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Physical quantities are expressed through a number and a unit. Dimensional analysis tests equations and converts units, but cannot determine dimensionless constants.
A position-time graph has slope equal to velocity; the slope of a velocity-time graph gives acceleration. The area under a velocity-time graph gives displacement.
Resolve vectors into perpendicular components before combining them. Projectile motion has constant horizontal velocity and vertical acceleration g.
Isolate the body and draw every external force before applying ΣF = ma. Static friction adjusts up to its limiting value μsN.
The work-energy theorem relates net work to change in kinetic energy. Conservative forces allow mechanical energy conservation when non-conservative work is zero.
Torque changes angular momentum and moment of inertia plays the role of mass in rotational dynamics. Pure rolling connects translation and rotation by v = ωR.
Newton's inverse-square law governs attraction between masses. A circular satellite's orbital speed depends on its orbital radius, not its own mass.
Stress is force per area and strain is fractional deformation. In static fluids pressure increases with depth; buoyant force equals the weight of displaced fluid.
The first law accounts for heat supplied as change in internal energy plus work done by the system. In an isothermal ideal-gas process ΔU is zero.
Pressure arises from molecular collisions and temperature measures mean translational kinetic energy. For an ideal gas PV = nRT.
Restoring acceleration in SHM is proportional to and opposite displacement. Energy swaps between kinetic and potential while total mechanical energy remains fixed.
A travelling wave transports energy without net transport of the medium. Wave speed is frequency times wavelength and standing waves have nodes and antinodes.
Electric field is force per unit positive test charge; potential is energy per unit charge. Gauss's law is most useful with high symmetry.
Capacitance measures charge stored per unit potential difference. A dielectric raises capacitance by reducing the effective field between the plates.
Current is charge flow per unit time. Resistances combine differently in series and parallel; Kirchhoff's laws apply charge and energy conservation to circuits.
A magnetic field exerts force q(v × B) on moving charge. A current loop acts as a magnetic dipole and experiences torque in a field.
Changing magnetic flux produces an induced emf. Lenz's law sets its direction so the induced effect opposes the change in flux.
In sinusoidal AC, rms values relate to peak values by 1/√2. At series LCR resonance inductive and capacitive reactances cancel.
Changing electric and magnetic fields sustain electromagnetic waves. They travel in vacuum at c and are transverse.
Refraction follows Snell's law and image formation follows mirror or lens equations. A sign convention is essential for image distance and focal length.
Interference redistributes intensity through coherent superposition. In YDSE the fringe width is β = λD/d for small angles.
Photons carry energy hν; photoelectric emission requires a threshold frequency. Matter has wave character with de Broglie wavelength h/p.
Bohr's hydrogen model uses quantized energy levels. Radioactive decay is statistical with constant half-life; mass defect corresponds to binding energy.
Doping changes majority carriers; a p-n junction conducts readily in forward bias. Logic gates implement Boolean operations.
The mole connects particle count to macroscopic mass. Balance the chemical equation before comparing reactants in moles to find the limiting reagent.
Electrons occupy orbitals specified by quantum numbers. Aufbau, Pauli exclusion and Hund's rule guide ground-state configurations.
Lewis structures track valence electrons; VSEPR predicts shape from electron-pair repulsion. Hybridisation describes directional bonding, while MO theory describes bond order.
Enthalpy describes heat at constant pressure. Gibbs energy ΔG = ΔH − TΔS predicts spontaneity at constant temperature and pressure.
Colligative properties depend on the number of solute particles, not their identity in ideal dilute solutions. Raoult's law relates vapor pressure to mole fraction.
At equilibrium forward and reverse rates are equal, not necessarily concentrations. The equilibrium constant depends only on temperature.
Oxidation is electron loss; reduction is gain. In a galvanic cell electrons travel externally from anode to cathode.
Reaction order comes from experiment and need not match stoichiometric coefficients. Temperature affects rate constants through activation energy.
Ideal gas behavior assumes negligible particle volume and interactions. Deviations grow at high pressure and low temperature.
Adsorption concentrates molecules on a surface; physisorption and chemisorption differ in bonding strength. Colloids scatter light in the Tyndall effect.
Purity can be checked by melting/boiling range or chromatography. Separation methods exploit differences in solubility, volatility or affinity.
Inductive, resonance and hyperconjugation effects explain stability and reactivity. Distinguish carbocations, carbanions and free radicals by electron count.
Alkanes commonly undergo substitution; alkenes and alkynes undergo addition. Aromatic compounds favor substitution that retains aromaticity.
SN2 occurs in one step with backside attack; SN1 proceeds via a carbocation. Substrate, nucleophile and solvent affect their competition.
Phenol is more acidic than typical alcohols because phenoxide is resonance-stabilized. Alcohols undergo oxidation, dehydration and substitution.
The polarized C=O bond attracts nucleophiles. Aldehydes oxidize more readily than ketones; carboxylate resonance explains acid strength.
Amines act as bases due to nitrogen's lone pair. Aromatic diazonium salts are versatile intermediates for substitutions and coupling.
Proteins are polymers of amino acids linked by peptide bonds. DNA and RNA contain nucleotide units; carbohydrates include mono-, oligo- and polysaccharides.
Practical identification combines observation with selective tests. Use controls and distinguish evidence for a functional group from proof of a unique molecule.
Properties recur with electron configuration. Across a period effective nuclear charge generally increases and atomic radius generally decreases.
Valence-shell configuration controls trends within p-block groups. Oxidation states, bonding and acidic/basic character vary systematically across periods.
Partially filled d orbitals support variable oxidation states, coloured ions and complex formation. Lanthanide contraction affects size trends.
A coordination complex has a central metal and surrounding ligands. Geometry and ligand-field splitting help explain isomerism, colour and magnetism.
Hydrogen forms ionic, covalent and interstitial hydrides. Hydrogen peroxide can act as both an oxidizing and reducing agent.
s-block metals lose valence electrons easily and form mostly ionic compounds. Reactivity and hydration enthalpy vary down each group.
Environmental chemistry connects chemical reactions to air, water and soil systems. Green chemistry aims to reduce hazards and waste at the source.
A function assigns each domain element exactly one output. Composition requires the inner function's range to lie in the outer domain.
Complex numbers combine real and imaginary parts. The discriminant determines the nature of quadratic roots; conjugates simplify division.
Matrix multiplication is order-dependent. A square matrix is invertible only when its determinant is nonzero.
Order matters in permutations but not combinations. Count complementary cases when a direct case split becomes messy.
The binomial theorem expands non-negative integer powers with coefficients nCr. The (r+1)th term contains x^(n−r)y^r.
An AP has constant difference and a GP has constant ratio. Summation formulas depend on the number of terms and indexing convention.
Continuity requires the limit to equal the function value. Differentiability implies continuity, but the converse need not hold.
Integration reverses differentiation. Definite integrals accumulate signed area, so geometry needs care when the graph crosses the axis.
A differential equation connects an unknown function with derivatives. Separation of variables works when x and y terms can be isolated.
The slope measures rise over run. Equivalent line forms are useful for intersections, perpendicular distances and angle conditions.
A circle is the set of points at fixed distance from a centre. Its standard equation makes centre and radius immediate.
Conics can be defined using focus, directrix and eccentricity. A parabola has e=1; ellipse e<1; hyperbola e>1.
Direction cosines encode a line's orientation in 3D. Dot products test perpendicularity and compute angles between directions.
The dot product gives a scalar related to angle and projection; the cross product gives a perpendicular vector with area magnitude.
Variance measures spread around the mean. Conditional probability updates the sample space; independence means P(A∩B)=P(A)P(B).
Trig identities link angle ratios and enable equation solving. Inverse trig functions use restricted principal-value ranges.
Induction proves a statement for integers by establishing a starting case and a step from k to k+1.
Logical connectives combine propositions. The negation of 'for all' is 'there exists a counterexample'.
Linear programming optimizes a linear objective over a feasible region set by inequalities. If an optimum exists at a vertex, evaluate corner points.
The derivative gives an instantaneous rate and tangent slope. Critical points are candidates for extrema; sign changes decide their nature.