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⚛️ Physics
🧪 Chemistry
📐 Mathematics
🔢 Constants
1. Mechanics - Kinematics
Equations of Motion (Uniform Acceleration)
v = u + at
s = ut + ½at²
v² = u² + 2as
u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement
SI Units: m/s, m/s², s, m
Average Velocity & Acceleration
v_avg = (u + v) / 2
a_avg = (v - u) / t
For uniformly accelerated motion
Projectile Motion
Time of Flight: T = 2u sinθ / g
Maximum Height: H = u² sin²θ / 2g
Range: R = u² sin2θ / g
u = initial velocity, θ = angle of projection, g = 9.8 m/s²
2. Laws of Motion
Newton's Second Law
F = ma
F = dp/dt (General form)
F = force, m = mass, a = acceleration, p = momentum
SI Unit: Newton (N) = kg⋅m/s²
Friction
f = μN
f_max = μ_s N (Static)
f_k = μ_k N (Kinetic)
μ = coefficient of friction, N = normal force
Momentum & Impulse
p = mv
Impulse: J = Δp = FΔt
Conservation: p₁ + p₂ = constant (isolated system)
3. Work, Energy & Power
Work
W = F⋅s = Fs cosθ
θ = angle between force and displacement
SI Unit: Joule (J) = N⋅m
Kinetic Energy
KE = ½mv²
KE = p²/2m
Work-Energy Theorem: W = ΔKE
Potential Energy
PE = mgh (Gravitational)
PE = ½kx² (Spring)
h = height, k = spring constant, x = displacement
Power
P = W/t = F⋅v
P = dW/dt
SI Unit: Watt (W) = J/s
4. Rotational Motion
Angular Kinematics
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
ω = angular velocity, α = angular acceleration, θ = angular displacement
Torque & Moment of Inertia
τ = Iα = r × F
I = Σmr² (Discrete)
I = ∫r²dm (Continuous)
τ = torque, I = moment of inertia, r = perpendicular distance
Angular Momentum
L = Iω = r × p
τ = dL/dt
Conservation: L = constant (no external torque)
Rotational Kinetic Energy
KE_rot = ½Iω²
KE_total = ½mv² + ½Iω² (Rolling)
5. Gravitation
Universal Law of Gravitation
F = GMm/r²
G = 6.67 × 10⁻¹¹ N⋅m²/kg²
Acceleration due to Gravity
g = GM/R²
g' = g(1 - 2h/R) (at height h)
g' = g(1 - d/R) (at depth d)
Orbital Velocity & Escape Velocity
v_orbital = √(GM/r)
v_escape = √(2GM/R) = √(2gR)
For Earth: v_escape ≈ 11.2 km/s
Kepler's Laws
T² ∝ r³
T² = (4π²/GM)r³
T = time period, r = orbital radius
6. Electrostatics
Coulomb's Law
F = kq₁q₂/r²
k = 1/(4πε₀) = 9 × 10⁹ N⋅m²/C²
ε₀ = 8.85 × 10⁻¹² C²/N⋅m² (permittivity of free space)
Electric Field
E = F/q = kQ/r²
E = -dV/dr (relation with potential)
SI Unit: N/C or V/m
Electric Potential
V = kQ/r
V = W/q
ΔV = -∫E⋅dr
SI Unit: Volt (V) = J/C
Capacitance
C = Q/V
C = ε₀A/d (Parallel plate)
Energy: U = ½CV² = ½Q²/C
SI Unit: Farad (F) = C/V
7. Current Electricity
Ohm's Law
V = IR
R = ρL/A
ρ = resistivity, L = length, A = cross-sectional area
Power
P = VI = I²R = V²/R
Energy: E = Pt = VIt
SI Unit: Watt (W)
Series & Parallel Resistances
Series: R_eq = R₁ + R₂ + R₃ + ...
Parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ...
Kirchhoff's Laws
KCL: ΣI_in = ΣI_out
KVL: ΣV = 0 (closed loop)
8. Thermodynamics
First Law of Thermodynamics
ΔU = Q - W
dU = dQ - dW
ΔU = change in internal energy, Q = heat added, W = work done by system
Ideal Gas Law
PV = nRT
PV = NkT
R = 8.314 J/(mol⋅K), k = 1.38 × 10⁻²³ J/K
Work Done
W = ∫PdV
W = nRT ln(V₂/V₁) (Isothermal)
W = P(V₂ - V₁) (Isobaric)
Heat Capacity
C_v = (f/2)R (Molar heat capacity at constant volume)
C_p = C_v + R
γ = C_p/C_v
f = degrees of freedom (3 for monoatomic, 5 for diatomic)
Carnot Efficiency
η = 1 - T₂/T₁ = W/Q₁
T₁ = source temperature, T₂ = sink temperature (in Kelvin)
1. Atomic Structure
Bohr's Model
E_n = -13.6Z²/n² eV
r_n = 0.529n²/Z Å
Z = atomic number, n = principal quantum number
Rydberg Formula
1/λ = R(1/n₁² - 1/n₂²)
R = 1.097 × 10⁷ m⁻¹ (Rydberg constant)
de Broglie Wavelength
λ = h/mv = h/p
h = 6.626 × 10⁻³⁴ J⋅s (Planck's constant)
Heisenberg Uncertainty Principle
Δx⋅Δp ≥ h/4π
Cannot simultaneously determine position and momentum with absolute precision
2. Chemical Thermodynamics
Enthalpy
ΔH = ΔU + ΔnRT
ΔH = H_products - H_reactants
Δn = change in number of moles of gas
Gibbs Free Energy
ΔG = ΔH - TΔS
ΔG° = -RT ln K
ΔG = ΔG° + RT ln Q
ΔG < 0: spontaneous, ΔG = 0: equilibrium, ΔG > 0: non-spontaneous
Entropy
ΔS = Q_rev/T
ΔS_universe = ΔS_system + ΔS_surroundings ≥ 0
3. Chemical Equilibrium
Equilibrium Constant
K_c = [C]^c[D]^d / [A]^a[B]^b
K_p = K_c(RT)^Δn
For aA + bB ⇌ cC + dD
pH & pOH
pH = -log[H⁺]
pOH = -log[OH⁻]
pH + pOH = 14 (at 25°C)
K_w = [H⁺][OH⁻] = 10⁻¹⁴
Henderson-Hasselbalch Equation
pH = pK_a + log([A⁻]/[HA])
For buffer solutions
4. Electrochemistry
Nernst Equation
E_cell = E°_cell - (RT/nF) ln Q
E_cell = E°_cell - (0.0591/n) log Q (at 25°C)
F = 96500 C/mol (Faraday constant)
Cell Potential
E°_cell = E°_cathode - E°_anode
ΔG° = -nFE°_cell
Faraday's Laws of Electrolysis
m = (M × i × t) / (n × F)
Q = i × t
m = mass deposited, M = molar mass, i = current, t = time, n = electrons transferred
5. Chemical Kinetics
Rate Law
Rate = k[A]^m[B]^n
k = rate constant, m,n = order of reaction
Integrated Rate Laws
Zero Order: [A] = [A]₀ - kt
First Order: ln[A] = ln[A]₀ - kt
Second Order: 1/[A] = 1/[A]₀ + kt
Half-Life
Zero Order: t₁/₂ = [A]₀/2k
First Order: t₁/₂ = 0.693/k
Second Order: t₁/₂ = 1/(k[A]₀)
Arrhenius Equation
k = Ae^(-E_a/RT)
ln(k₂/k₁) = (E_a/R)(1/T₁ - 1/T₂)
E_a = activation energy, A = frequency factor
1. Algebra
Quadratic Equation
x = [-b ± √(b² - 4ac)] / 2a
Discriminant: D = b² - 4ac
D > 0: two real roots, D = 0: one real root, D < 0: complex roots
Binomial Theorem
(a + b)^n = Σ(nCr)a^(n-r)b^r
nCr = n!/(r!(n-r)!)
General term: T_(r+1) = nCr × a^(n-r) × b^r
Arithmetic Progression (AP)
a_n = a + (n-1)d
S_n = n/2[2a + (n-1)d]
S_n = n/2(a + l)
a = first term, d = common difference, l = last term
Geometric Progression (GP)
a_n = ar^(n-1)
S_n = a(r^n - 1)/(r - 1) for r ≠ 1
S_∞ = a/(1-r) for |r| < 1
a = first term, r = common ratio
2. Trigonometry
Fundamental Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Compound Angle Formulas
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)
Double Angle Formulas
sin2θ = 2sinθ cosθ
cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan2θ = 2tanθ/(1 - tan²θ)
3. Calculus
Differentiation Rules
d/dx(x^n) = nx^(n-1)
d/dx(e^x) = e^x
d/dx(ln x) = 1/x
d/dx(sinx) = cosx
d/dx(cosx) = -sinx
Product & Quotient Rules
d/dx(uv) = u(dv/dx) + v(du/dx)
d/dx(u/v) = [v(du/dx) - u(dv/dx)]/v²
Chain Rule
d/dx[f(g(x))] = f'(g(x)) × g'(x)
Integration Formulas
∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
∫e^x dx = e^x + C
∫(1/x) dx = ln|x| + C
∫sinx dx = -cosx + C
∫cosx dx = sinx + C
4. Coordinate Geometry
Distance & Section Formula
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Section Formula: (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Straight Line
Slope: m = (y₂-y₁)/(x₂-x₁)
Point-Slope: y - y₁ = m(x - x₁)
Slope-Intercept: y = mx + c
General Form: ax + by + c = 0
Circle
Standard: (x-h)² + (y-k)² = r²
General: x² + y² + 2gx + 2fy + c = 0
Center: (-g, -f), Radius: √(g² + f² - c)
Conic Sections
Parabola: y² = 4ax
Ellipse: x²/a² + y²/b² = 1
Hyperbola: x²/a² - y²/b² = 1
Universal Constants
Speed of Light
c = 3 × 10⁸ m/s
Planck's Constant
h = 6.626 × 10⁻³⁴ J⋅s
Gravitational Constant
G = 6.67 × 10⁻¹¹ N⋅m²/kg²
Avogadro's Number
N_A = 6.022 × 10²³ mol⁻¹
Gas Constant
R = 8.314 J/(mol⋅K)
R = 0.0821 L⋅atm/(mol⋅K)
Boltzmann Constant
k = 1.38 × 10⁻²³ J/K
Electron Charge
e = 1.6 × 10⁻¹⁹ C
Electron Mass
m_e = 9.1 × 10⁻³¹ kg
Proton Mass
m_p = 1.67 × 10⁻²⁷ kg
Faraday Constant
F = 96500 C/mol
Mathematical Constants
π (Pi)
π = 3.14159...
e (Euler's Number)
e = 2.71828...
√2
√2 = 1.414
√3
√3 = 1.732
log 2
log₁₀ 2 = 0.301
log 3
log₁₀ 3 = 0.477