Core Concepts
Build from basics to advanced. Every concept explained with physical intuition, not just formulas. Derivations included where exam-relevant.
What is Work?
In Physics, work is done when a force causes displacement in its own direction. The common English meaning of "effort" is irrelevant here. This distinction alone separates toppers from average students.
Work is a scalar quantity — it has no direction. But its sign (positive or negative) carries critical physical meaning.
W = F⃗ · d⃗ is a dot product. This is why it's scalar. JEE Advanced sometimes asks you to prove W is invariant under coordinate rotation — directly using the dot product definition.
Sign Convention — Critical for Exams
θ = 0°–90°: Force has component in direction of displacement.
Example: Pushing a box forward.
θ = 90°–180°: Force opposes motion.
Example: Friction, air resistance.
θ = 90°: Force ⊥ displacement.
Example: Normal force, centripetal force.
d = 0: No work regardless of force.
Example: Holding a heavy book stationary.
Work by Variable Force
When force is not constant, you cannot directly use W = Fd·cosθ. You must integrate.
If you see an F-x graph in JEE, your job is to find the AREA under the curve between two points. Positive area = positive work. Triangles, rectangles, trapezoids — be fast with geometry.
Work by Spring Force
Spring force F = −kx (Hooke's Law). The negative sign means it opposes extension/compression.
Work done by the spring and work done against the spring are negatives of each other. This distinction causes 30% of mistakes in this topic. Always specify the agent doing the work.
Area under F-x graph = Work done by spring
In NEET, work problems are mostly direct formula. In JEE Main, you'll face F-x graphs and spring combinations. In JEE Advanced, work is integrated with energy methods — never solve independently.
Kinetic Energy
Kinetic energy is the energy possessed by an object by virtue of its motion. The formula appears simple, but its implications run deep.
The formula KE = p²/(2m) is tested more in JEE than KE = ½mv². When two objects have the same momentum, the lighter one has MORE kinetic energy. When they have the same KE, the heavier one has MORE momentum.
Derivation of KE Formula
This derivation is asked in CBSE boards (3 marks). Know it step by step.
- 1
Start with Newton's 2nd law: F = ma
- 2
Work done by F over displacement s:
W = F·s = ma·s - 3
Use kinematics: v² = u² + 2as → as = (v² − u²)/2
W = m·(v² − u²)/2 = ½mv² − ½mu² - 4
For object starting from rest (u = 0):
W = ½mv² = KE gained
Key Relations — Very Important
KE = p²/2m → p = √(2mKE)
If KE is same: p ∝ √m → heavier body has larger momentum.
NEET asks: "A truck and a car have same KE. Which has more momentum?" Answer: Truck (heavier).
KE = p²/2m
If p is same: KE ∝ 1/m → lighter body has larger KE.
JEE asks: Two bodies with same momentum. Lighter one has more KE. In explosion problems, fragments often have equal momentum but different KE.
KE ∝ v² → If v increases by x%, KE increases by approximately 2x% (for small x).
Exact: New KE/Old KE = (1 + x/100)²
If v doubles: KE becomes 4 times. If v halves: KE becomes ¼. Memorize this instantly.
KE = ½mv² uses speed, not velocity. KE is always positive. If you get a negative KE in calculation, you made a sign error somewhere.
Relativistic KE (JEE Advanced Context)
Work-Energy Theorem
Net work done by ALL forces on an object equals the change in its kinetic energy.
The Work-Energy Theorem bypasses Newton's 2nd Law. When forces are variable, or when you don't need acceleration explicitly — this is your tool. JEE Advanced regularly tests situations where WET is the only elegant approach.
When to Apply WET
- ✅ Variable force problems (no constant acceleration)
- ✅ Finding speed at a point given forces
- ✅ Finding work done when motion is known
- ✅ Checking if motion is possible (KE < 0 means impossible)
Solved Example — WET Application
A block of mass 2 kg starts from rest on a rough surface (μ = 0.2). A horizontal force F = 10 N is applied for 5 m. Find final velocity. (g = 10 m/s²)
- 1
Work by applied force:
W_F = 10 × 5 = 50 J - 2
Work by friction (opposes motion):
W_f = −μmg × d = −0.2 × 2 × 10 × 5 = −20 J - 3
W_net = 50 + (−20) = 30 J
- 4
Apply WET: W_net = ΔKE = ½mv² − 0
30 = ½ × 2 × v² → v = √30 ≈ 5.47 m/s
Shortcut: Never compute acceleration when speed is asked at end of displacement. WET is faster.
Potential Energy
Potential energy is energy stored in a system due to position or configuration, not motion. It only exists for conservative forces.
Potential energy is a system property, not an object property. When you say "gravitational PE of a ball", you really mean the PE of the ball-Earth system. JEE Advanced tests this distinction.
Gravitational PE
Students always set reference at ground. This is fine. But in some problems, the reference is set at the top or midpoint to simplify calculation. The physics doesn't change — only numbers do.
Spring PE (Elastic PE)
Relation Between PE and Conservative Force
If U(x) = 3x² − 6x + 5, then F = −dU/dx = −(6x − 6) = 6 − 6x. At x = 1, F = 0 → equilibrium. Check d²U/dx² at equilibrium to determine stability.
PE Curve Analysis (JEE Advanced)
- 🟢 Minimum of U → Stable Equilibrium (F = 0, restoring)
- 🔴 Maximum of U → Unstable Equilibrium (F = 0, not restoring)
- 🔵 Where U = E_total: KE = 0, turning points
Power
Power is the rate of doing work. Same work can be done at different rates — power tells you how fast.
P = Fv is the most tested power formula in JEE Main. A vehicle moving at constant velocity on a level road: P = f × v where f is friction force. Engine force = friction at constant speed.
Efficiency
η < 100% always in real systems. If a pump has 80% efficiency and input is 100 W, output is 80 W. Never use total input power as output power.
Units & Dimensional Analysis
SI Unit: Joule (J) = kg·m²·s⁻²
Dimensional formula: [ML²T⁻²]
Other units: eV (electron-volt) = 1.6 × 10⁻¹⁹ J, erg = 10⁻⁷ J, kWh = 3.6 × 10⁶ J
SI Unit: Watt (W) = J/s = kg·m²·s⁻³
Dimensional formula: [ML²T⁻³]
Other units: HP (horsepower) = 746 W ≈ 750 W
1 HP = 746 W is tested in NEET. Use 750 W for quick approximations in JEE.
Power Problem — Frequently Tested
A car of mass 1000 kg moves at constant 20 m/s. Coefficient of friction = 0.1. Find power of engine. (g = 10 m/s²)
At constant velocity: F_engine = f_friction = μmg = 0.1 × 1000 × 10 = 1000 N
Conservative vs Non-Conservative Forces
This is the conceptual backbone of energy methods. Get this wrong, and energy conservation will fail you in every problem.
- • Gravity
- • Spring force
- • Electrostatic force
- • Magnetic force on charge
- • Friction
- • Air resistance
- • Normal force (moving surface)
- • Tension (in some cases)
The defining test: If an object travels a closed loop and the work done by a force is zero, that force is conservative. For gravity: go up 10m, come back → net W = 0. For friction: go up 10m, come back → W is negative (energy lost). Therefore friction is non-conservative.
Modified Energy Conservation
When friction is present, energy is not conserved — but total energy including thermal energy IS. For exams:
For 80% of JEE problems: KE_initial + PE_initial − Energy_lost_to_friction = KE_final + PE_final. Write this down and fill in the blanks. It works for inclines, rough surfaces, springs, and pulley systems.
Normal Force: Zero Work Agent
Normal force is always perpendicular to surface and hence to displacement. Therefore W_N = 0 always. This is why on any frictionless surface, only gravity and spring do work.
Centripetal force also does zero work because it is always perpendicular to velocity (which equals displacement direction).