Yamanashi Entrance Science cheat sheet

Newton's second law: ΣF = ma; for a system of particles use the center of mass, and for rotation replace force with torque τ = Iα about a fixed axis.

Conservation of mechanical energy applies when only conservative forces do work: (1/2)mv² + mgh + (1/2)kx² = constant; friction converts mechanical energy into heat Q = f·d.

Momentum is conserved in collisions with no net external impulse: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'; perfectly inelastic collisions merge masses, elastic collisions also conserve kinetic energy.

Uniform circular motion requires centripetal acceleration a = v²/r directed toward the center; the centripetal force is supplied by friction, tension, gravity, or normal force depending on context.

Coulomb's law: F = kq₁q₂/r² with k ≈ 9.0×10⁹ N·m²/C²; the electric field E = F/q, and field lines point away from positive and toward negative charges.

Ohm's law V = IR; resistors in series add (R = R₁ + R₂), in parallel reciprocals add (1/R = 1/R₁ + 1/R₂); power dissipated P = VI = I²R = V²/R.

Faraday's law: induced emf = −N ΔΦ/Δt, where magnetic flux Φ = BA cos θ; Lenz's law fixes the induced current direction so it opposes the change in flux.

Wave relation v = fλ; for a string fixed at both ends, the n-th harmonic has λₙ = 2L/n; refraction obeys Snell's law n₁ sin θ₁ = n₂ sin θ₂ with n = c/v.

First law of thermodynamics: ΔU = Q − W (W done by the gas); isothermal processes keep T constant (PV = constant), adiabatic processes have Q = 0, and efficiency of a heat engine is η = 1 − T_cold/T_hot.

The mole connects mass and particle count: n = m/M = N/N_A, with N_A ≈ 6.02×10²³ mol⁻¹; molar volume of an ideal gas at 0 °C and 1 atm is about 22.4 L.

Ideal gas law PV = nRT with R ≈ 8.31 J/(mol·K); at constant temperature P₁V₁ = P₂V₂ (Boyle), at constant pressure V₁/T₁ = V₂/T₂ (Charles).

Molarity M = moles of solute / liters of solution; dilution obeys M₁V₁ = M₂V₂; pH = −log[H⁺] and pH + pOH = 14 at 25 °C.

Oxidation is loss of electrons (oxidation number increases); reduction is gain. In redox balancing, electrons lost equal electrons gained.

Periodic trends: atomic radius decreases left to right across a period and increases down a group; ionization energy and electronegativity show the opposite trend.

Strong acids (HCl, HNO₃, H₂SO₄) and strong bases (NaOH, KOH) dissociate completely; weak acids ionize only partially and require Ka for equilibrium calculations.

Organic functional groups: alcohols −OH, aldehydes −CHO, ketones C=O between carbons, carboxylic acids −COOH, esters −COO−, amines −NH₂; oxidation of a primary alcohol gives an aldehyde then a carboxylic acid.

Addition reactions occur at C=C double bonds (e.g., H₂, Br₂, HBr); substitution occurs on saturated alkanes under light; esterification of an acid with an alcohol is catalyzed by strong acid and produces water.

Le Chatelier's principle: raising pressure shifts equilibrium toward the side with fewer gas moles; raising temperature shifts an exothermic reaction toward reactants.

Prokaryotic cells lack a membrane-bound nucleus and organelles; eukaryotic cells contain a nucleus, mitochondria, and (in plants) chloroplasts and a cell wall of cellulose.

Mitosis produces two genetically identical diploid cells for growth and repair; meiosis produces four haploid gametes with crossing over in prophase I and independent assortment in metaphase I.

Mendel's law of segregation: alleles separate during gamete formation; a cross Aa × Aa gives a 3:1 phenotype ratio and a 1:2:1 genotype ratio for complete dominance.

DNA replication is semiconservative; transcription copies DNA into mRNA, and translation at the ribosome reads codons (three bases) into amino acids using tRNA.

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplast thylakoids (light reactions) and stroma (Calvin cycle); cellular respiration reverses this in mitochondria.

Enzyme activity rises with temperature up to an optimum, then denatures; pH outside the optimum also reduces activity because the active-site shape changes.

Population growth: exponential growth follows dN/dt = rN, while logistic growth adds carrying capacity K and follows dN/dt = rN(1 − N/K).

Natural selection requires heritable variation in a population, differential reproductive success, and change in allele frequency over generations; genetic drift is random change, strongest in small populations.

Energy flows through ecosystems in one direction with roughly 10% transfer between trophic levels; matter such as carbon and nitrogen cycles repeatedly through biotic and abiotic pools.

Earth's interior is layered: crust, mantle, outer core (liquid, generates the magnetic field by dynamo action), and inner core (solid iron-nickel).

Plate tectonics: divergent boundaries create new crust at mid-ocean ridges, convergent boundaries produce subduction and mountain belts, and transform boundaries cause strike-slip earthquakes.

Seismic waves: P waves are longitudinal and travel through solids and liquids; S waves are transverse and cannot pass through liquids, which reveals the liquid outer core.

The Richter scale is logarithmic: each whole-number increase means about 10 times greater amplitude and roughly 32 times more energy released.

Igneous rocks form from cooling magma (intrusive = coarse-grained, extrusive = fine-grained); sedimentary rocks form by deposition and compaction; metamorphic rocks form under heat and pressure.

Atmospheric pressure decreases with altitude; wind blows from high to low pressure, deflected by the Coriolis effect (right in the Northern Hemisphere, left in the Southern).

Relative humidity is the ratio of actual vapor content to saturation content at that temperature; cooling air to the dew point causes condensation and cloud formation.

Kepler's third law: T² ∝ a³ for orbits around the same body; Earth's axial tilt of 23.4° combined with revolution causes the seasons, not distance from the Sun.

Stellar brightness: apparent magnitude measures observed brightness (smaller is brighter); absolute magnitude measures intrinsic luminosity at a standard distance of 10 parsecs.

A controlled experiment changes one independent variable while holding all other conditions constant; the dependent variable is what is measured as a result.

A hypothesis must be falsifiable: it predicts an observable outcome that could prove it wrong if the data disagree.

Correlation does not establish causation; a confounding variable can produce an apparent relationship between two measured quantities.

Accuracy is closeness to the true value; precision is repeatability of measurements; random errors scatter both ways, while systematic errors shift all readings in one direction.

Significant figures: in multiplication and division, keep the fewest significant figures of the inputs; in addition and subtraction, keep the fewest decimal places.

Percent error = |measured − accepted| / accepted × 100; percent yield = actual yield / theoretical yield × 100.

When graphing, plot the independent variable on the x-axis and the dependent variable on the y-axis; a linear fit yields slope with units of y per x.

A control group receives no treatment or a placebo so the effect of the treatment can be isolated from other influences.

A conclusion is valid only if the sample is representative and large enough; a single trial is insufficient to establish a reliable trend.

Newton's second law: ΣF = ma; for a system of particles use the center of mass, and for rotation replace force with torque τ = Iα about a fixed axis.

Conservation of mechanical energy applies when only conservative forces do work: (1/2)mv² + mgh + (1/2)kx² = constant; friction converts mechanical energy into heat Q = f·d.

Momentum is conserved in collisions with no net external impulse: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'; perfectly inelastic collisions merge masses, elastic collisions also conserve kinetic energy.

Uniform circular motion requires centripetal acceleration a = v²/r directed toward the center; the centripetal force is supplied by friction, tension, gravity, or normal force depending on context.

Coulomb's law: F = kq₁q₂/r² with k ≈ 9.0×10⁹ N·m²/C²; the electric field E = F/q, and field lines point away from positive and toward negative charges.

Ohm's law V = IR; resistors in series add (R = R₁ + R₂), in parallel reciprocals add (1/R = 1/R₁ + 1/R₂); power dissipated P = VI = I²R = V²/R.

Faraday's law: induced emf = −N ΔΦ/Δt, where magnetic flux Φ = BA cos θ; Lenz's law fixes the induced current direction so it opposes the change in flux.

Wave relation v = fλ; for a string fixed at both ends, the n-th harmonic has λₙ = 2L/n; refraction obeys Snell's law n₁ sin θ₁ = n₂ sin θ₂ with n = c/v.

First law of thermodynamics: ΔU = Q − W (W done by the gas); isothermal processes keep T constant (PV = constant), adiabatic processes have Q = 0, and efficiency of a heat engine is η = 1 − T_cold/T_hot.

The mole connects mass and particle count: n = m/M = N/N_A, with N_A ≈ 6.02×10²³ mol⁻¹; molar volume of an ideal gas at 0 °C and 1 atm is about 22.4 L.

Ideal gas law PV = nRT with R ≈ 8.31 J/(mol·K); at constant temperature P₁V₁ = P₂V₂ (Boyle), at constant pressure V₁/T₁ = V₂/T₂ (Charles).

Molarity M = moles of solute / liters of solution; dilution obeys M₁V₁ = M₂V₂; pH = −log[H⁺] and pH + pOH = 14 at 25 °C.

Oxidation is loss of electrons (oxidation number increases); reduction is gain. In redox balancing, electrons lost equal electrons gained.

Periodic trends: atomic radius decreases left to right across a period and increases down a group; ionization energy and electronegativity show the opposite trend.

Strong acids (HCl, HNO₃, H₂SO₄) and strong bases (NaOH, KOH) dissociate completely; weak acids ionize only partially and require Ka for equilibrium calculations.

Organic functional groups: alcohols −OH, aldehydes −CHO, ketones C=O between carbons, carboxylic acids −COOH, esters −COO−, amines −NH₂; oxidation of a primary alcohol gives an aldehyde then a carboxylic acid.

Addition reactions occur at C=C double bonds (e.g., H₂, Br₂, HBr); substitution occurs on saturated alkanes under light; esterification of an acid with an alcohol is catalyzed by strong acid and produces water.

Le Chatelier's principle: raising pressure shifts equilibrium toward the side with fewer gas moles; raising temperature shifts an exothermic reaction toward reactants.

Prokaryotic cells lack a membrane-bound nucleus and organelles; eukaryotic cells contain a nucleus, mitochondria, and (in plants) chloroplasts and a cell wall of cellulose.

Mitosis produces two genetically identical diploid cells for growth and repair; meiosis produces four haploid gametes with crossing over in prophase I and independent assortment in metaphase I.

Mendel's law of segregation: alleles separate during gamete formation; a cross Aa × Aa gives a 3:1 phenotype ratio and a 1:2:1 genotype ratio for complete dominance.

DNA replication is semiconservative; transcription copies DNA into mRNA, and translation at the ribosome reads codons (three bases) into amino acids using tRNA.

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplast thylakoids (light reactions) and stroma (Calvin cycle); cellular respiration reverses this in mitochondria.

Enzyme activity rises with temperature up to an optimum, then denatures; pH outside the optimum also reduces activity because the active-site shape changes.

Population growth: exponential growth follows dN/dt = rN, while logistic growth adds carrying capacity K and follows dN/dt = rN(1 − N/K).

Natural selection requires heritable variation in a population, differential reproductive success, and change in allele frequency over generations; genetic drift is random change, strongest in small populations.

Energy flows through ecosystems in one direction with roughly 10% transfer between trophic levels; matter such as carbon and nitrogen cycles repeatedly through biotic and abiotic pools.

Earth's interior is layered: crust, mantle, outer core (liquid, generates the magnetic field by dynamo action), and inner core (solid iron-nickel).

Plate tectonics: divergent boundaries create new crust at mid-ocean ridges, convergent boundaries produce subduction and mountain belts, and transform boundaries cause strike-slip earthquakes.

Seismic waves: P waves are longitudinal and travel through solids and liquids; S waves are transverse and cannot pass through liquids, which reveals the liquid outer core.

The Richter scale is logarithmic: each whole-number increase means about 10 times greater amplitude and roughly 32 times more energy released.

Igneous rocks form from cooling magma (intrusive = coarse-grained, extrusive = fine-grained); sedimentary rocks form by deposition and compaction; metamorphic rocks form under heat and pressure.

Atmospheric pressure decreases with altitude; wind blows from high to low pressure, deflected by the Coriolis effect (right in the Northern Hemisphere, left in the Southern).

Relative humidity is the ratio of actual vapor content to saturation content at that temperature; cooling air to the dew point causes condensation and cloud formation.

Kepler's third law: T² ∝ a³ for orbits around the same body; Earth's axial tilt of 23.4° combined with revolution causes the seasons, not distance from the Sun.

Stellar brightness: apparent magnitude measures observed brightness (smaller is brighter); absolute magnitude measures intrinsic luminosity at a standard distance of 10 parsecs.

A controlled experiment changes one independent variable while holding all other conditions constant; the dependent variable is what is measured as a result.

A hypothesis must be falsifiable: it predicts an observable outcome that could prove it wrong if the data disagree.

Correlation does not establish causation; a confounding variable can produce an apparent relationship between two measured quantities.

Accuracy is closeness to the true value; precision is repeatability of measurements; random errors scatter both ways, while systematic errors shift all readings in one direction.

Significant figures: in multiplication and division, keep the fewest significant figures of the inputs; in addition and subtraction, keep the fewest decimal places.

Percent error = |measured − accepted| / accepted × 100; percent yield = actual yield / theoretical yield × 100.

When graphing, plot the independent variable on the x-axis and the dependent variable on the y-axis; a linear fit yields slope with units of y per x.

A control group receives no treatment or a placebo so the effect of the treatment can be isolated from other influences.

A conclusion is valid only if the sample is representative and large enough; a single trial is insufficient to establish a reliable trend.

Newton's second law: ΣF = ma; for a system of particles use the center of mass, and for rotation replace force with torque τ = Iα about a fixed axis.

Conservation of mechanical energy applies when only conservative forces do work: (1/2)mv² + mgh + (1/2)kx² = constant; friction converts mechanical energy into heat Q = f·d.

Momentum is conserved in collisions with no net external impulse: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'; perfectly inelastic collisions merge masses, elastic collisions also conserve kinetic energy.

Uniform circular motion requires centripetal acceleration a = v²/r directed toward the center; the centripetal force is supplied by friction, tension, gravity, or normal force depending on context.

Coulomb's law: F = kq₁q₂/r² with k ≈ 9.0×10⁹ N·m²/C²; the electric field E = F/q, and field lines point away from positive and toward negative charges.

Ohm's law V = IR; resistors in series add (R = R₁ + R₂), in parallel reciprocals add (1/R = 1/R₁ + 1/R₂); power dissipated P = VI = I²R = V²/R.

Faraday's law: induced emf = −N ΔΦ/Δt, where magnetic flux Φ = BA cos θ; Lenz's law fixes the induced current direction so it opposes the change in flux.

Wave relation v = fλ; for a string fixed at both ends, the n-th harmonic has λₙ = 2L/n; refraction obeys Snell's law n₁ sin θ₁ = n₂ sin θ₂ with n = c/v.

First law of thermodynamics: ΔU = Q − W (W done by the gas); isothermal processes keep T constant (PV = constant), adiabatic processes have Q = 0, and efficiency of a heat engine is η = 1 − T_cold/T_hot.

The mole connects mass and particle count: n = m/M = N/N_A, with N_A ≈ 6.02×10²³ mol⁻¹; molar volume of an ideal gas at 0 °C and 1 atm is about 22.4 L.

Ideal gas law PV = nRT with R ≈ 8.31 J/(mol·K); at constant temperature P₁V₁ = P₂V₂ (Boyle), at constant pressure V₁/T₁ = V₂/T₂ (Charles).

Molarity M = moles of solute / liters of solution; dilution obeys M₁V₁ = M₂V₂; pH = −log[H⁺] and pH + pOH = 14 at 25 °C.

Oxidation is loss of electrons (oxidation number increases); reduction is gain. In redox balancing, electrons lost equal electrons gained.

Periodic trends: atomic radius decreases left to right across a period and increases down a group; ionization energy and electronegativity show the opposite trend.

Strong acids (HCl, HNO₃, H₂SO₄) and strong bases (NaOH, KOH) dissociate completely; weak acids ionize only partially and require Ka for equilibrium calculations.

Organic functional groups: alcohols −OH, aldehydes −CHO, ketones C=O between carbons, carboxylic acids −COOH, esters −COO−, amines −NH₂; oxidation of a primary alcohol gives an aldehyde then a carboxylic acid.

Addition reactions occur at C=C double bonds (e.g., H₂, Br₂, HBr); substitution occurs on saturated alkanes under light; esterification of an acid with an alcohol is catalyzed by strong acid and produces water.

Le Chatelier's principle: raising pressure shifts equilibrium toward the side with fewer gas moles; raising temperature shifts an exothermic reaction toward reactants.

Prokaryotic cells lack a membrane-bound nucleus and organelles; eukaryotic cells contain a nucleus, mitochondria, and (in plants) chloroplasts and a cell wall of cellulose.

Mitosis produces two genetically identical diploid cells for growth and repair; meiosis produces four haploid gametes with crossing over in prophase I and independent assortment in metaphase I.

Mendel's law of segregation: alleles separate during gamete formation; a cross Aa × Aa gives a 3:1 phenotype ratio and a 1:2:1 genotype ratio for complete dominance.

DNA replication is semiconservative; transcription copies DNA into mRNA, and translation at the ribosome reads codons (three bases) into amino acids using tRNA.

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplast thylakoids (light reactions) and stroma (Calvin cycle); cellular respiration reverses this in mitochondria.

Enzyme activity rises with temperature up to an optimum, then denatures; pH outside the optimum also reduces activity because the active-site shape changes.

Population growth: exponential growth follows dN/dt = rN, while logistic growth adds carrying capacity K and follows dN/dt = rN(1 − N/K).

Natural selection requires heritable variation in a population, differential reproductive success, and change in allele frequency over generations; genetic drift is random change, strongest in small populations.

Energy flows through ecosystems in one direction with roughly 10% transfer between trophic levels; matter such as carbon and nitrogen cycles repeatedly through biotic and abiotic pools.

Earth's interior is layered: crust, mantle, outer core (liquid, generates the magnetic field by dynamo action), and inner core (solid iron-nickel).

Plate tectonics: divergent boundaries create new crust at mid-ocean ridges, convergent boundaries produce subduction and mountain belts, and transform boundaries cause strike-slip earthquakes.

Seismic waves: P waves are longitudinal and travel through solids and liquids; S waves are transverse and cannot pass through liquids, which reveals the liquid outer core.

The Richter scale is logarithmic: each whole-number increase means about 10 times greater amplitude and roughly 32 times more energy released.

Igneous rocks form from cooling magma (intrusive = coarse-grained, extrusive = fine-grained); sedimentary rocks form by deposition and compaction; metamorphic rocks form under heat and pressure.

Atmospheric pressure decreases with altitude; wind blows from high to low pressure, deflected by the Coriolis effect (right in the Northern Hemisphere, left in the Southern).

Relative humidity is the ratio of actual vapor content to saturation content at that temperature; cooling air to the dew point causes condensation and cloud formation.

Kepler's third law: T² ∝ a³ for orbits around the same body; Earth's axial tilt of 23.4° combined with revolution causes the seasons, not distance from the Sun.

Stellar brightness: apparent magnitude measures observed brightness (smaller is brighter); absolute magnitude measures intrinsic luminosity at a standard distance of 10 parsecs.

A controlled experiment changes one independent variable while holding all other conditions constant; the dependent variable is what is measured as a result.

A hypothesis must be falsifiable: it predicts an observable outcome that could prove it wrong if the data disagree.

Correlation does not establish causation; a confounding variable can produce an apparent relationship between two measured quantities.

Accuracy is closeness to the true value; precision is repeatability of measurements; random errors scatter both ways, while systematic errors shift all readings in one direction.

Significant figures: in multiplication and division, keep the fewest significant figures of the inputs; in addition and subtraction, keep the fewest decimal places.

Percent error = |measured − accepted| / accepted × 100; percent yield = actual yield / theoretical yield × 100.

When graphing, plot the independent variable on the x-axis and the dependent variable on the y-axis; a linear fit yields slope with units of y per x.

A control group receives no treatment or a placebo so the effect of the treatment can be isolated from other influences.

A conclusion is valid only if the sample is representative and large enough; a single trial is insufficient to establish a reliable trend.