Laxmi

Pressure Definition Physics:
Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.

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What is Pressure in Physics? | Definition, Formula, Examples Units – Hydrostatics

Pressure Formula:
Pressure of liquid at a point is

p = \(\frac{\text { Thrust }}{\text { Area }}=\frac{F}{A}\)

Pressure is a scalar quantity

Pressure Unit:
SI unit is Pascal denoted bt pa
1 pa = Nm^-2

Pressure Dimensional Formula:
Dimensional formula [ML^-1T^–²].

Examples of Pressure in Physics

Pressure Examples in everyday life – High Pressure:
1. Nails, needles and pins have very sharp ends with very small surface areas. When a force is applied to the head of a nail, the pressure will drive its sharp end into a piece of wood easily.

2. The studs on a football boot have only a small area of contact with the ground.

3. The pressure under the studs is high enough for them to sink into the ground, which gives extra grip.

4. If you turn the blade so the cutting edge is pressed into the fruit.

Pressure Examples in everyday life – Low Pressure:
1. Skis have a large area to reduce the pressure on the snow so that they do not sink-in too far.

2. A tractor moving on soft ground has wide tires to reduce the pressure on the ground so that they will not sink into the ground

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

Fluids Definition Physics:
1. Fluid, any liquid or gas or generally any material that cannot sustain a tangential, or shearing, force when at rest and that undergoes a continuous change in shape when subjected to such a stress.

2. Fluids are those substances which can flow when an external force is applied on them.

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What are Fluids in Physics? | Definition, Examples, Properties, Types – Hydrostatics

Liquids and gases are fluids. The key property of fluids is that they offer very little resistance to shear stress. Hence, fluids do not have finite shap but takes the shape of the containing vessel.

Fluid Properties:
In fluid mechanics, the following properties of fluid would be considered

• When the fluid is at rest- hydrostatics
• When the fluid is in motion — hydrodynamics

Types of Fluids in Physics:

• Ideal fluid.
• Real fluid.
• Newtonian fluid.
• Non-Newtonian fluid.
• Ideal plastic fluid.

Examples of Fluids in Physics:

1. A river flowing down a mountain
2. Air passing over a bird’s wing
3. Blood moving through a circulatory system
4. Fuel moving through an engine

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

The Density of a Mixture of Substances – Hydrostatics

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Density of a Mixture of Two Liquids:
1. When two liquids of masses m₁ and m₂ having densities ρ₁ and ρ₂ are mixed together, then density of mixture is

ρ = \(\frac{m_{1}+m_{2}}{\left(\frac{m_{1}}{\rho_{1}}\right)+\left(\frac{m_{2}}{\rho_{2}}\right)}=\frac{\rho_{1} \rho_{2}\left(m_{1}+m_{2}\right)}{\left(m_{1} \rho_{2}+m_{2} \rho_{1}\right)}\)

2. When two liquids of same mass m but of different densities ρ₁ and ρ₂ are mixed together, then density of mixture is

ρ = \(\frac{2 \rho_{1} \rho_{2}}{\rho_{1}+\rho_{2}}\)

3. When two liquids of same volume V but of different densities ρ₁ and ρ₂ are mixed together, then density of mixture is

ρ = \(\frac{\rho_{1}+\rho_{2}}{2}\).

Density of a liquid varies with pressure,

ρ = \(\rho_{0}\left[1+\frac{\Delta p}{K}\right]\)

where,
ρ₀ = initial density of the liquid,
K = bulk modulus of elasticity of the liquid
Δp = change in pressure.

4. With rise in temperature (ΔT) due to thermal expansion of a given body, volume will increase while mass will remains constant, so density will decrease.

ρ = \(\frac{\rho_{0}}{(1+\gamma \cdot \Delta T)}\) ≃ ρ₀ (1 – γ . ΔT); where γ is volumetric expansion.

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

Relative Density Definition Physics:
Relative density of a substance is defined as the ratio of its density to the density of water at 4°C.

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What is Relative Density in Physics? | Definition, Formula, Units – Hydrostatics

Relative Density Formula in Physics:

Relative density = \(\frac{\text { Density of substance }}{\text { Density of water at } 4^{\circ} \mathrm{C}}\)
= \(\frac{\text { Weight of substance in air }}{\text { Loss of weight in water}}\)

Relative Density unit:
Relative density has no unit because it is the ratio of same units which gets cancelled.

Relative Density Dimensional Formula:

[M⁰ L⁰ T⁰]

Where,
M = Mass
L = Length
T = Time

• Relative density also known as specific gravity has no unit, no dimensions.
• For a solid body, density of body = density of substance.
• While for a hollow body, density of body is lesser than that of substance.
• When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while that of lowest density at the top and interfaces will be plane.

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

Density in Physics Definition:
1. Density of a substance is defined as the ratio of its mass to its volume.

2. Density is the mass per unit volume of any object. It is calculated by dividing the mass of an object by its volume.

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What is Density in Physics? | Definition, Formula, Units – Hydrostatics

Density Formula in Physics:

Density of a liquid = \(\frac{\text { Mass }}{\text { Volume }}\)
ρ = \(\frac{m}{V}\)

Where,
ρ = Density
m = mass
V = volume

Density of water = 1 g/cm³ or 10³ kg/m³

Density Units:
Density is commonly expressed in units of grams per cubic centimetre.
kg/m³

In case of homogneous (isotropic) substance it has no directional properties, so it is scalar quantity.

Density Dimensional Formula:
Dimensional formula is [ML^-3].

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

Archimedes’ Principle Definition:
Archimedes’ Principle states that when a body is partially or fully immersed in a liquid, it loses some of its weight and it is equal to the weight of the liquid displaced by the immersed part of the body.

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Archimedes Principle | Definition, Formula, Examples – Hydrostatics

If a is loss of weight of a body in water and b is loss of weight in another liquid, then

\(\frac{a}{b}=\frac{w_{\text {air }}-w_{\text {liquid }}}{w_{\text {air }}-w_{\text {water }}}\)

If T is the observed weight of a body of density σ when it is fully immersed in a liquid of density ρ, then real weight of the body

\(w=\frac{T}{\left(1-\frac{\rho}{\sigma}\right)}\)

Archimedes Principle Formula:
If w₁ = weight of body in air, w₂ = weight of body in liquid,
V_i = immersed of volume of liquid, ρ_L= density of liquid and
g = acceleration due to gravity

⇒ \(V_{i}=\frac{w_{1}-w_{2}}{\rho_{L} g}\)

Archimedes Principle of Buoyancy:
The Buoyant force applied by the fluid is equal to the weight of the displaced fluid.

Archimedes Principle Examples in our Daily Life

1. On entering a bathtub filled with water, an equal quantity of water is displaced as one weight.
2.  A ship floats in the sea because of the counter-acting buoyant force from the water based on which safe depth is calculated.
3. A small lead shot is embedded in a big lump of ice floating in a jar of water.
4. The mug floats when we try to push it in in water.

Hydrostatics:
Hydrostatics is a property of liquid or fluid in mechanics. A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. Here we will study the fluids in motion.

Properties of Fluids	Thrust
Pressure	Pressure Exerted by Liquid
Buoyant Force	Pascal’s Law
Archimedes’ Principle	Law of Floatation
Density	Relative Density
Density of a Mixture

Thermal Stress Definition:
When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress.

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What is Thermal Stress in Physics? | Definition, Formula – Elasticity

“Stress caused due to the change in temperature”

Thermal Stress Formula:

Thermal stress = \(\frac{F}{A}\) = Y α Δθ

where, α = Coefficient of linear expansion of the material of the rod.
When temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to change in pressure (Δp) of the gas.

Thermal stress = Δp = Kγ Δθ

where,
K = Bulk modulus of elasticity
γ = Coefficient of cubical expansion of the gas.

Elasticity:
Elasticity defines a property of an object that has the ability to regain its original shape after being stretched or compressed. Learn about the deforming force applied on an elastic object and how the stress and strain works on an object. What is a Hooke’s law and how it is applicable for the concept of elasticity.

Deforming Force	Elasticity
Stress	Strain
Hooke’s Law	Elastic Modulus
Types of Modulus of Elasticity	Poisson’s Ratio
Stress and Strain Curve	Thermal Stress
Cantilever Beam	Torsion of a Cylinder

Torsion of a Cylinder Definition:
1. Torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius.

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Torsion of a Cylinder | Definition, Formula, Units – Elasticity

2. If the upper end of a cylinder is clamped and a torque is applied at the lower end the cylinder get twisted by angle θ, then

Couple per unit twist,

C = \(\frac{\pi \eta r^{4}}{2 l}\)

where,
η = Modulus of rigidity of the material of cylinder,
r = Radius of cylinder and
l = length of cylinder.

Work done in twisting the cylinder through an angle θ

W = \(\frac{1}{2} C \theta^{2}\)

Relation between angle of twist (θ) and angle of shear (Φ)
rθ = lΦ
or
Φ = \(\frac{r}{l} \theta\)

Elasticity:
Elasticity defines a property of an object that has the ability to regain its original shape after being stretched or compressed. Learn about the deforming force applied on an elastic object and how the stress and strain works on an object. What is a Hooke’s law and how it is applicable for the concept of elasticity.

Deforming Force	Elasticity
Stress	Strain
Hooke’s Law	Elastic Modulus
Types of Modulus of Elasticity	Poisson’s Ratio
Stress and Strain Curve	Thermal Stress
Cantilever Beam	Torsion of a Cylinder

Cantilever Definition in Physics:
A beam clamped at one end and loaded at free end is called a cantilever.

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What is Cantilever Beam in Physics? | Definition, Example, Formula – Elasticity

Cantilever Formula Physics:
Depression (δ) at the free end of a cantilever is given by

δ = \(\frac{w l^{3}}{3 Y I_{G}}\)

where,
w = Load,
l = Length of the cantilever,
Y = Young’s modulus of elasticity
I_G = Geometrical moment of inertia.

For a beam of rectangular cross-section having breadth b and thicknes d,

\(I_{G}=\frac{b d^{3}}{12}\)

For a beam of circular cross-section area having radius r,

\(I_{G}=\frac{\pi r^{4}}{4}\)

Cantilever Beam Example:
A good example of a cantilever beam is a balcony. A balcony is supported on one end only, the rest of the beam extends over open space; there is nothing supporting it on the ther side.

Beam Supported at Two Ends and Loaded at the Middle

Depression at middle, δ = \(\frac{w l^{3}}{48 Y I_{G}}\)

Elasticity:
Elasticity defines a property of an object that has the ability to regain its original shape after being stretched or compressed. Learn about the deforming force applied on an elastic object and how the stress and strain works on an object. What is a Hooke’s law and how it is applicable for the concept of elasticity.

Deforming Force	Elasticity
Stress	Strain
Hooke’s Law	Elastic Modulus
Types of Modulus of Elasticity	Poisson’s Ratio
Stress and Strain Curve	Thermal Stress
Cantilever Beam	Torsion of a Cylinder

Stress-Strain Curve Explanation:
When a wire is stretched by a load as in Fig. (a), it is seen that for small value of load, the extension produced in the wire is proportional to the load as shown in Fig. (b). Hence,

Stress ∝ Strain

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Stress and Strain Curve | Explanation, Formula, Examples – Elasticity

Beyond the limit of elasticity, the stress and strain are not proportional to each other, on increasing the load further, the wire breaks at point D, known as feature point.

Stress-strain Curve Example:
The stress-strain curve illustrates the mechanical properties of materials, for example, steel, stainless steel or aluminium. This is the area in which the material is stressed beyond its elastic loadability and the first plastic deformations occur.

Breaking Stress Definition:
The minimum value of stress required to break a wire, is called breaking stress. Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.

Safety factor = \(\frac{\text { Breaking stress }}{\text { Working stress }}\)

Elastic Relaxation Time:
The time delay in restoring the original configuration after removal of deforming force is called elastic relaxation time. For quartz and phospher bronze this time is negligible.

Elastic After Effect:
The elastic after effect shows that the temporary delay in regaining the original configuration by the elastic body after the removal of deforming force is called elastic after effect.

Elastic Fatigue Meaning:
The property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.

Ductile Materials are Defined as:
The materials which show large plastic range beyond elastic limits are called ductile materials.

Ductile Materials Examples

• Copper
• Silver
• Iron
• Aluminum

Ductile materials are used for making springs and sheets.

Brittle Materials Definition:
The materials which show very small plastic range beyond elastic limits are called brittle materials.

Brittle Materials Examples

• Glass
• Cast
• Iron

Elastomers Definition:
The materials for which strain produced is much larger, than the stress applied, within the limit of elasticity are called elastomers.

Elastomers Examples
Rubber, the elastic tissue of arota, the large vessel carrying blood from heart etc. Elastomers have no plastic range.

Malleability
When a solid is compressed, a stage is reached beyond which it cannot regains its original shape after the deforming force is removed. This quality is called malleability of solid substance.

Elastic hysteresis
As a natural consequence of the elastic after-effect, the strain in the body tends to lag behind the stress applied to the body so that during a rapidly changing stress, the strain is greater for the same value of stress. This lag of strain behind the stress is called elastic hysteresis.

Elastic Potential Energy in a Stretched Wire

The work done in stretching a wire is stored in the form of potential energy of the wire.
Potential energy

U = Average force × Increase in length = \(\frac{1}{2}\) FΔl
= \(\frac{1}{2}\) Stress × Strain × Volume of the wire

Elastic potential energy per unit volume

U = \(\frac{1}{2}\) × Stress × Strain = \(\frac{1}{2}\) (Young’s modulus) × (Strain)²

Elastic potential energy of a stretched spring = \(\frac{1}{2}\) kx²
where, k = Force constant of spring and x = Change in length.

Elasticity:
Elasticity defines a property of an object that has the ability to regain its original shape after being stretched or compressed. Learn about the deforming force applied on an elastic object and how the stress and strain works on an object. What is a Hooke’s law and how it is applicable for the concept of elasticity.

Deforming Force	Elasticity
Stress	Strain
Hooke’s Law	Elastic Modulus
Types of Modulus of Elasticity	Poisson’s Ratio
Stress and Strain Curve	Thermal Stress
Cantilever Beam	Torsion of a Cylinder