LC Applied Maths cheat sheet

Constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as; valid only when a is constant, so never apply them to SHM or variable-force motion.

Use s = ½(u + v)t when time is unknown or unwanted; it follows from eliminating a between the first two equations.

Variable acceleration: write a = dv/dt = v·dv/ds, then integrate. Choose the form matching the variable given (t or s).

Newton's second law in vector form: F_net = ma = m·dv/dt; resolve along and perpendicular to motion rather than horizontally/vertically when the path curves.

Friction: limiting value F = μR at the point of slipping; static friction takes whatever value (≤ μR) is needed for equilibrium.

Connected particles over a pulley: apply F = ma to each body separately, with the same magnitude of acceleration and tension; the tension is not equal to either weight when accelerating.

Impulse = ∫F dt = change in momentum; momentum is conserved in collisions and explosions when no external impulsive force acts along the line of motion.

Projectiles: horizontal component of velocity is constant (a = 0), vertical component has a = −g; time of flight comes from the vertical equation alone.

A particle is in equilibrium iff the vector sum of all forces is zero; equivalently the force polygon closes and ΣFx = 0, ΣFy = 0.

An extended rigid body needs both ΣF = 0 and Σ(moments about any point) = 0; a third equation is available because moment equilibrium holds about every point.

Moment of a force about a point = F·d, where d is the perpendicular distance from the point to the line of action; sign convention must be fixed before summing.

A couple consists of two equal, opposite, parallel forces; its moment is F·d (d = perpendicular separation) and is the same about every point, so it produces rotation only.

Resultant of two forces: use the parallelogram law R² = P² + Q² + 2PQ·cos θ, where θ is the angle between the lines of action; direction from the sine rule.

Three-force equilibrium: the lines of action of three non-parallel coplanar forces in equilibrium are concurrent, which locates unknown reaction directions geometrically.

Toppling versus sliding: the body topples when the line of action of its weight falls outside the base, and slides when the required friction exceeds μR.

Tension in a light inextensible string is uniform throughout its length unless a mass, pulley friction, or an external force acts on it; a light smooth pulley changes direction only.

Uniform circular motion: v = rω, period T = 2π/ω, centripetal acceleration a = v²/r = rω² directed toward the centre; the speed is constant but velocity is not.

Centripetal force is the resultant of the real forces (tension, weight, normal reaction, friction), never an extra force to be added to the diagram.

Vertical circle on a string: at the top T = mv²/r − mg; the string goes slack when v² < gr, and the minimum speed at the lowest point for full rotation is √(5gr).

Conical pendulum: T·cos θ = mg and T·sin θ = mrω², so tan θ = ω²r/g and the period depends only on the height of the cone.

SHM definition: acceleration is proportional to displacement from a fixed point and directed toward it, a = −ω²x; solutions are x = A·cos(ωt + φ) with period 2π/ω.

SHM speed: v = ±ω√(A² − x²), maximum ωA at the centre and zero at the extremes; maximum acceleration ω²A occurs at the extremes.

Newton's law of gravitation: F = GMm/r², with potential energy U = −GMm/r and potential V = −GM/r, taking zero at infinity.

Orbital motion: GMm/r² = mv²/r gives v = √(GM/r) and T² = 4π²r³/(GM); geostationary orbits require T = 24 h and an equatorial plane.

Differentiate from first principles: f′(x) = lim(h→0)[f(x + h) − f(x)]/h; this is the required method when the question names it.

Chain rule: dy/dx = dy/du · du/dx; product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².

Derivatives to know: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(ln x) = 1/x, d/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ ln a.

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, ∫dx/x = ln|x| + C, ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C, ∫eˣ dx = eˣ + C.

Definite integrals give signed area; for total area split where the curve crosses the axis and take absolute values of each part.

Volume of revolution about the x-axis: V = π∫y² dx between the limits; about the y-axis use π∫x² dy with x expressed in terms of y.

Stationary points: solve dy/dx = 0, then classify by the sign change of dy/dx or by d²y/dx² (negative = maximum, positive = minimum, zero = test further).

Trapezoidal rule with n strips of width h: ∫f dx ≈ (h/2)[f₀ + 2(f₁ + … + fₙ₋₁) + fₙ]; the error decreases as h².

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Multiplication rule: P(A ∩ B) = P(A)·P(B|A); for independent events P(A ∩ B) = P(A)·P(B) and P(B|A) = P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0; rearrange to P(A ∩ B) = P(B)·P(A|B) when the joint probability is unknown.

Total probability: P(B) = ΣP(Aᵢ)·P(B|Aᵢ) over a partition; combine with Bayes' theorem P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/P(B) to reverse conditioning.

Binomial distribution B(n, p): P(X = r) = C(n, r)·pʳ·(1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p); requires a fixed number of independent trials with constant p.

Normal distribution: standardise with z = (x − μ)/σ, then use tables for Φ(z); for a sample mean use σ/√n as the standard error.

Mean and variance of a discrete variable: E(X) = Σx·P(x), Var(X) = E(X²) − [E(X)]²; for aX + b the mean is aE(X) + b and the variance is a²Var(X).

Hypothesis test: state H₀ and H₁, choose the significance level, compute the test statistic or tail probability, and reject H₀ only when the p-value is below the level.

Constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as; valid only when a is constant, so never apply them to SHM or variable-force motion.

Use s = ½(u + v)t when time is unknown or unwanted; it follows from eliminating a between the first two equations.

Variable acceleration: write a = dv/dt = v·dv/ds, then integrate. Choose the form matching the variable given (t or s).

Newton's second law in vector form: F_net = ma = m·dv/dt; resolve along and perpendicular to motion rather than horizontally/vertically when the path curves.

Friction: limiting value F = μR at the point of slipping; static friction takes whatever value (≤ μR) is needed for equilibrium.

Connected particles over a pulley: apply F = ma to each body separately, with the same magnitude of acceleration and tension; the tension is not equal to either weight when accelerating.

Impulse = ∫F dt = change in momentum; momentum is conserved in collisions and explosions when no external impulsive force acts along the line of motion.

Projectiles: horizontal component of velocity is constant (a = 0), vertical component has a = −g; time of flight comes from the vertical equation alone.

A particle is in equilibrium iff the vector sum of all forces is zero; equivalently the force polygon closes and ΣFx = 0, ΣFy = 0.

An extended rigid body needs both ΣF = 0 and Σ(moments about any point) = 0; a third equation is available because moment equilibrium holds about every point.

Moment of a force about a point = F·d, where d is the perpendicular distance from the point to the line of action; sign convention must be fixed before summing.

A couple consists of two equal, opposite, parallel forces; its moment is F·d (d = perpendicular separation) and is the same about every point, so it produces rotation only.

Resultant of two forces: use the parallelogram law R² = P² + Q² + 2PQ·cos θ, where θ is the angle between the lines of action; direction from the sine rule.

Three-force equilibrium: the lines of action of three non-parallel coplanar forces in equilibrium are concurrent, which locates unknown reaction directions geometrically.

Toppling versus sliding: the body topples when the line of action of its weight falls outside the base, and slides when the required friction exceeds μR.

Tension in a light inextensible string is uniform throughout its length unless a mass, pulley friction, or an external force acts on it; a light smooth pulley changes direction only.

Uniform circular motion: v = rω, period T = 2π/ω, centripetal acceleration a = v²/r = rω² directed toward the centre; the speed is constant but velocity is not.

Centripetal force is the resultant of the real forces (tension, weight, normal reaction, friction), never an extra force to be added to the diagram.

Vertical circle on a string: at the top T = mv²/r − mg; the string goes slack when v² < gr, and the minimum speed at the lowest point for full rotation is √(5gr).

Conical pendulum: T·cos θ = mg and T·sin θ = mrω², so tan θ = ω²r/g and the period depends only on the height of the cone.

SHM definition: acceleration is proportional to displacement from a fixed point and directed toward it, a = −ω²x; solutions are x = A·cos(ωt + φ) with period 2π/ω.

SHM speed: v = ±ω√(A² − x²), maximum ωA at the centre and zero at the extremes; maximum acceleration ω²A occurs at the extremes.

Newton's law of gravitation: F = GMm/r², with potential energy U = −GMm/r and potential V = −GM/r, taking zero at infinity.

Orbital motion: GMm/r² = mv²/r gives v = √(GM/r) and T² = 4π²r³/(GM); geostationary orbits require T = 24 h and an equatorial plane.

Differentiate from first principles: f′(x) = lim(h→0)[f(x + h) − f(x)]/h; this is the required method when the question names it.

Chain rule: dy/dx = dy/du · du/dx; product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².

Derivatives to know: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(ln x) = 1/x, d/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ ln a.

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, ∫dx/x = ln|x| + C, ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C, ∫eˣ dx = eˣ + C.

Definite integrals give signed area; for total area split where the curve crosses the axis and take absolute values of each part.

Volume of revolution about the x-axis: V = π∫y² dx between the limits; about the y-axis use π∫x² dy with x expressed in terms of y.

Stationary points: solve dy/dx = 0, then classify by the sign change of dy/dx or by d²y/dx² (negative = maximum, positive = minimum, zero = test further).

Trapezoidal rule with n strips of width h: ∫f dx ≈ (h/2)[f₀ + 2(f₁ + … + fₙ₋₁) + fₙ]; the error decreases as h².

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Multiplication rule: P(A ∩ B) = P(A)·P(B|A); for independent events P(A ∩ B) = P(A)·P(B) and P(B|A) = P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0; rearrange to P(A ∩ B) = P(B)·P(A|B) when the joint probability is unknown.

Total probability: P(B) = ΣP(Aᵢ)·P(B|Aᵢ) over a partition; combine with Bayes' theorem P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/P(B) to reverse conditioning.

Binomial distribution B(n, p): P(X = r) = C(n, r)·pʳ·(1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p); requires a fixed number of independent trials with constant p.

Normal distribution: standardise with z = (x − μ)/σ, then use tables for Φ(z); for a sample mean use σ/√n as the standard error.

Mean and variance of a discrete variable: E(X) = Σx·P(x), Var(X) = E(X²) − [E(X)]²; for aX + b the mean is aE(X) + b and the variance is a²Var(X).

Hypothesis test: state H₀ and H₁, choose the significance level, compute the test statistic or tail probability, and reject H₀ only when the p-value is below the level.