📚 Physics Basics – Study Notes for PCS‑J
(General Awareness & Current Affairs – Physics Section)
Audience – Candidates preparing for PCS‑J (and other competitive exams such as IIT‑JEE).
Objective – Build a solid, exam‑ready foundation: key concepts, derivations, important formulae, tricky points, and a challenging practice problem.
1️⃣ Introduction
- Physics is the science of matter, energy, space, and time.
- PCS‑J tests conceptual understanding, mathematical reasoning, and application of physics principles to real‑world problems.
- Focus on rigorous derivations and common pitfalls that often trip candidates.
2️⃣ Kinematics (Motion in 1‑D & 2‑D)
| Concept | Key Equations | Derivation / Insight |
|---|---|---|
| Displacement | (x = x_0 + vt + \frac12 a t^2) | From (v = u + at) integrate (v = dx/dt). |
| Average velocity | (\bar{v} = \frac{\Delta x}{\Delta t}) | Straight‑line average over finite interval. |
| Average acceleration | (\bar{a} = \frac{\Delta v}{\Delta t}) | Constant acceleration assumption. |
| Spherical polar coordinates | (r, \theta, \phi) | Useful for orbital mechanics. |
| Projectile motion | (x = v_0 \cos\theta,t) (y = v_0 \sin\theta,t - \frac12 g t^2) |
Decompose velocity; vertical motion under constant (g). |
| Maximum height | (h_{\max} = \frac{v_0^2 \sin^2\theta}{2g}) | Set vertical velocity to zero. |
| Range | (R = \frac{v_0^2 \sin 2\theta}{g}) | Derived by solving for time of flight. |
| Time of flight | (t_f = \frac{2 v_0 \sin\theta}{g}) | From vertical displacement zero again. |
Tricky Points
- Sign conventions: In vertical motion, take upward as positive, (g = +9.81,\text{m/s}^2) downwards (so (a_y = -g)).
- Units: Always keep SI units, especially when combining vectors.
- Range maximization: (\theta = 45^\circ) yields maximum range only for horizontal launch from ground level; for non‑horizontal launch, adjust accordingly.
3️⃣ Newton’s Laws & Dynamics
| Law | Equation | Practical Implication |
|---|---|---|
| First Law | ( \sum \vec{F} = 0 \Rightarrow \vec{a}=0) | Inertia; motion persists unless acted upon. |
| Second Law | ( \vec{F} = m\vec{a}) | Force = mass × acceleration; basis for all dynamics. |
| Third Law | (\vec{F}{12} = -\vec{F}{21}) | Action–reaction pairs; conservation. |
Derivation of (F = ma)
Start from definition of momentum: (\vec{p} = m\vec{v}).
By Newton’s 2nd law (definition of force): (\vec{F} = \frac{d\vec{p}}{dt}).
If (m) is constant: (\vec{F} = m \frac{d\vec{v}}{dt} = m\vec{a}).
Common Pitfalls
- Treating weight as a force of gravity rather than applied force.
- Ignoring normal force in inclined‑plane problems.
- Misapplying friction: static vs kinetic, coefficient of kinetic friction (\mu_k) vs static (\mu_s).
4️⃣ Work, Energy & Power
| Quantity | Symbol | Formula | Units |
|---|---|---|---|
| Work | (W) | (W = \vec{F}\cdot\vec{d}) | J |
| Kinetic Energy | (K) | (K = \frac12 m v^2) | J |
| Potential Energy (grav.) | (U_g) | (U_g = mgh) | J |
| Power | (P) | (P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}) | W |
Work-Energy Theorem
(W_{\text{net}} = \Delta K).
Derivation: (W = \int \vec{F}\cdot d\vec{r} = m\int \vec{a}\cdot d\vec{v} = \frac12 m(v_f^2 - v_i^2)).
Tricky Points
- Non‑conservative forces (e.g., friction) do not conserve mechanical energy; account for work done separately.
- Potential energy is relative. Choose reference point appropriately.
- Power can be negative if force opposes motion (e.g., braking).
5️⃣ Momentum & Collisions
| Concept | Formula | Notes |
|---|---|---|
| Linear Momentum | (\vec{p} = m\vec{v}) | Conserved if net external force zero. |
| Impulse | (\vec{J} = \int \vec{F},dt = \Delta \vec{p}) | High force |