Physics Basics – MIT‑Level Study Notes
(Prepared for the Puducherry TET – General Awareness & Current Affairs)
These notes start from first principles and progressively build a rigorous mathematical framework, suitable for a university‑level understanding while remaining focused on concepts that appear in the TET syllabus.
Where appropriate, we point out subtle edge cases and give a challenging problem at the end.
1. Introduction
| Topic | Why It Matters for TET | Typical Question Types |
|---|---|---|
| Classical mechanics | Foundations of motion, kinematics, dynamics | Multiple‑choice, short‑answer, derivation |
| Thermodynamics | Heat, temperature, energy transfer | Conceptual, calculation |
| Electromagnetism | Electricity & magnetism, Maxwell’s equations | Vector calculus, circuit analysis |
| Waves & optics | Light, sound, interference | Wave equations, resonance |
| Relativity & Quantum | Modern physics, current research | Conceptual explanations, derivations |
2. Classical Mechanics (First Principles)
2.1 Newton’s Laws
- Inertia – An object remains at rest or in uniform motion unless acted upon by a net external force.
- Dynamics –
[ \mathbf{F}=m,\mathbf{a} ] - Action‑Reaction –
[ \mathbf{F}{12}=-\mathbf{F}{21} ]
Edge case – Zero mass: (m=0) gives (\mathbf{F}=0) regardless of acceleration; physically impossible for a massive particle.
2.2 Lagrangian & Hamiltonian Formulations
-
Lagrangian (L=T-U)
Euler–Lagrange equation:[ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} -\frac{\partial L}{\partial q_i}=0 ]
-
Hamiltonian (H=\sum_i p_i\dot{q}_i - L)
Generates equations of motion via Poisson brackets.
Edge case – Constrained systems: For a bead on a wire, coordinates reduce; Lagrange multipliers enforce constraints.
2.3 Conservation Laws
- Linear momentum (\mathbf{p}=m\mathbf{v}) conserved in isolated systems.
- Angular momentum (\mathbf{L}=\mathbf{r}\times\mathbf{p}); torque (\boldsymbol{\tau}=\frac{d\mathbf{L}}{dt}).
- Energy (E=T+U) conserved if potential is time‑independent.
3. Work, Energy, and Power
| Concept | Formula | Physical Insight |
|---|---|---|
| Work | (W=\int \mathbf{F}\cdot d\mathbf{r}) | Path independence for conservative forces |
| Kinetic Energy | (T=\tfrac12 m v^2) | Scalar invariant of motion |
| Potential Energy | (U=-\int \mathbf{F}\cdot d\mathbf{r}) | Defined up to a constant |
| Power | (P=\frac{dW}{dt}=\mathbf{F}\cdot\mathbf{v}) | Energy transfer rate |
Edge case – Impulse–momentum theorem: For impulsive forces (F(t)), ( \Delta \mathbf{p} = \int F(t),dt ).
4. Thermodynamics
4.1 Laws
| Law | Statement | Mathematical Form |
|---|---|---|
| 0 | Temperature equalizes | (T_1=T_2) at equilibrium |
| 1 | Energy conservation | ( \Delta U = Q - W ) |
| 2 | Entropy never decreases | ( \Delta S \ge 0) |
| 3 | Absolute zero unattainable | (T\rightarrow 0) impossible |
4.2 Entropy & Statistical Mechanics
- Boltzmann's formula
[ S = k_B \ln \Omega ] where (\Omega) is the number of microstates. - Canonical distribution
[ P_i = \frac{e^{-\beta E_i}}{Z},\quad Z=\sum_i e^{-\beta E_i} ]
Edge case – Negative temperatures: In systems with bounded spectra, population inversion leads to (\beta<0), implying higher energy states are more populated.
5. Electromagnetism
5.1 Coulomb’s Law
[ \mathbf{F} = k_e \frac{