, liquids and gases at rest cannot resist the tangential force acting on them, and due to the application of even a slight tangential force, these substances begin to flow. Since liquids and gases can flow easily, they are called fluids.

• The branch of physics in which the characteristic properties of fluids at rest are studied is called hydrostatics.
• A liquid has a volume but no definite shape. It takes the shape of the container in which it is kept. But gases have neither a definite shape nor a volume of their own they usually take the shape and the volume of their container.
• Consequently, a gas can be compressed easily by applying pressure on it, while a liquid is almost incompressible.
• Any substance which has no definite shape and has the ability to flow is called fluid. Thus, both liquids and gases are fluids.

Fluids are everywhere around us. Earth has an envelope of air, and two third of earth’s surface is covered with water. Fluids are a phases of matter and include liquids, gases, plasmas and to some extent, plastic solids.

The fundamental difference between solid and liquid:

1. Shearing stress causes a change in the shape of a solid without changing its volume whereas fluids offer little resistance to shearing stress. The shearing stress of fluids is about million times smaller than that of solids.
2. A fluid can exert or withstand a force in a direction perpendicular to its surface. So a fluid does have a bulk modulus of rigidity.

The fundamental difference between liquid and gas: A liquid is incompressible and has a definite volume and a free surface of its own. However, a gas is compressible and it expands to occupy all the space available to it.

Hydrostatics Archimedes Principle

Hydrostatic Force on Submerged Objects

When a body is totally or partly immersed in a liquid or gas at rest, it appears to lose a part of its weight. This apparent loss of weight is equal to the weight of the liquid or gas displaced by the body. This is Archimedes’ principle.

It is to be noted, that Archimedes’ principle is related to the weight of a body. So, this principle is not applicable for a body in a weightless condition. The weight of the body in an artificial satellite or in a freely falling situation is zero. So in these cases, Archimedes’ principle is not applicable.

Application of Archimedes’ Principle: We can determine the following with the help of Archimedes’ principle:

1. The volume of a solid of any shape
2. The density of a substance
3. The amount of the constituent elements in a piece of alloy made of two elements

For the determination of the above things, the choice of the liquid used should be such that

1. The body should not be soluble in that liquid and
2. The body under experiment should not react chemically with the liquid.

Hydrostatics Principles with Examples

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1. Determination of the volume of a solid of any shape: The method described below is the simplest one to find the volume of a body of any shape.

The body is heavier than the liquid: Let the weight of the body in air be W₁ and its weight when totally immersed in the liquid be W₂.

According to Archimedes’ principle,

W₁ – W₂ = apparent loss in the weight of the body

= weight of the liquid displaced by the body

= weight of the liquid equal to the volume of the body

If the density of the liquid is ρ, then volume of the displaced liquid = \(\frac{W_1-W_2}{\rho g}\)

= volume of the body……(1)

The body is lighter than the liquid: in this case, a sinker is used to immerse the lighter body experiment completely in the liquid.

Let the weogth of the body in air be W₁

the weight of the sinker inside the liquid be W₂ and the weight the body with the sinker inside the liquid = W₃

According to Archimedes’ principle, W₁ – (W₃ – W₂)

= apparent weight loss of the body

= weight of the liquid displaced by the body

= weight of equal volume of the liquid If the density of the liquid is ρ, then the volume of the displaced liquid = \(\frac{W_1-\left(W_3-W_2\right)}{\rho g}\)

∴ volume of the body….. (2)

2. Determination of density of a substance: Let the weight of a body in air be W₁ when it is immersed in a liquid of density ρ, it weighs W₂.

According to Archimedes’ principle, from equation (1) we get the volume of the body, V = \(\frac{W_1-W_2}{\rho g}\)

∴ Density of the material of the body,

D = \(\frac{\text { mass of the body }}{\text { volume of the body }}=\frac{W_1 / g}{V}=\frac{W_1 \rho}{W_1-W_2}\)…..(3)

From the measured value of density, the purity of a metal can be known. If the measured value of the density is equal to the actual density of that metal, then we can say that the metal is pure; otherwise it is impure.

3. Determination of the amounts of constituent elements in a piece of alloy made of two elements: Let us assume that an alloy is made of two metals A and B. Let the mass of the alloy in air be W₁ and its weight when immersed completely in water be W₂.

According to Archimedes’ principle, volume of the alloy V = \(\frac{W_1-W_2}{\rho g}\)

Let us assume that the amount of masses and the densities of metals A and B in the alloy are m_a, m_b, and p_a, p_b respectively.

∴ Volume of metal \(A=\frac{m_a}{\rho_a}\)

and volume of metal B = \(\frac{m_b}{\rho_b}=\frac{\left(W_1 / g\right)-m_a}{\rho_b}\)

∴ \(\frac{m_a}{\rho_a}+\frac{\left(W_1 / g\right)-m_a}{\rho_b}=\frac{W_1-W_2}{\rho g}\)

or, \(m_a\left(\frac{1}{\rho_a}-\frac{1}{\rho_b}\right)=\frac{W_1-W_2}{\rho g}-\frac{W_1}{\rho_b g}\)

or, \(m_a=\frac{1}{g}\left(\frac{W_1-W_2}{\rho}-\frac{W_1}{\rho_b}\right) \frac{\rho_a \rho_b}{\rho_b-\rho_a}\)

So, if the values of W₁ , W₂, ρ, ρ_a, and ρ_b are known, then the amount of metal A can be determined from equation (4), and from this, the amount of metal B can be determined.

Hydrostatics Synopsis

Density Definition:

The mass per unit volume of a substance is called its density.

• The ratio of the density of a solid or a liquid substance to the density of pure water at 4°C is called the specific gravity of that substance.
• , The ratio of the mass of a certain volume of any solid or liquid substance to the mass of an equal volume of pure water at 4°C is called the specific gravity of that substance.
• The force acting normally on unit area of a surface is called pressure.
• The normal force exerted by a liquid on any surface in contact with it is called the thrust of the liquid.

The characteristics of pressure at a point in a liquid at rest are:

1. The pressure at a point in a liquid at rest is the same in all directions.
2. The pressure at all points on the same horizontal level in a liquid at rest is the same.
3. The pressure at a point in a liquid at rest is directly proportional to the depth of that point inside the liquid.

The free surface of a liquid at rest is always horizontal.

If two immiscible liquids in a U-tube are in equilibrium, then the heights of the liquids from the plane of separation are inversely proportional to the densities of the liquids.

The free surface of a liquid at rest in connected vessels remains in the same horizontal plane.

Key Formulas in Hydrostatics

Pascal’s law: The pressure applied at any point of a confined fluid is transmitted with undiminished magnitude in all directions throughout the fluid and acts normally on the surface in contact with the fluid.

• The ability of a liquid or gas at rest to exert an upward force on a body immersed in that fluid is called buoyancy.
• The upward thrust exerted on a body by a liquid or gas immersed partly or totally in it is called the buoyant force.
• If the liquid with the body remains in a weightless state, then no buoyant force acts on the body.
• The point where the centre of gravity of the liquid or gas lies before it is displaced by a body immersed in it is the centre of buoyancy or centre of floatation of the immersed body.

Archimedes’ principle: When a body is immersed partly or totally in a liquid or gas at rest, the body appears to lose a part of its weight. This apparent loss in weight is equal to the weight of the liquid or gas displaced by the body.

In the case of a body immersed in a liquid at rest, let the weight of the body be W₁ and the buoyant force acting on the body be W₂.

If W₁ > W₂, then the body sinks in the liquid.

If W₁ = W₂, then the body remains floating at any position inside the liquid being totally immersed in the liquid.

If W₁ <W₂, the body moves up through the liquid and remains floating partly submerged in the liquid.

A floating body has no apparent weight.

1. Condition of floatation: The weight of a floating body must be equal to the weight of the liquid displaced by the body.
2. Condition of equilibrium: The centre of gravity and the centre of buoyancy must lie on the same vertical plane.

• For a floating body in a tilted position, the vertical line drawn through the centre of buoyancy cuts the central line at a point called the metacentre of the body.
• For stable equilibrium of a floating body, the metacentre of the body should lie above its centre of gravity.