Physics Basics – Study Notes
(Advanced, MIT‑level treatment from first principles – suitable for an in‑depth review for the Chandigarh Police Sub‑Inspector exam)
Audience – Candidates who already know the high‑school syllabus but want a rigorous, conceptual foundation that will help you solve exam questions with confidence.
Goal – Provide a concise yet complete framework, with derivations, edge‑case discussions, and a challenging practice problem that ties several topics together.
1. Introduction
Physics is the science of matter, energy, and the interactions that govern them. In this guide we start from the first principles (axioms or laws) and use them to derive the familiar equations. We will
- State the postulates or axioms (e.g., Newton's laws, Maxwell’s equations).
- Derive the conservation laws (energy, momentum, charge).
- Discuss edge cases (non‑inertial frames, relativistic corrections, quantum effects).
- Provide a challenging problem that tests integration of several concepts.
Tip: When you see an exam question, first identify the conservation law or principle that governs it. Then apply the appropriate equations.
2. Classical Mechanics
| Label | Content |
|---|---|
| Postulate 1 | In an inertial frame, the motion of a particle is governed by Newton’s laws. |
| Postulate 2 | For a system of particles, the net external force equals the time derivative of the total momentum. |
| Postulate 3 | In the absence of external forces, the center of mass of an isolated system remains at rest or moves with constant velocity. |
2.1 Kinematics (1‑D)
- Definitions
[ v(t) = \frac{dx}{dt}, \qquad a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2} ] - Uniform acceleration
[ v = v_0 + at, \qquad x = x_0 + v_0t + \frac{1}{2}at^2 ] - General case
[ v^2 = v_0^2 + 2a(x - x_0) ]
2.2 Dynamics
- Newton’s Second Law (vector form)
[ \mathbf{F} = m\mathbf{a} ] - Forces
- Gravitational: (\mathbf{F}_g = m\mathbf{g})
- Normal & friction: (\mathbf{F}_N, \mathbf{F}_f)
- Spring (Hooke’s law): (\mathbf{F}_s = -k\mathbf{x})
- Drag (Stokes & quadratic): (\mathbf{F}_d = -b\mathbf{v}) or (-c v^2 \hat{\mathbf{v}})
2.3 Conservation Laws
| Law | Statement |
|---|---|
| Linear Momentum | (\frac{d\mathbf{p}}{dt} = \mathbf{F}_{\text{ext}}) |
| Angular Momentum | (\frac{d\mathbf{L}}{dt} = \boldsymbol{\tau}_{\text{ext}}) |
| Energy | (\frac{d}{dt}\left(\frac{1}{2}mv^2 + U\right) = \mathbf{F}_{\text{non‑conservative}}\cdot\mathbf{v}) |
2.4 Rotational Dynamics
- Moment of Inertia
[ I = \sum m_i r_i^2 ] - Rotational kinetic energy
[ K_{\text{rot}} = \frac{1}{2}I\omega^2 ] - Torque
[ \boldsymbol{\tau} = \mathbf{r}\times\mathbf{F} ]
2.5 Lagrangian Mechanics
- Lagrangian
[ L = T - V ] - Euler–Lagrange Equation
[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i}=0 ] where (q_i) are generalized coordinates. - Edge Case – Non‑holonomic constraints: Lagrange multipliers are needed; standard Euler–Lagrange fails.
Why it matters: Lagrangian formulation is the bridge to quantum mechanics (path integrals)