Physics Fundamentals

1. Units and Dimensions

Physics is a quantitative science. To measure physical quantities, we need units. A unit is a standard, arbitrarily chosen, and internationally accepted reference used for measuring a physical quantity. The system of units defines the base units and rules for deriving other units.

1.1 Systems of Units

Historically, several systems of units have been used. The most common ones are:

• CGS System: Centimetre, Gram, Second. This system is based on length in centimetres (cm), mass in grams (g), and time in seconds (s).
• FPS System: Foot, Pound, Second. This system uses length in feet (ft), mass in pounds (lb), and time in seconds (s). This system is primarily used in some English-speaking countries.
• MKS System: Metre, Kilogram, Second. This system uses length in metres (m), mass in kilograms (kg), and time in seconds (s).

1.2 SI System (International System of Units)

The MKS system was later expanded and modernized to become the International System of Units (SI), which is the most widely used system globally. The SI system has seven base units and two supplementary units.

Physical Quantity	SI Unit	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric Current	Ampere	A
Thermodynamic Temperature	Kelvin	K
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

The two supplementary units are:

• Plane Angle: Radian (rad)
• Solid Angle: Steradian (sr)

1.3 Dimensions

Dimensions are the powers to which the base units are raised to represent a derived unit. The fundamental dimensions are usually represented by symbols: [L] for length, [M] for mass, [T] for time, [A] for electric current, [Θ] for temperature, [N] for amount of substance, and [J] for luminous intensity.

For example, the dimensions of velocity (distance/time) are [L]/[T] = [L¹T^-1]. The dimensions of force (mass × acceleration = mass × distance/time²) are [M] × [L]/[T²] = [M¹L¹T^-2].

2. Motion

Motion is the change in position of an object with respect to its surroundings over time. Understanding motion is fundamental to classical mechanics.

2.1 Distance and Displacement

Distance is the total path length covered by an object. It is a scalar quantity (magnitude only).

Displacement is the shortest distance between the initial and final positions of an object, measured in a straight line. It is a vector quantity (magnitude and direction).

Example: If a person walks 5 meters east and then 5 meters west, the total distance covered is 10 meters. However, their displacement is 0 meters because they end up at their starting point.

2.2 Speed and Velocity

Speed is the rate at which an object covers distance. It is a scalar quantity.
Speed = Distance / Time

Velocity is the rate at which an object changes its displacement. It is a vector quantity.
Velocity = Displacement / Time

If the direction of motion is constant, speed and the magnitude of velocity are the same. However, if the direction changes, velocity changes even if the speed is constant.

2.3 Acceleration

Acceleration is the rate at which velocity changes. It is a vector quantity.
Acceleration (a) = Change in Velocity / Time taken
a = (v - u) / t where 'v' is the final velocity, 'u' is the initial velocity, and 't' is the time taken.

If velocity increases, acceleration is positive. If velocity decreases (deceleration or retardation), acceleration is negative. If velocity is constant, acceleration is zero.

2.4 Laws of Motion

Sir Isaac Newton formulated three fundamental laws of motion that describe the relationship between an object and the forces acting upon it.

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

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced external force.

Inertia is the property of an object to resist changes in its state of motion. Mass is a measure of inertia. The more massive an object, the greater its inertia.

Examples:

• When a bus suddenly starts, passengers tend to fall backward due to the inertia of rest.
• When a moving bus suddenly stops, passengers tend to fall forward due to the inertia of motion.
• Dust particles are removed from a carpet by beating it; the carpet moves, but the dust particles tend to remain at rest.

2.4.2 Newton's Second Law of Motion

The rate of change of momentum of an object is directly proportional to the applied unbalanced force and takes place in the direction in which the force acts.

Momentum (p) = mass (m) × velocity (v)

Mathematically, F ∝ Δp / Δt. If we consider a constant force, F = m × a.

This law establishes the relationship between force, mass, and acceleration. A larger force produces a larger acceleration for a given mass. A larger mass requires a larger force to produce the same acceleration.

Unit of Force: The SI unit of force is the Newton (N). 1 Newton is the force required to accelerate a mass of 1 kilogram by 1 meter per second squared (1 N = 1 kg⋅m/s²).

2.4.3 Newton's Third Law of Motion

For every action, there is an equal and opposite reaction.

This means that forces always occur in pairs. If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. These forces act on different objects.

Examples:

• When you push a wall, the wall pushes back on you with an equal force.
• A rocket expels gases downwards (action), and the gases push the rocket upwards (reaction).
• A boat moves forward because the oars push the water backward (action), and the water pushes the oars (and boat) forward (reaction).

2.5 Conservation of Momentum

If no external force acts on a system of two or more interacting bodies, the total momentum of the system remains conserved (constant).

Total initial momentum = Total final momentum

This principle is a direct consequence of Newton's Third Law. It is crucial in understanding collisions and explosions.

Example: In a collision between two billiard balls, the total momentum of the two balls just before the collision is equal to their total momentum just after the collision, assuming no friction or air resistance.

3. Work, Energy, and Power

These concepts are closely related and describe the capacity to do work and the rate at which work is done.

3.1 Work

In physics, work is done when a force causes a displacement. Work is a scalar quantity.
Work (W) = Force (F) × Displacement (d) × cos(θ) where θ is the angle between the force vector and the displacement vector.

If the force is in the same direction as the displacement (θ=0°, cos(0°)=1), then W = F × d. If the force is perpendicular to the displacement (θ=90°, cos(90°)=0), then W = 0. If the force is opposite to the displacement (θ=180°, cos(180°)=-1), then W = -F × d (negative work).

Unit of Work: The SI unit of work is the Joule (J). 1 Joule is the work done when a force of 1 Newton moves an object by 1 meter in its direction. (1 J = 1 N⋅m).

3.2 Energy

Energy is the capacity to do work. It exists in various forms, such as kinetic energy, potential energy, thermal energy, chemical energy, etc. Energy can be transformed from one form to another, but it cannot be created or destroyed (Law of Conservation of Energy).

3.2.1 Kinetic Energy (KE)

Kinetic energy is the energy possessed by an object due to its motion.
KE = 1/2 × mass (m) × velocity (v)²
KE = 1/2 mv²

The unit of energy is the Joule (J).

3.2.2 Potential Energy (PE)

Potential energy is the energy stored in an object due to its position or configuration.

Gravitational Potential Energy: Energy stored due to an object's height above a reference point.
PE = mass (m) × acceleration due to gravity (g) × height (h)
PE = mgh

The unit of potential energy is also the Joule (J).

3.3 Power

Power is the rate at which work is done or energy is transferred. It is a scalar quantity.
Power (P) = Work (W) / Time (t)
or P = Energy transferred / Time

Unit of Power: The SI unit of power is the Watt (W). 1 Watt is equal to 1 Joule per second (1 W = 1 J/s).

Another common unit of power is Horsepower (hp). 1 hp ≈ 746 W.

4. Heat and Temperature

While often used interchangeably in everyday language, heat and temperature are distinct physical concepts.

4.1 Temperature

Temperature is a measure of the degree of hotness or coldness of a substance. It is related to the average kinetic energy of the molecules within a substance.

Scales of Temperature:

• Celsius (°C): Water freezes at 0°C and boils at 100°C (at standard atmospheric pressure).
• Fahrenheit (°F): Water freezes at 32°F and boils at 212°F.
• Kelvin (K): The SI unit. Absolute zero (the theoretical lowest possible temperature) is 0 K. Water freezes at 273.15 K and boils at 373.15 K.

Conversion Formulas:

• °C to °F: °F = (°C × 9/5) + 32
• °F to °C: °C = (°F - 32) × 5/9
• °C to K: K = °C + 273.15
• K to °C: °C = K - 273.15

4.2 Heat

Heat is a form of energy that is transferred between systems or objects due to a temperature difference. Heat flows spontaneously from a hotter body to a colder body.

Units of Heat: The SI unit of heat is the Joule (J). Other common units include calorie (cal) and kilocalorie (kcal).
1 calorie (cal) is the amount of heat required to raise the temperature of 1 gram of water by 1°C.
1 kilocalorie (kcal) = 1000 calories.
1 calorie ≈ 4.184 Joules.

4.3 Heat Transfer

Heat can be transferred through three main mechanisms:

• Conduction: Transfer of heat through direct contact, primarily in solids. Energy is transferred from molecule to molecule without actual movement of the molecules themselves. Metals are good conductors of heat.
• Convection: Transfer of heat through the movement of fluids (liquids or gases). Warmer, less dense fluid rises, and cooler, denser fluid sinks, creating convection currents. This is how water heats up in a pot or air circulates in a room.
• Radiation: Transfer of heat through electromagnetic waves. This method does not require a medium and can occur through a vacuum. The Sun's heat reaches Earth via radiation. All objects emit thermal radiation.

5. Light

Light is a form of electromagnetic radiation that makes things visible. It exhibits wave-particle duality.

5.1 Properties of Light

• Rectilinear Propagation: Light travels in straight lines in a uniform medium. This explains phenomena like shadows.
• Reflection: The bouncing back of light rays when they strike a surface.
• Refraction: The bending of light rays as they pass from one medium to another (e.g., from air to water).
• Diffraction: The spreading of light waves as they pass through narrow openings or around obstacles.
• Interference: The superposition of two or more waves, resulting in a new wave pattern.
• Polarization: The phenomenon where light waves are restricted to vibrate in a particular plane.

5.2 Reflection of Light

When light strikes a surface, some of it bounces back.

• Laws of Reflection:
  1. The angle of incidence is equal to the angle of reflection (∠i = ∠r).
  2. The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
• Types of Reflection:
  • Regular Reflection: Occurs on smooth surfaces (like mirrors), where parallel incident rays are reflected as parallel rays.
  • Irregular Reflection (or Diffuse Reflection): Occurs on rough surfaces, where parallel incident rays are reflected in many different directions. This is why we can see most objects around us.
• Mirrors:
  • Plane Mirror: Forms a virtual, erect, and laterally inverted image of the same size as the object.
  • Spherical Mirrors: These are parts of a sphere.
    • Concave Mirror: A mirror that curves inward. It converges parallel rays of light to a focal point. Used in headlights, torches, and shaving mirrors (to get a magnified image).
    • Convex Mirror: A mirror that curves outward. It diverges parallel rays of light. Forms virtual, erect, and diminished images. Used as rear-view mirrors in vehicles because they provide a wider field of view.

5.3 Refraction of Light

When light passes from one medium to another, its speed changes, causing it to bend.

• Laws of Refraction (Snell's Law):
  1. The incident ray, the refracted ray, and the normal to the surface at the point of incidence all lie in the same plane.
  2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media and a given wavelength of light. This constant is called the refractive index (n) of the second medium with respect to the first.
     n = sin(i) / sin(r)
     Also, n = Speed of light in vacuum (c) / Speed of light in the medium (v)
• Refractive Index (n): A measure of how much light bends when entering a medium. A higher refractive index means light bends more. The refractive index of vacuum is 1. For air, it's approximately 1.0003. For water, it's about 1.33. For glass, it's about 1.5.
• Lenses: Optical devices that refract light.
  • Converging Lens (Convex Lens): Thicker in the middle, thinner at the edges. Converges parallel rays of light to a focal point. Used in magnifying glasses, cameras, and the human eye.
  • Diverging Lens (Concave Lens): Thinner in the middle, thicker at the edges. Diverges parallel rays of light. Used in spectacles to correct myopia (nearsightedness).

5.4 Dispersion of Light

When white light passes through a prism, it splits into its constituent colors (VIBGYOR - Violet, Indigo, Blue, Green, Yellow, Orange, Red). This phenomenon is called dispersion. It occurs because the refractive index of the prism material is slightly different for each color (wavelength) of light. Violet light is deviated the most, and red light is deviated the least.

6. Sound

Sound is a form of energy that travels as waves, typically longitudinal waves, through a medium (solid, liquid, or gas). It is produced by vibrations.

6.1 Properties of Sound Waves

• Frequency: The number of oscillations or cycles per second. Measured in Hertz (Hz). Frequency determines the pitch of the sound. Higher frequency means higher pitch.
• Amplitude: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. Amplitude determines the loudness or intensity of the sound. Larger amplitude means louder sound.
• Wavelength (λ): The distance between two successive crests or troughs of a wave.
• Speed of Sound: The distance sound travels per unit time. Speed of sound depends on the medium and temperature. It is generally faster in solids than in liquids, and faster in liquids than in gases. In air at 20°C, the speed of sound is approximately 343 meters per second.

The relationship between speed (v), frequency (f), and wavelength (λ) is:
v = f × λ

6.2 Range of Hearing

The human ear can typically detect sound frequencies ranging from about 20 Hz to 20,000 Hz (20 kHz).

• Infrasound: Sound waves with frequencies below the human audible range (below 20 Hz).
• Audible Sound: Sound waves with frequencies within the human audible range (20 Hz to 20 kHz).
• Ultrasound: Sound waves with frequencies above the human audible range (above 20 kHz).

Ultrasound has many applications, including medical imaging (sonography), sonar, and non-destructive testing.

6.3 Echo

An echo is the reflection of sound waves off a surface. For an echo to be heard distinctly, the time interval between the original sound and the reflected sound must be at least 0.1 seconds. This requires the reflecting surface to be a sufficient distance away.

Minimum distance for a distinct echo = (Speed of sound × 0.1 seconds) / 2.
Example: If the speed of sound is 340 m/s, the minimum distance for a distinct echo is (340 m/s × 0.1 s) / 2 = 17 meters.

7. Electricity and Magnetism

This section covers the fundamental concepts of electric charges, currents, and their relationship with magnetic fields.

7.1 Electric Charge

Electric charge is a fundamental property of matter. There are two types of electric charges: positive and negative. Like charges repel each other, and unlike charges attract each other.

The SI unit of electric charge is the Coulomb (C).

The smallest unit of free charge is the charge of an electron, which is approximately -1.602 × 10^-19 C. Protons have a positive charge of the same magnitude.

7.2 Electric Current

Electric current is the rate of flow of electric charge through a conductor.
Current (I) = Charge (Q) / Time (t)
I = Q / t

The SI unit of electric current is the Ampere (A). 1 Ampere = 1 Coulomb per second (1 A = 1 C/s).

7.3 Voltage (Electric Potential Difference)

Voltage is the electric potential energy per unit charge. It is the driving force that causes electric charges to flow.
Voltage (V) = Work done (W) / Charge (Q)
V = W / Q

The SI unit of voltage is the Volt (V). 1 Volt = 1 Joule per Coulomb (1 V = 1 J/C).

7.4 Resistance

Resistance is the opposition to the flow of electric current in a conductor.

Ohm's Law: States that the current flowing through a conductor is directly proportional to the voltage across its ends and inversely proportional to its resistance, provided the temperature remains constant.
V = I × R where V is voltage, I is current, and R is resistance.

The SI unit of resistance is the Ohm (Ω). 1 Ohm = 1 Volt per Ampere (1 Ω = 1 V/A).

Factors affecting resistance:

• Length (L): Resistance is directly proportional to the length of the conductor (R ∝ L).
• Area of Cross-section (A): Resistance is inversely proportional to the area of cross-section (R ∝ 1/A).
• Nature of Material: Different materials have different resistivities (ρ).
• Temperature: Resistance of most conductors increases with temperature.

Formula: R = ρ (L/A)

7.5 Electric Power

Electric power is the rate at which electrical energy is consumed or dissipated.
Power (P) = Voltage (V) × Current (I)
P = VI Using Ohm's law, we can also write:
P = I²R
P = V²/R

The SI unit of electric power is the Watt (W).

7.6 Magnetism

Magnetism is a force of attraction or repulsion between magnetic poles. Every magnet has two poles: North (N) and South (S). Like poles repel, and unlike poles attract.

Magnetic Field: The region around a magnet where its magnetic effect can be detected. Magnetic field lines represent the direction and strength of the magnetic field. They emerge from the North pole and enter the South pole, forming closed loops.

Electromagnetism: The phenomenon where electric currents create magnetic fields.

• Oersted's Experiment: Showed that an electric current flowing through a wire creates a magnetic field around it.
• Electromagnet: A temporary magnet made by passing an electric current through a coil of wire wrapped around an iron core. The strength of an electromagnet can be increased by increasing the current, the number of turns in the coil, or by using a soft iron core.
• Electric Motor: A device that converts electrical energy into mechanical energy, based on the principle that a current-carrying conductor placed in a magnetic field experiences a force.
• Electric Generator: A device that converts mechanical energy into electrical energy, based on the principle of electromagnetic induction (Faraday's Law).
• Electromagnetic Induction: The production of an electromotive force (voltage) across an electrical conductor in a changing magnetic field.

8. Modern Physics (Brief Introduction)

While classical physics explains most macroscopic phenomena, modern physics deals with the behavior of matter and energy at the atomic and subatomic levels.

8.1 Atomic Structure

Atoms consist of a nucleus (containing protons and neutrons) surrounded by electrons orbiting the nucleus.

• Protons: Positively charged particles.
• Neutrons: Neutral particles.
• Electrons: Negatively charged particles.

The number of protons in the nucleus defines the atomic number (Z) and determines the element. The sum of protons and neutrons is the mass number (A).

8.2 Radioactivity

The spontaneous emission of radiation (alpha particles, beta particles, or gamma rays) from the nucleus of an unstable atom.

8.3 Nuclear Energy

Energy released from the nucleus of an atom, typically through nuclear fission (splitting of a heavy nucleus) or nuclear fusion (combining of light nuclei). This is the basis for nuclear power plants and nuclear weapons.

8.4 Relativity (Einstein's Theory)

Special relativity deals with the relationship between space and time, particularly at speeds close to the speed of light. A key outcome is E=mc², showing the equivalence of mass and energy. General relativity describes gravity as a curvature of spacetime.

8.5 Quantum Mechanics

Deals with the behavior of matter and energy at the atomic and subatomic levels, where energy exists in discrete packets called quanta. Phenomena like wave-particle duality and quantum entanglement are key concepts.