IMAT Physics Master Handbook & Compendium
The ultimate, exhaustive study guide containing full theoretical principles, detailed formula derivations, step-by-step worked numerical examples, IMAT exam shortcuts, and common trap warnings.
🧭 Quick Navigator - 8 Core Syllabus Modules
1. Physical Quantities, Units & Vector Operations
1.1 Fundamental vs Derived Physical Quantities
Physics is an empirical science based on precise measurements. Physical quantities are divided into two fundamental classes: Base (Fundamental) Quantities which are defined by arbitrary operational standards, and Derived Quantities which are expressed algebraically in terms of base quantities.
- 7 Base SI Units: Meter (m) for length, Kilogram (kg) for mass, Second (s) for time, Ampere (A) for electric current, Kelvin (K) for thermodynamic temperature, Mole (mol) for amount of substance, and Candela (cd) for luminous intensity.
- Dimensional Analysis Technique: The dimensions of any physical quantity can be written as . Every physically valid equation must satisfy the Principle of Dimensional Homogeneity (both sides of an equality must have identical dimensions).
| Comprehensive SI Base and Derived Units Table | ||
|---|---|---|
| Quantity | SI Unit Name (Symbol) | Expression in SI Base Units |
| Basic SI Units | ||
| Length | meter (m) | |
| Mass | kilogram (kg) | |
| Time | second (s) | |
| Electric Current | ampere (A) | |
| Thermodynamic Temperature | kelvin (K) | |
| Amount of Substance | mole (mol) | |
| Luminous Intensity | candela (cd) | |
| Common Derived SI Units | ||
| Velocity / Speed | meter per second (m/s) | |
| Acceleration | meter per second squared (m/s²) | |
| Force | newton (N) | |
| Pressure / Stress | pascal (Pa) | |
| Energy / Work / Heat | joule (J) | |
| Power / Radiant Flux | watt (W) | |
| Electric Charge | coulomb (C) | |
| Electric Potential / Voltage | volt (V) | |
| Electrical Resistance | ohm (Ω) | |
| Capacitance | farad (F) | |
| Magnetic Flux Density | tesla (T) | |
| Frequency | hertz (Hz) | |
1.2 Vector Algebra & Component Resolution
A Scalar is specified entirely by a real number magnitude and unit (e.g., Mass, Distance, Speed, Work, Electric Potential). A Vector requires both magnitude AND spatial direction (e.g., Displacement, Velocity, Acceleration, Force, Momentum, Electric Field).
2D Vector Decomposition & Reconstitution
📌 Concept & Application: Any 2D vector A can be broken into independent perpendicular components along the x and y axes. Angle θ is measured counterclockwise from the positive x-axis.
Dot Product (Scalar Product) vs Cross Product (Vector Product)
📌 Concept & Application: Dot product yields a scalar scalar (max when vectors are parallel θ=0°). Cross product yields a vector perpendicular to both (max when vectors are orthogonal θ=90°).
📊 Diagram Visual: Head-to-Tail vector addition method.
📊 Diagram Visual: Parallelogram law of vector addition.
A crate is acted upon by two forces simultaneously: Force due East and Force due North. Find the magnitude and direction angle of the net resultant force.
1. Set up perpendicular components: , .
2. Calculate resultant magnitude using the Pythagorean theorem:
3. Calculate angle θ North of East:
Final Answer: Net Force is directed at North of East.
2. Kinematics (Describing Motion)
2.1 Uniform Rectilinear Motion & SUVAT Derivations
Kinematics analyzes the motion of points and bodies without considering the forces causing the motion. Position , velocity , and acceleration are linked via calculus and graphs.
- Displacement-Time Graph (s-t): Slope (gradient) = Velocity.
- Velocity-Time Graph (v-t): Slope = Acceleration. Area under curve = Displacement ().
- Acceleration-Time Graph (a-t): Area under curve = Change in Velocity ().
Full Derivation of the 4 SUVAT Equations
For constant acceleration :
- Equation 1 Derivation: By definition of average acceleration, .
- Equation 2 Derivation: For linear velocity change, average velocity is , so displacement is .
- Equation 3 Derivation: Substitute Eq 1 into Eq 2: .
- Equation 4 Derivation: Solve Eq 1 for time and substitute into Eq 2: .
2.2 Free Fall & 2D Projectile Motion
Free fall is vertical motion influenced solely by gravity (neglecting air resistance), where .
2D Projectile Motion Decomposition
📌 Concept & Application: Horizontal velocity remains constant (a_x = 0). Vertical motion undergoes free fall acceleration (a_y = -g).
Key Trajectory Formulas (Flight Time, Max Height, Range)
📌 Concept & Application: Max range R occurs at θ = 45°. Complementary launch angles (e.g., 30° and 60°) yield identical horizontal ranges.
A car travelling at applies brakes, causing a uniform deceleration of . Calculate the braking distance required for the car to come to a complete stop.
1. Convert speed to SI units: .
2. Identify known variables: Initial velocity , final velocity , acceleration .
3. Use the timeless SUVAT equation :
Final Answer: The car requires to come to a full stop.
2.3 Uniform Circular Motion & Simple Harmonic Motion (SHM)
In Uniform Circular Motion (UCM), an object travels in a circular path of radius at constant linear speed . Because velocity direction changes continuously, there is a centripetal acceleration pointing toward the center.
Uniform Circular Motion Relations
📌 Concept & Application: Angular velocity ω (rad/s), period T (seconds per revolution), frequency f (revolutions per second in Hz).
📊 Diagram Visual: Centripetal acceleration vector directed toward center, perpendicular to tangential velocity.
Simple Harmonic Motion (SHM)
SHM is periodic motion driven by a restoring force proportional to displacement: .
SHM Displacement, Velocity, Acceleration & Periods
📌 Concept & Application: Maximum speed v_max = Aω occurs at equilibrium (x=0). Maximum acceleration a_max = Aω² occurs at extreme amplitudes (x=±A).
3. Dynamics, Work, Energy & Momentum
3.1 Newton's Laws of Motion & Friction
- Newton's 1st Law (Law of Inertia): An object remains in its state of rest or uniform rectilinear motion unless acted upon by a non-zero net external force ().
- Newton's 2nd Law (Fundamental Law of Dynamics): The acceleration of a body is directly proportional to the net force applied and inversely proportional to its mass: .
- Newton's 3rd Law (Action-Reaction): When body A exerts a force on body B, body B simultaneously exerts an equal and opposite force on body A (). Action and reaction forces act on DIFFERENT bodies!
📊 Diagram Visual: Free-body diagram showing Normal force balancing Gravitational Weight on a flat surface.
Friction Forces (Static vs Kinetic)
📌 Concept & Application: Static friction f_s balances applied pushing force up to a maximum limit μ_s N. Once sliding begins, kinetic friction f_k remains constant.
- Parallel component pulling down slope: .
- Perpendicular component pressing against slope: .
- Critical angle where block just starts sliding: .
3.2 Moment of Force (Torque) & Mechanical Equilibrium
A rigid body is in complete mechanical equilibrium when both translational and rotational accelerations are zero.
Conditions for Mechanical Equilibrium & Torque Formula
📌 Concept & Application: Torque τ is the turning effect of a force. It equals force magnitude F times perpendicular distance from pivot (moment arm r sin θ).
📊 Diagram Visual: Torque calculation around a pivot axis with lever arm r.
3.3 Work, Energy, Power & Impulse-Momentum
Work Definition & Work-Kinetic Energy Theorem
📌 Concept & Application: Work is a scalar (Joule). Work is positive if θ < 90°, negative if θ > 90° (e.g. friction), and zero if force is perpendicular (θ = 90°).
Potential Energies & Conservation of Mechanical Energy
📌 Concept & Application: If only conservative forces (gravity, springs) do work, total mechanical energy E = K + U is conserved.
Mechanical Power & Efficiency
📌 Concept & Application: Power is the rate of doing work (1 Watt = 1 J/s). Electric or mechanical power can also be written as Force × Velocity.
Linear Momentum, Impulse & Collisions
📌 Concept & Application: Total momentum is conserved in all isolated collisions (∑p_initial = ∑p_final). Elastic collisions conserve kinetic energy (e=1). Completely inelastic collisions stick together (e=0).
Cart A of mass moving right at collides with stationary Cart B of mass . If they couple together upon collision, calculate their combined final velocity.
1. Use Conservation of Linear Momentum ():
2. Calculate Kinetic Energy lost during collision:
Final Answer: Combined velocity is to the right ( lost as heat/sound).
4. Fluid Mechanics
4.1 Hydrostatics: Pressure, Stevin's Law & Pascal's Principle
A fluid (liquid or gas) is a substance that deforms continuously under applied shear stress. Density is mass per unit volume: (kg/m³). Pressure is perpendicular force per unit area: (Pascal Pa = N/m²).
Stevin's Law (Hydrostatic Law)
📌 Concept & Application: Hydrostatic pressure at depth h in a liquid equals surface atmospheric pressure P_0 plus column pressure ρgh. Pressure depends only on depth h, NOT container shape!
Pascal's Principle (Hydraulics)
📌 Concept & Application: Pressure applied to an enclosed fluid is transmitted undiminished throughout. Mechanical advantage multiplies force by area ratio, while conserving total work done.
📊 Diagram Visual: Hydraulic system demonstrating Pascal's Principle.
4.2 Archimedes' Principle & Floating Conditions
Archimedes' Buoyancy Law
📌 Concept & Application: Any object completely or partially submerged experiences an upward buoyant force F_b equal to the weight of the fluid it displaces.
- Sinks (): Weight exceeds max buoyant force (). Apparent weight = .
- Neutral Buoyancy (): Remains suspended at any depth.
- Floats (): Submerged volume fraction equals density ratio:
An iceberg has a density of and floats in seawater of density . What percentage of the iceberg's total volume remains submerged underwater?
1. Use the floating equilibrium condition ():
2. Substitute values:
Final Answer: Approximately of the iceberg is submerged (only ~ is visible above water).
4.3 Fluid Dynamics: Continuity Equation & Bernoulli's Principle
An Ideal Fluid is incompressible (), non-viscous (zero internal friction), and undergoes steady, laminar flow.
Continuity Equation (Volume Flow Rate)
📌 Concept & Application: Mass flow rate is conserved. When pipe cross-section constricts (A↓), fluid speed must increase proportionally (v↑).
Bernoulli's Equation (Energy Conservation in Fluids)
📌 Concept & Application: Sum of static pressure P, dynamic pressure (0.5ρv²), and hydrostatic energy density (ρgh) is constant along any streamline. High velocity implies lower pressure!
Torricelli's Law (Efflux Speed from Tank)
📌 Concept & Application: The speed of liquid flowing out of an orifice at depth h below an open surface equals free-fall speed from height h.
5. Thermodynamics & Kinetic Theory of Gases
5.1 Temperature, Heat Capacity & Calorimetry
Temperature reflects the average translational kinetic energy of molecules. Absolute Zero is .
Sensible Heat & Latent Heat Transformations
📌 Concept & Application: Specific heat c (J/(kg·K)) governs temperature changes. Latent heat L (J/kg) governs phase transitions (fusion L_f or vaporization L_v) at CONSTANT temperature.
Calorimetry Thermal Equilibrium Principle
📌 Concept & Application: In an insulated calorimeter, total heat lost by hotter substances equals total heat gained by cooler substances until final equilibrium temperature T_f is reached.
A piece of copper () at is dropped into of water () at . Find the final equilibrium temperature .
1. Set up heat balance equation ():
2. Substitute values:
Final Answer: Final equilibrium temperature is .
5.2 Kinetic Theory & Ideal Gas Laws
An Ideal Gas consists of point-like particles undergoing elastic collisions with no intermolecular attractive forces.
Ideal Gas Law & Microscopic Kinetic Energy
📌 Concept & Application: Universal gas constant R = 8.314 J/(mol·K). Boltzmann constant k_B = R/N_A = 1.38 × 10⁻²³ J/K. Internal energy U of a monatomic ideal gas depends ONLY on temperature T!
| Summary of 4 Special Thermodynamic Gas Processes | |||
|---|---|---|---|
| Process | Constant Feature | Governing Gas Law | Work Done W |
| Isothermal | Temperature T = const (ΔU = 0) | (Boyle) | |
| Isobaric | Pressure P = const | (Charles) | |
| Isochoric | Volume V = const (W = 0) | (Gay-Lussac) | |
| Adiabatic | No Heat Exchange (Q = 0) | ||
5.3 1st & 2nd Laws of Thermodynamics & Heat Engines
First Law of Thermodynamics
📌 Concept & Application: Energy conservation: change in internal energy ΔU equals heat Q added to gas minus work W done BY gas on surroundings.
Thermal Engine Efficiency & Carnot Limit
📌 Concept & Application: Carnot efficiency is the theoretical upper limit for any heat engine operating between hot reservoir T_H and cold reservoir T_C (Temperatures MUST be in Kelvin!).
6. Waves & Optics
6.1 Wave Mechanics & Doppler Effect
Fundamental Wave Equation & Frequency
📌 Concept & Application: Wave speed v depends entirely on the medium properties (e.g., tension and density for strings).
General Doppler Effect Formula
📌 Concept & Application: Use top signs (+ in numerator, - in denominator) when observer and source approach each other (higher apparent pitch f').
An ambulance driving at emits a siren frequency of . If speed of sound in air is , calculate the frequency heard by a stationary pedestrian as the ambulance approaches.
1. Known values: Sound speed , Source speed , Observer speed , Emitted frequency .
2. Use approaching source formula ():
Final Answer: The pedestrian hears a higher frequency of .
6.2 Geometric Optics: Refraction, Lenses & Mirrors
📊 Diagram Visual: Snell's Law of Refraction across optical boundary.
Snell's Law of Refraction & Critical Angle
📌 Concept & Application: Light refracts toward normal when entering denser medium (n2 > n1). Total internal reflection occurs when incident angle exceeds critical angle θ_c.
Thin Lens & Spherical Mirror Equations
📌 Concept & Application: Sign convention: Convex lens has positive focal length (f>0). Real image has positive image distance (d_i>0). Negative m indicates inverted image.
7. Electricity & Electromagnetism
7.1 Electrostatics: Coulomb, Electric Fields & Capacitors
📊 Diagram Visual: Coulomb's Law electrostatic interaction between point charges.
Coulomb's Law & Electric Field Intensity
📌 Concept & Application: Electrostatic force is an inverse-square force. Electric field E represents vector force per unit positive test charge.
Electric Potential & Uniform Field Work
📌 Concept & Application: Electric potential V is scalar potential energy per unit charge. In uniform field between parallel plates, voltage difference is E · d.
Capacitance & Stored Electrostatic Energy
📌 Concept & Application: Capacitance increases with larger plate area A and smaller separation d. Dielectric material κ multiplies capacitance by factor κ.
7.2 Direct Current (DC) Circuits & Kirchhoff's Rules
Ohm's Law, Resistance & Joule Power
📌 Concept & Application: Resistivity ρ depends on material. Resistance R is proportional to wire length L and inversely proportional to area A. Power dissipates as heat.
📊 Diagram Visual: Series Circuit: Current I is identical through all components.
📊 Diagram Visual: Parallel Circuit: Voltage V is identical across all branches.
| Complete Circuit Combination Comparison Table | ||
|---|---|---|
| Parameter | Series Circuit | Parallel Circuit |
| Current (I) | ||
| Voltage (V) | ||
| Equivalent Resistance (R) | ||
| Equivalent Capacitance (C) | ||
Two resistors of and are connected in parallel across a battery. Calculate the equivalent resistance and total current drawn from the battery.
1. Use parallel resistance formula ():
2. Calculate total current using Ohm's Law ():
Final Answer: Equivalent resistance is and total current is .
7.3 Magnetism & Faraday's Law of Induction
Lorentz Magnetic Force & Motion Radius
📌 Concept & Application: Magnetic force is perpendicular to velocity. In a uniform field, a moving charge undergoes circular motion with radius r = mv/(qB).
Faraday's Law of Electromagnetic Induction & Lenz's Law
📌 Concept & Application: Changing magnetic flux through a loop induces an electromotive force (EMF). The negative sign (Lenz's Law) indicates induced current opposes the flux change.
8. Modern & Nuclear Physics
8.1 Quantum Physics & Photoelectric Effect
Photon Energy & Photoelectric Effect Equation
📌 Concept & Application: Planck constant h = 6.63 × 10⁻³⁴ J·s. Incident photon energy hf must exceed work function W_0 to eject electrons with max kinetic energy K_max.
De Broglie Matter Wavelength
📌 Concept & Application: Wave-particle duality: every particle with momentum p exhibits an associated matter wavelength λ.
8.2 Nuclear Physics & Half-Life Decay Law
Mass Defect & Nuclear Binding Energy
📌 Concept & Application: Mass defect Δm during nuclear fusion or fission releases nuclear binding energy ΔE.
Radioactive Decay Law & Half-Life Formula
📌 Concept & Application: Half-life T_1/2 is the time taken for half of the radioactive parent nuclei N_0 to decay.
A radioactive isotope has a half-life of . If initial mass is , calculate the remaining mass after .
1. Determine the number of half-lives elapsed ():
2. Calculate remaining mass using :
Final Answer: of the isotope remains after 24 hours.
9. Practice & Interactive Challenge Modules
Put your formula knowledge into practice with our interactive drills and challenge exams.