Physics

Equilibrium of a Particle Physics | Definition – Laws of Motion

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Equilibrium of a Particle in Physics:
A particle is in equilibrium if the vector sum of the external forces acting on it is zero.
When the vector sum of the forces acting on a body is zero, then the body is said to be in equilibrium.

If two forces F₁ and F₂ act on a particle, then they will be in equilibrium if F₁ + F₂ = 0.

Lami’s theorem:
It states that, if three forces acting on a particle are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

\(3 \frac{F_{1}}{\sin \alpha}=\frac{F_{2}}{\sin \beta}=\frac{F_{3}}{\sin \gamma}\)

Laws of Motion:
There are various laws in Physics that define the motion of the object. When an object is in motion whether it is linear or circular there is some force which is always imposed on it.

What is Inertia of Motion	Force
Law of Conservation of Linear Momentum	Impulse
Laws of Motion	Rocket
Equilibrium of a Particle	Weight
Friction	Motion on a Rough Inclined Plane
Motion of Bodies in Contact	Pulley Mass System

Perpendicular Axis Theorem Statement:
The moment of inertia of any two dimensional body about an axis perpendicular to its plane (I_z) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersecting each other at a point, where the perpendicular axis passes through it.

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Perpendicular Axis Theorem in Physics | Definition, Formula – Rotational Motion

Perpendicular axis Theorem Diagram:

Mathematically, I_Z =I_X + I_Y
where,
I_X and I_Y are the moments of inertia of plane lamina about the perpendicular axes X and Y, respectively which lie in the plane of lamina and intersect each other.

Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two dimensional bodies only.

Rotational Motion:
In this portion, we will learn about the rotational motion of the objects. A body moves completely in rotational motion when each particle of the body moves in a circle about a single line. When a force is applied on a body about an axis it causes a rotational motion. The force applied here is called the torque. The axis of the rotation usually goes through the body. Also, learn the two theorems such as parallel axes and perpendicular theorem explained with respect to rotational motion of objects.

Centre of Mass	Linear Momentum of a System of Particles
Rigid Body	Moment of Inertia
Radius of Gyration	Parallel Axis Theorem
Perpendicular Axis Theorem	Moment of Inertia of Rigid Body
Torque	Angular Momentum
Centre of Gravity	Angular Impulse
Rotational Kinetic Energy

Types of Modulus of Elasticity:
Modulus of elasticity is of three types

• Young’s Modulus of Elasticity
• Bulk Modulus of Elasticity
• Modulus of Rigidity
  

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Types of Modulus of Elasticity in Physics | Definition, Formulas, Units – Elasticity

1. Young’s Modulus of Elasticity Definition:
Young’s Modulus of Elasticity is defined as the ratio of normal stress to the longitudinal strain within the elastic limit.

Young’s Modulus of Elasticity Formula:

Y = \(\frac{\text { Normal stress }}{\text { Longitudinal strain }}\)
Y = \(\frac{F \Delta l}{A l}=\frac{M g \Delta l}{\pi r^{2} l}\)

Young’s Modulus of Elasticity unit:
Young’s Modulus of Elasticity SI unit is N/m² or pascal

Young’s Modulus of Elasticity Dimensional Formula:
Its dimensional formula is [ML^-1T^-2].

Force Constant of Wire
Force required to produce unit elongation in a wire is called force constant of a material of wire. It is denoted by k

K = \(\frac{Y A}{l}\)

where, Y = Young’s modulus of elasticity
and A = cross-section area of wire.

2. Bulk Modulus of Elasticity Definition:
It is defined as the ratio of volumetric stress to the volumetric strain within the elastic limit.

Bulk Modulus of Elasticity Formula:

K = \(\frac{\text { Volumetric stress }}{\text { Volumetric strain }}\)
K = \(-\frac{F V}{A \Delta V}=-\frac{\Delta p V}{\Delta V}\)

where, Δp = F / A = Change in pressure.
Negative sign implies that when the pressure increases volume decreases and vice-versa.

Bulk Modulus of Elasticity unit:
Bulk Modulus of Elasticity SI unit is N/m² or pascal

Bulk Modulus of Elasticity Dimensional Formula:
The dimensional formula is [ML^-1T^-2].

Compressibility
Compressibility of a material is the reciprocal of its bulk modulus of elasticity.

Compressibility Formula:

Compressibility (C) = \(\frac{1}{K}\)

Compressibility unit:
Its SI unit is N^-1m² and CGS unit is dyne^-1 cm².

3. Modulus of Rigidity Definition: (η) (Shear Modulus)
It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit. Modulus of Rigidity is also known as Shear Modulus.

Rigidity Modulus Formula:

η = \(\frac{\text { Tangential stress }}{\text { Shearing strain }}\)
η = \(\frac{F}{A \theta}\)

Modulus of Rigidity unit:
SI unit is N/m² or pascal

Modulus of Rigidity Dimensional Formula:
Its Dimensional formula is [ML^-1T^-2].

Factors Affecting Elasticity Physics

Factors Affecting Elasticity of a Material:

1. Modulus of elasticity of materials decreases with the rise in temperature, except for invar.
2. By annealing elasticity of material decreases.
3. By hammering or rolling elasticity of material increases.
4. Addition of impurities affects elastic properties depending on whether impurities are themselves more or less elastic.

Note:

• For liquids, modulus of rigidity is zero.
• Young’s modulus (V) and modulus of rigidity (rj) are possessed by solid materials only.

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

Rounding Off: The process of omitting the non significant digits and retaining only the desired number of significant-digits, incorporating the required modifications to the last significant digit is called rounding off the number.

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Rounding Off – Units and Measurement

Rounding Off – Definition, Rules

Rules for Rounding Off a Measurement

(i) If the digit to be dropped is less than 5, then the preceding digit is left unchanged, e.g. 1.54 is rounded off to 1.5.
(ii) If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g. 2.49 is rounded off to 2.5.
(iii) If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g. 3.55 is rounded off to 3.6.
(iv) If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g. 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6.

Units and Measurement Topics:
Measurement requires tools to provide scientists with a quantity. A quantity describes how much of something there is and how many there are.

Physical Quantities and Their Units	Systems of Units
Dimensions	Significant Figures
Rounding off	Error
Combinations of Errors

Angular Momentum Definition:
The moment of linear momentum is called angular momentum.

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Angular Momentum in Physics | Definition, Formula, Symbol, Units – Rotational Motion

Angular Momentum Symbol:
Angular Momentum is denoted by L.

Angular Momentum Formula:
Angular momentum, L = Iω = mvr
Where,
L = angular momentum
m = mass
v = velocity
r = radius
In vector form, L = I ω = r x mv

Angular Momentum Units:
SI unit is J-s

Angular Momentum Dimensional Formula:
Dimensional formula is [ML²T^-1].

Torque, t = \(\frac{d \mathbf{L}}{d t}\)

Principle of Moment:
When an object is in rotational equilibrium, then algebraic sum of all torques acting on it is zero. Clockwise torques are taken negative and anti-clockwise torques are taken positive.

Conservation of Angular Momentum Equation:
If the external torque acting on a system is zero, then its angular momentum remains conserved.
If T_ext = 0, then L = Iω = constant ⇒ I₁ω₁ = I₂ ω₂

Torque and Angular Momentum for a System of Particles:
The rate of change of the total angular momentum of a system of particles about a point is equal to the sum of the external torques acting on the system taken about the same point.
\(\frac{d \mathbf{L}}{d t}\) = T_ext

Equilibrium of Rigid Body:
A rigid body is said to be in equilibrium, if both of its linear momentum and angular momentum are not changing with time. Thus, for equilibrium body does not possess linear acceleration or angular acceleration.

Couple in Physics Meaning:
A pair of equal and opposite forces with parallel lines of action is called couple. It produces rotation without translation.

Rotational Motion:
In this portion, we will learn about the rotational motion of the objects. A body moves completely in rotational motion when each particle of the body moves in a circle about a single line. When a force is applied on a body about an axis it causes a rotational motion. The force applied here is called the torque. The axis of the rotation usually goes through the body. Also, learn the two theorems such as parallel axes and perpendicular theorem explained with respect to rotational motion of objects.

Centre of Mass	Linear Momentum of a System of Particles
Rigid Body	Moment of Inertia
Radius of Gyration	Parallel Axis Theorem
Perpendicular Axis Theorem	Moment of Inertia of Rigid Body
Torque	Angular Momentum
Centre of Gravity	Angular Impulse
Rotational Kinetic Energy

Gravitational Potential Definition Physics:
Gravitational potential at any point in gravitational field is equal to the work done per unit mass in bringing a very light body from infinity to that point. It is denoted by V_g.

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Gravitational Potential Energy | Definition, Derivations, Formulas, Units – Gravitation

Gravitational Potential Formula:

PE_G  = mgΔh

Where,
PE_G = potential energy due to gravity.
g = acceleration due to gravity.
m = mass.
Δh = distance above a surface.

Gravitational Potential Derivation:

Gravitational potential, V_g = \(\frac{W}{m}=-\frac{G M}{r}\)

Gravitational Potential Units:
Its SI unit is J/kg and it is a scalar quantity.

Gravitational Potential Dimensional Formula:
Its dimensional formula is [L² T^-2].
Since, work W is obtained, i.e. it is negative, the gravitational potential is always negative.

Gravitational Potential Energy Definition:
Gravitational potential energy of any object at any point in gravitational field is equal to the work done in bringing it from infinity to that point. It is denoted by U.

Gravitational Potential Energy Derivation:

Gravitational potential energy, U = \(-\frac{G M m}{r}\)

The negative sign shows that the gravitational potential energy decreases with increase in distance.
Gravitational potential energy at height h from surface of earth

U_h = \(-\frac{G M m}{R+h}=\frac{m g R}{1+\frac{h}{R}}\)

Gravitational Potential Energy of a Two Particle System

The gravitational potential energy of two particles of masses m₁ and m₂ separated by a distance r is given by

U = \(-\frac{G m_{1} m_{2}}{r}\)

Gravitational Potential Energy for a System of More than Two Particles
The gravitational potential energy for a system of particles (say m₁, m₂, m₃ and m₄) is given by

Thus, for a n particle system there are \(\frac{n(n-1)}{2}\) pairs and the potential energy is calculated for each pair and added to get the total potential energy of the system.

Gravitational Potential for Different Bodies

1. Gravitational Potential due to a Point Mass:
Suppose a point mass M is situated at a point O, the gravitational potential due to this mass at point P is given by

V = \(-\frac{G M}{r}\)

2. Gravitational Potential due to Ring:

3. Gravitational Potential due to Spherical Shell:

4. Gravitational Potential due to Solid Sphere:

Gravitation:
Have you ever thought, when we throw a ball above the ground level, why it returns back to the ground. It’s because of gravity. When a ball is thrown above the ground in the opposite direction, a gravitational force acts on it which pulls it downwards and makes it fall. This phenomena is called gravitation.

Learn relation between gravitational field and potential field, Kepler’s law of planetary, weightlessness of objects in absence of gravitation, etc.

Newton’s Law of Gravitation	Central Force
Acceleration Due to Gravity	Factors Affecting Acceleration Due to Gravity
Gravitational Field Intensity	Gravitational Potential Energy
Relation between Gravitational Field and Potential	Kepler s Laws of Planetary Motion
Earth’s Satellite	Orbital Velocity
Escape Velocity	Weightlessness

Elasticity Definition Physics:
Elasticity is a measure of a variable’s sensitivity to a change in another variable, most commonly this sensitivity is the change in price relative to changes in other factors.

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Elasticity in Physics | Definition, Types – Elasticity

Elasticity is that property of the object by virtue of which it regain its original configuration after the removal of the deforming force.

The modulus of elasticity is simply the ratio between stress and strain. There are three types of modulus of elasticity, Young’s modulus, Shear modulus, and Bulk modulus.

Elastic Limit Definition:
Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity and get permanently deformed.

Perfectly Elastic Bodies:
Those bodies which regain its original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies, e.g. quartz, phospher bronze etc.

Perfectly Plastic Bodies:
Those bodies which does not regain its original configuration at all on the removal of deforming force are called perfectly plastic bodies, e.g. putty, paraffin, wax etc.

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

Torque Definition Physics:
Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point, and will label ‘O’. We will call the force ‘F’.

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What is Torque in Physics? | Definition, Formula, Symbol, Units – Rotational Motion

Torque or moment of a force about the axis of rotation

τ = r x F = rF sin θ \(\hat{\mathbf{n}}\)

It is a vector quantity. It is also known as moment of force or couple.

If the nature of the force is to rotate the object clockwise, then torque is called negative torque and if rotate the object anti-clockwise, then it is called positive torque.

Torque Units:
SI unit is N-m

Torque Symbol:
τ

Torque Dimensional Formula:
Dimensional formula is [ML²T^-2].

Torque in Rotational Motion:
In rotational motion, torque, τ = I α
where,
α is angular acceleration
I is moment of inertia.

Rotational Motion:
In this portion, we will learn about the rotational motion of the objects. A body moves completely in rotational motion when each particle of the body moves in a circle about a single line. When a force is applied on a body about an axis it causes a rotational motion. The force applied here is called the torque. The axis of the rotation usually goes through the body. Also, learn the two theorems such as parallel axes and perpendicular theorem explained with respect to rotational motion of objects.

Centre of Mass	Linear Momentum of a System of Particles
Rigid Body	Moment of Inertia
Radius of Gyration	Parallel Axis Theorem
Perpendicular Axis Theorem	Moment of Inertia of Rigid Body
Torque	Angular Momentum
Centre of Gravity	Angular Impulse
Rotational Kinetic Energy

Thrust Physics Definition:
Total force acting perpendicular direction to the surface is called Thrust. The total normal force exerted by liquid at rest on a given surface is called thrust of liquid.

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What is Thrust in Physics? | Definition, Example, Units – Hydrostatics

Example of Thrust in Daily Life:

• Thrust is to move forward as a crowd entering a stadium.
• Thrust is to force one’s self into a conversation.
• An example of thrust is a fish being expelled from the ocean by a strong wave.
• It is easy to carry or to walk with high heels, and cause thrust acting here (a perpendicular force).
• A pistole engine can produces thrust on its own.

Thrust Unit:
Thrust = Force
The SI unit of thrust is newton.
The CGS unit of thrust is dyne.

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

Deforming Force Definition in Physics:
1. A force which produces a change in configuration of the object on applying it, is called a deforming force.

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Deforming Force | Definition, Examples, Types – Elasticity

2. Firstly we take a body, Now we apply the external force to the body. After some time you see that the shape and the size of the body changes this property is known as Deforming Force.

Deforming Force Types:

• Elastic deformation is reversed when the force is removed.
• Inelastic deformation is not fully reversed when the force is removed – there is a permanent change in shape.
• Temporary deformation is also called elastic deformation, while the permanent deformation is called plastic deformation.

Deforming Force Examples:

• A pencil which is broken by applying force.
• Mild steel rods
• Iron rods

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