NEET Physics Cheatsheet - Part 1

# Dimensional Formulas
Length [L], Mass [M], Time [T]

Force: [MLT⁻²]
Energy: [ML²T⁻²]
Power: [ML²T⁻³]
Pressure: [ML⁻¹T⁻²]
Momentum: [MLT⁻¹]

# Significant Figures Rules
- All non-zero digits are significant
- Zeros between non-zeros are significant
- Trailing zeros after decimal are significant
- Leading zeros are NOT significant

# Error Analysis
Absolute Error = |Measured - True Value|
Relative Error = Absolute Error / True Value
Percentage Error = Relative Error × 100%

# Dimensional Analysis Applications
- Check correctness of equations
- Derive relationships between quantities
- Convert units between systems

# Equations of Motion (Uniform Acceleration)
v = u + at
s = ut + (1/2)at²
v² = u² + 2as
sₙₜₕ = u + a(n - 1/2)

Where:
u = initial velocity
v = final velocity
a = acceleration
t = time
s = displacement
sₙₜₕ = displacement in nth second

# Special Cases
Free Fall: a = g = 9.8 m/s² (downward)
Vertical Throw Up: a = -g

# Relative Motion
v_AB = v_A - v_B (velocity of A w.r.t B)

# Vector Addition
Triangle Law: R² = A² + B² + 2AB cos θ
Parallelogram Law: Same as above

If θ = 0°: R = A + B (parallel, same direction)
If θ = 180°: R = |A - B| (antiparallel)
If θ = 90°: R = √(A² + B²) (perpendicular)

# Vector Resolution
Aₓ = A cos θ (x-component)
Aᵧ = A sin θ (y-component)

# Dot Product (Scalar Product)
A · B = AB cos θ = AₓBₓ + AᵧBᵧ + A_zB_z
Work = F · s

# Cross Product (Vector Product)
|A × B| = AB sin θ
Direction: Right-hand rule
Torque = r × F

# Initial Velocity Components
uₓ = u cos θ (horizontal)
uᵧ = u sin θ (vertical)

# Time of Flight
T = (2u sin θ)/g

# Maximum Height
H = (u² sin² θ)/(2g)

# Range (Horizontal Distance)
R = (u² sin 2θ)/g
Maximum Range: θ = 45°, R_max = u²/g

# Velocity at any time t
vₓ = u cos θ (constant)
vᵧ = u sin θ - gt

# Position at time t
x = (u cos θ)t
y = (u sin θ)t - (1/2)gt²

# Equation of Trajectory
y = x tan θ - (gx²)/(2u² cos² θ)

# Newton's Second Law
F = ma (when mass is constant)
F = dp/dt (general form)
Impulse: J = FΔt = Δp = m(v - u)

# Momentum
p = mv
Conservation: p_initial = p_final (isolated system)

# Applications
Free Body Diagram: Essential for problem solving
Connected Bodies: Same acceleration
Pulley Systems: Tension analysis

# Equilibrium Conditions
ΣF = 0 (no linear acceleration)
Στ = 0 (no rotational acceleration)

# Common Forces
Weight: W = mg
Normal Force: N (perpendicular to surface)
Tension: T (in strings/ropes)
Friction: f (opposes motion)

# Friction Formulas
Static: f_s ≤ μ_s N (maximum = μ_s N)
Kinetic: f_k = μ_k N
Rolling: f_r = μ_r N (μ_r << μ_k)

Where:
μ = coefficient of friction
N = Normal force

# Laws of Friction
1. Independent of contact area
2. Proportional to normal force
3. Direction: Opposite to motion/tendency

# Angle of Friction
tan λ = μ
λ = angle of repose (for incline)

# Motion on Incline
Parallel component: mg sin θ
Perpendicular: mg cos θ
Motion starts when: tan θ > μ_s

# Work Formulas
Constant Force: W = Fs cos θ
Variable Force: W = ∫F·ds
By gravity: W = mgh (downward)
By spring: W = (1/2)kx²

# Kinetic Energy
KE = (1/2)mv²
Work-Energy Theorem: W_net = ΔKE

# Potential Energy
Gravitational: PE = mgh
Elastic (spring): PE = (1/2)kx²

# Conservation of Mechanical Energy
KE + PE = constant (conservative forces)
E_initial = E_final

# Power
P = W/t (average)
P = F·v = Fv cos θ (instantaneous)

# Conservative vs Non-conservative
Conservative: Path independent (gravity, spring)
Non-conservative: Path dependent (friction)

# Conservation Laws
Momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (always)
KE: (1/2)m₁u₁² + (1/2)m₂u₂² = (1/2)m₁v₁² + (1/2)m₂v₂² (elastic only)

# Elastic Collision (1D)
v₁ = [(m₁-m₂)u₁ + 2m₂u₂]/(m₁+m₂)
v₂ = [(m₂-m₁)u₂ + 2m₁u₁]/(m₁+m₂)

Special Cases:
- Equal masses: velocities exchange
- m₁ >> m₂, m₂ at rest: v₂ ≈ 2u₁
- m₂ >> m₁, m₂ at rest: v₁ ≈ -u₁ (bounces back)

# Coefficient of Restitution
e = (v₂ - v₁)/(u₁ - u₂)
e = 1: Elastic
e = 0: Perfectly inelastic
0 < e < 1: Inelastic

# Perfectly Inelastic
v_common = (m₁u₁ + m₂u₂)/(m₁ + m₂)
KE lost = (1/2)[m₁m₂/(m₁+m₂)](u₁-u₂)²

# Rotational Kinematics (analog to linear)
ω = ω₀ + αt
θ = ω₀t + (1/2)αt²
ω² = ω₀² + 2αθ

# Torque and Angular Momentum
τ = Iα (Newton's 2nd law analog)
L = Iω
τ = dL/dt

# Rotational KE
KE_rot = (1/2)Iω²

# Total KE (rolling)
KE_total = (1/2)mv² + (1/2)Iω²
For rolling: v = rω

# Radius of Gyration
k = √(I/M)
I = Mk²

# Conservation of Angular Momentum
L_initial = L_final
I₁ω₁ = I₂ω₂

Applications:
- Ice skater spinning (I↓ → ω↑)
- Diver tucking (I↓ → ω↑)
- Earth-Moon system

# Equilibrium Conditions
Translational: ΣF_x = 0, ΣF_y = 0
Rotational: Στ = 0 (about any point)

# Rolling Motion
Pure rolling: v_cm = rω
KE = (1/2)mv_cm² + (1/2)Iω²
   = (1/2)mv_cm²[1 + k²/r²]

# Linear vs Rotational Comparison
Linear          Rotational
s ←→ θ
v ←→ ω
a ←→ α
m ←→ I
F ←→ τ
p ←→ L

# Newton's Law of Gravitation
F = G(m₁m₂)/r²
G = 6.67 × 10⁻¹¹ Nm²/kg²

# Acceleration due to Gravity
g = GM/R² (at surface)
Standard: g = 9.8 m/s²

# Variation with Height
g_h = g(1 - 2h/R) [h << R]
g_h = g[R²/(R+h)²] [general]

# Variation with Depth
g_d = g(1 - d/R)
At center (d = R): g = 0

# Important Relations
Weight: W = mg
g ∝ 1/R² (at surface)
g ∝ M (mass of planet)

# Kepler's Third Law
T² ∝ r³
T²/r³ = 4π²/(GM) = constant

# Satellite Orbital Velocity
v_o = √(GM/r) = √(gR²/r)
At surface: v_o = √(gR) ≈ 7.9 km/s

# Time Period
T = 2πr/v_o = 2π√(r³/GM)
At surface: T = 2π√(R/g) ≈ 85 min

# Energy of Satellite
KE = GMm/(2r)
PE = -GMm/r
Total E = -GMm/(2r) (negative = bound)

# Geostationary Satellite
T = 24 hours
Height ≈ 36,000 km above surface
Always above same point on equator

# Relation: v_e = √2 × v_o

# Hooke's Law
Stress ∝ Strain
Stress = E × Strain

# Young's Modulus (Y)
Y = (F/A)/(ΔL/L) = Longitudinal Stress/Strain
Unit: N/m² or Pa

# Bulk Modulus (K)
K = -ΔP/(ΔV/V) = Volume Stress/Strain
Compressibility: 1/K

# Modulus of Rigidity (η)
η = (F/A)/(θ) = Shear Stress/Shear Strain

# Poisson's Ratio
σ = -Lateral Strain/Longitudinal Strain
Range: 0 to 0.5

# Elastic Potential Energy
U = (1/2) × Stress × Strain × Volume
U = (1/2)(F²L)/(AY)

# Important Points
- Higher modulus → more rigid
- Stress-strain curve: elastic → yield → plastic

# Fluid Pressure
P = P₀ + ρgh
Where:
P₀ = atmospheric pressure
ρ = density
g = acceleration due to gravity
h = depth

# Pascal's Law Applications
- Hydraulic lift: F₁/A₁ = F₂/A₂
- Hydraulic brakes

# Buoyancy (Archimedes)
F_b = ρ_fluid × V_displaced × g
F_b = Weight of displaced fluid

# Floating Condition
Weight of body = Buoyant force
ρ_body × V_body × g = ρ_fluid × V_immersed × g

# Apparent Weight
W_apparent = W_actual - F_b

# Relative Density
RD = ρ_substance/ρ_water
For floating: ρ_body/ρ_fluid = V_immersed/V_total

# Viscous Force (Stokes' Law)
F = 6πηrv
Where:
η = coefficient of viscosity
r = radius of sphere
v = velocity

# Terminal Velocity
v_t = (2r²g(ρ - σ))/(9η)
ρ = density of sphere
σ = density of fluid

# Equation of Continuity
A₁v₁ = A₂v₂ (incompressible fluid)
Mass flow rate = ρAv = constant

# Bernoulli's Equation
P + (1/2)ρv² + ρgh = constant
Pressure + KE/volume + PE/volume = const

# Applications of Bernoulli
- Venturi meter: flow measurement
- Airplane wing lift
- Atomizer/sprayer
- Magnus effect

# Energy Interpretation
Pressure energy + Kinetic energy + Potential energy = constant

# Surface Tension
γ = F/L = Energy/Area
Unit: N/m or J/m²

# Excess Pressure
Spherical drop: ΔP = 2γ/r
Soap bubble: ΔP = 4γ/r (two surfaces)
Cylindrical: ΔP = γ/r

# Angle of Contact (θ)
- Water-glass: θ < 90° (concave meniscus)
- Mercury-glass: θ > 90° (convex meniscus)

# Capillary Rise/Fall
h = (2γ cos θ)/(rρg)
Rise (θ < 90°): h is positive
Fall (θ > 90°): h is negative

# Work Done in Blowing Bubble
W = 8πr²γ (soap bubble, two surfaces)
W = 4πr²γ (drop, one surface)

# Factors Affecting γ
- Decreases with temperature
- Impurities can increase/decrease
- Detergents decrease γ

# First Law of Thermodynamics
ΔQ = ΔU + ΔW
ΔQ = heat supplied to system
ΔU = change in internal energy
ΔW = work done by system

# Work Done by Gas
W = ∫PdV = PΔV (isobaric)

# Thermodynamic Processes
1. Isothermal (ΔT = 0)
   - ΔU = 0 (for ideal gas)
   - ΔQ = ΔW = nRT ln(V₂/V₁)
   - PV = constant

2. Adiabatic (ΔQ = 0)
   - ΔU = -ΔW
   - PVᵞ = constant
   - TVᵞ⁻¹ = constant
   - ΔW = (P₁V₁ - P₂V₂)/(γ-1)

3. Isochoric (ΔV = 0)
   - ΔW = 0
   - ΔQ = ΔU = nCᵥΔT

4. Isobaric (ΔP = 0)
   - ΔQ = nCₚΔT
   - ΔW = PΔV = nRΔT

# Specific Heat Capacities
Cₚ - Cᵥ = R
γ = Cₚ/Cᵥ

# Ideal Gas Equation
PV = nRT = NkT
Where:
R = 8.314 J/(mol·K) (universal gas constant)
k = 1.38 × 10⁻²³ J/K (Boltzmann constant)
k = R/N_A

# Kinetic Interpretation
P = (1/3)ρv²_rms = (1/3)nm/V × v²_rms
Average KE per molecule = (3/2)kT

# Molecular Speeds
RMS speed: v_rms = √(3RT/M) = √(3kT/m)
Average speed: v_avg = √(8RT/πM)
Most probable: v_mp = √(2RT/M)

Relation: v_rms > v_avg > v_mp
√3 : √(8/π) : √2

# Internal Energy
U = (f/2)nRT
f = degrees of freedom

# Degrees of Freedom
Monoatomic: f = 3 (translational)
Diatomic: f = 5 (3 trans + 2 rot)
Polyatomic: f = 6 (3 trans + 3 rot)

# Coulomb's Law
F = k(q₁q₂)/r² = (1/4πε₀)(q₁q₂)/r²
k = 9 × 10⁹ Nm²/C²
ε₀ = 8.85 × 10⁻¹² C²/Nm² (permittivity)

# Superposition Principle
F_net = F₁ + F₂ + F₃ + ... (vector sum)

# Electric Field
E = F/q₀ (force per unit positive charge)
E = kQ/r² (point charge)
Direction: Away from +ve, towards -ve

# Electric Field Lines
- Start from +ve, end at -ve
- Never cross each other
- Density ∝ field strength
- Perpendicular to surface

# Electric Dipole
p = q × 2a (dipole moment)
Direction: -ve to +ve charge

# Field due to Dipole
Axial: E = 2kp/r³
Equatorial: E = kp/r³
Torque: τ = p × E = pE sin θ