1. Mechanics

Mechanics is the branch of physics dealing with the motion of bodies under the action of forces, and with forces themselves. It is the oldest and most examined branch of physics in competitive examinations because its laws describe phenomena visible in daily life — falling objects, moving vehicles, floating ships, and flowing liquids.

1.1 Newton's Laws of Motion

Sir Isaac Newton published his three laws of motion in his landmark work Philosophiae Naturalis Principia Mathematica (commonly called the Principia) in 1687. These three laws form the foundation of classical (Newtonian) mechanics and describe the relationship between a body and the forces acting upon it, and its motion in response to those forces.

1.1.1 Newton's First Law of Motion (Law of Inertia)

Scientist: Sir Isaac Newton (building on earlier ideas of Galileo Galilei)

Year / Era: 1687 (Principia)

Statement: A body continues to remain in its state of rest or of uniform motion in a straight line unless it is acted upon by an external unbalanced force.

Explanation: This law introduces the concept of inertia — the natural tendency of an object to resist a change in its state of motion. Inertia is directly proportional to the mass of the body; heavier objects have greater inertia and are harder to start moving or to stop. The first law essentially defines what a force is: it is that which changes, or tends to change, a body's state of rest or uniform motion.

Formula: No single formula; conceptually expressed as: if net external force F = 0, then acceleration a = 0 (velocity remains constant).

Application/Example: When a moving bus stops suddenly, passengers standing inside are jerked forward because their bodies, in motion, tend to continue moving forward due to inertia. Similarly, a coin placed on a card balanced on a glass falls into the glass when the card is flicked away quickly, because the coin's inertia keeps it in place momentarily.

1.1.2 Newton's Second Law of Motion

Scientist: Sir Isaac Newton

Year / Era: 1687 (Principia)

Statement: The rate of change of momentum of a body is directly proportional to the applied external force and takes place in the direction in which the force acts.

Explanation: This law quantifies force. Momentum (p) is the product of mass and velocity (p = mv). When a force acts on a body, it changes the body's momentum. For a body of constant mass, this reduces to the familiar relation that force equals mass times acceleration. The second law allows us to calculate the exact acceleration produced by a given force on a given mass.

Formula: F = dp/dt = m·a (for constant mass), where F is force in newtons, m is mass in kilograms, and a is acceleration in metres per second squared.

Application/Example: A cricket ball hit harder (greater force) accelerates faster and travels farther. Rocket engines are designed using this law: a large mass of exhaust gas is expelled at high acceleration to generate the thrust force needed to lift the rocket.

1.1.3 Newton's Third Law of Motion

Scientist: Sir Isaac Newton

Year / Era: 1687 (Principia)

Statement: For every action, there is an equal and opposite reaction, and these two forces act on two different bodies.

Explanation: Whenever one body exerts a force (the 'action') on a second body, the second body simultaneously exerts a force of equal magnitude but opposite direction (the 'reaction') on the first body. These action-reaction pairs never cancel each other because they act on different objects, not on the same object.

Formula: F(A on B) = −F(B on A)

Application/Example: When a swimmer pushes water backward with their hands and legs, the water pushes the swimmer forward with equal force, propelling them ahead. Similarly, a rocket expels burning gases downward (action), and the gases push the rocket upward (reaction), enabling it to lift off.

1.2 Newton's Law of Universal Gravitation

1.2 Law of Universal Gravitation

Scientist: Sir Isaac Newton

Year / Era: 1687 (Principia); inspired, according to popular legend, by observing a falling apple around 1665–66

Statement: Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

Explanation: This single law unified terrestrial and celestial mechanics — it showed that the same force pulling an apple to the ground also keeps the Moon in orbit around the Earth and the planets in orbit around the Sun. The constant of proportionality, G (the universal gravitational constant), was later measured experimentally by Henry Cavendish in 1798.

Formula: F = G·(m1·m2)/r², where G = 6.674 × 10⁻¹¹ N·m²/kg².

Application/Example: This law explains the orbits of planets and satellites, the phenomenon of ocean tides caused by the Moon's and Sun's gravitational pull, and why objects have weight. It is fundamental to space science, enabling calculation of satellite orbits and interplanetary trajectories used by ISRO and other space agencies.

1.3 Kepler's Laws of Planetary Motion

Johannes Kepler, a German astronomer and mathematician, formulated three laws describing the motion of planets around the Sun, based on the precise observational data collected by the Danish astronomer Tycho Brahe. Kepler's laws, published between 1609 and 1619, preceded Newton's law of gravitation and were later shown by Newton to be a direct consequence of it.

1.3.1 Kepler's First Law (Law of Orbits)

Scientist: Johannes Kepler

Year / Era: 1609

Statement: Every planet moves around the Sun in an elliptical orbit, with the Sun located at one of the two foci of the ellipse.

Explanation: Before Kepler, orbits were widely believed to be perfect circles. Kepler showed, using Tycho Brahe's precise data on Mars, that planetary orbits are actually ellipses, with the Sun not at the centre but at one focus. This meant a planet's distance from the Sun varies throughout its orbit.

Application/Example: This law explains why Earth is closer to the Sun in early January (perihelion) and farther in early July (aphelion), and is fundamental to plotting satellite and spacecraft trajectories.

1.3.2 Kepler's Second Law (Law of Areas)

Scientist: Johannes Kepler

Year / Era: 1609

Statement: The line joining a planet to the Sun sweeps out equal areas in equal intervals of time.

Explanation: This means a planet moves faster when it is closer to the Sun (perihelion) and slower when farther away (aphelion), because the area swept must remain constant per unit time even as the radius vector's length changes. This law is essentially a statement of the conservation of angular momentum.

Application/Example: It explains the varying orbital speed of comets, which move extremely fast near the Sun and very slowly at the outer reaches of their elongated orbits.

1.3.3 Kepler's Third Law (Law of Periods)

Scientist: Johannes Kepler

Year / Era: 1619

Statement: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Explanation: This law provides a precise mathematical relationship between a planet's distance from the Sun and the time it takes to complete one orbit, allowing comparison across all planets in the solar system.

Formula: T² ∝ a³, or T²/a³ = constant for all planets orbiting the Sun.

Application/Example: Used to calculate the orbital period of artificial satellites and to determine distances of exoplanets from their host stars based on observed orbital periods.

1.4 Hooke's Law

1.4 Hooke's Law of Elasticity

Scientist: Robert Hooke

Year / Era: 1660 (published 1678)

Statement: Within the elastic limit, the extension (or compression) produced in a spring or elastic material is directly proportional to the force (load) applied to it.

Explanation: When a material is stretched or compressed, it deforms; if the deforming force is removed and the material returns to its original shape, it is said to behave elastically. Hooke's law holds only up to a certain point called the elastic limit; beyond this, the material undergoes permanent deformation and no longer obeys the linear relationship.

Formula: F = k·x, where k is the spring (force) constant and x is the extension or compression.

Application/Example: This law underlies the working of spring balances (used to measure weight/force), mechanical weighing scales, the suspension systems of vehicles, and the calibration of measuring instruments in engineering.

1.5 Law of Conservation of Momentum

1.5 Law of Conservation of Linear Momentum

Scientist: Foundational work attributed to Sir Isaac Newton (a direct consequence of his laws of motion); the principle was refined through the work of scientists including René Descartes and Christiaan Huygens

Year / Era: 17th century

Statement: In the absence of an external force, the total momentum of a system of interacting bodies remains constant (conserved).

Explanation: Momentum, the product of mass and velocity, is transferred between bodies during collisions or interactions, but the total momentum of an isolated system before and after the interaction remains the same. This follows directly from Newton's third law: internal action-reaction force pairs between bodies of a system are equal and opposite, so they cancel out when summed over the whole system.

Formula: m1u1 + m2u2 = m1v1 + m2v2 (for a two-body collision, where u = initial velocity, v = final velocity).

Application/Example: This principle explains the recoil of a gun when fired (the backward momentum of the gun balances the forward momentum of the bullet), and is used to analyse collisions in vehicle crash-safety testing and in rocket propulsion.

1.6 Law of Conservation of Energy

1.6 Law of Conservation of Energy

Scientist: Julius Robert von Mayer, James Prescott Joule, and Hermann von Helmholtz are jointly credited with establishing this law in its modern form

Year / Era: Mid-19th century (1840s–1850s)

Statement: Energy can neither be created nor destroyed; it can only be transformed from one form to another, and the total energy of an isolated system remains constant.

Explanation: This is one of the most fundamental principles in all of physics. Energy may change form — mechanical energy can become heat, heat can become electrical energy, chemical energy can become kinetic energy — but the total quantity of energy in a closed system never changes. James Prescott Joule's careful experiments quantified the mechanical equivalent of heat, providing crucial experimental support for this law.

Formula: Total energy (initial) = Total energy (final); e.g., for a freely falling body, Potential Energy lost = Kinetic Energy gained.

Application/Example: This principle underlies the working of hydroelectric power plants (potential energy of stored water converts to kinetic energy, then to electrical energy), and explains energy transformations in engines, where chemical energy of fuel converts to heat and then mechanical work.

1.7 Archimedes' Principle

1.7 Archimedes' Principle

Scientist: Archimedes of Syracuse

Year / Era: c. 250 BCE

Statement: When a body is partially or wholly immersed in a fluid, it experiences an upward buoyant force (upthrust) equal to the weight of the fluid displaced by the body.

Explanation: According to popular legend, Archimedes discovered this principle while stepping into a bath and noticing the water level rise, then ran through the streets of Syracuse shouting 'Eureka!' ('I have found it!'). The principle explains why objects feel lighter in water and why some objects float while others sink: an object floats if the weight of fluid it displaces equals its own weight before it is fully submerged.

Formula: Upthrust (buoyant force) = weight of fluid displaced = ρ(fluid)·V(displaced)·g.

Application/Example: This principle is the basis for the design of ships and submarines (which displace enough water to support their weight), the working of the hydrometer (used to measure the density of liquids like milk or battery acid), and the lactometer used to test milk purity.

1.8 Pascal's Law

1.8 Pascal's Law

Scientist: Blaise Pascal

Year / Era: 1653

Statement: Pressure applied at any point on a confined, incompressible fluid is transmitted equally and undiminished in all directions throughout the fluid.

Explanation: Because liquids are (nearly) incompressible, any pressure exerted on an enclosed liquid is transmitted uniformly to every part of the liquid and to the walls of the containing vessel. This allows a small force applied over a small area to be converted into a much larger force over a larger area, giving a mechanical advantage.

Formula: P = F/A (pressure is transmitted equally, so F1/A1 = F2/A2 for connected pistons).

Application/Example: Pascal's law is the working principle behind hydraulic lifts (used in car service stations), hydraulic brakes in automobiles, and hydraulic jacks, all of which use a small applied force on a small piston to generate a much larger force on a larger piston.

1.9 Bernoulli's Principle

1.9 Bernoulli's Principle

Scientist: Daniel Bernoulli

Year / Era: 1738 (published in Hydrodynamica)

Statement: For a fluid flowing steadily and without friction, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline; consequently, as the speed of a fluid increases, its pressure decreases.

Explanation: Bernoulli's principle is a statement of the conservation of energy applied to flowing fluids. Where a fluid speeds up (for example, as it passes through a constriction), its pressure must drop to keep total energy constant, and vice versa. This inverse relationship between fluid speed and pressure has wide-ranging applications in aerodynamics and fluid engineering.

Formula: P + ½ρv² + ρgh = constant, where P is pressure, ρ is fluid density, v is fluid velocity, and h is height.

Application/Example: This principle explains how an aircraft wing generates lift (air moves faster over the curved upper surface, lowering pressure there compared to the lower surface, creating a net upward force), the working of a carburettor, and why a spray/perfume atomiser works by forcing air rapidly over a tube dipped in liquid.

1.10 Stokes' Law

1.10 Stokes' Law

Scientist: Sir George Gabriel Stokes

Year / Era: 1851

Statement: The viscous drag force experienced by a small spherical body moving through a viscous fluid is directly proportional to the radius of the sphere, the coefficient of viscosity of the fluid, and the velocity of the body.

Explanation: As a small sphere falls through a viscous fluid, it experiences an opposing viscous drag force that increases with its velocity. Eventually, this drag force (plus buoyant force) balances the force of gravity on the sphere, and the sphere falls at a constant maximum speed known as its 'terminal velocity.'

Formula: F = 6πηrv, where η is the coefficient of viscosity, r is the radius of the sphere, and v is its velocity.

Application/Example: Stokes' law explains why raindrops and parachutists reach a terminal (constant) velocity while falling through air, and is used in viscometers that measure the viscosity of liquids by timing the fall of a small ball through the liquid.