LC Applied Maths flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

v² = u² + 2as

Links initial and final speed to acceleration and displacement for constant acceleration only. Useful when time is neither given nor required, as in braking-distance problems.

Variable acceleration chain rule

When acceleration depends on displacement, write a = v·dv/ds and integrate with respect to s. When it depends on time, use a = dv/dt and integrate with respect to t.

Limiting friction

The maximum static friction a surface can provide, equal to μR where R is the normal reaction. Slipping begins only when the required friction reaches this value.

Impulse–momentum theorem

The impulse of a force, the integral of force over time, equals the change in momentum it produces. For a constant force this is simply F·t.

Projectile horizontal motion

With no air resistance the horizontal acceleration is zero, so horizontal velocity stays constant. Time of flight is found from the vertical motion alone.

Pulley with two hanging masses

Apply F = ma to each mass separately, using the same acceleration magnitude and a common tension. The tension lies between the two weights, not equal to either.

Moment of a force

The turning effect about a point, equal to force multiplied by the perpendicular distance from the point to the force's line of action. Units are newton-metres.

Moment of a couple

Two equal, opposite, parallel forces form a couple whose moment is F·d, with d the perpendicular separation. This moment is the same about every point.

Three-force equilibrium concurrency

If three non-parallel coplanar forces keep a body in equilibrium, their lines of action must all pass through a single point. This fixes unknown reaction directions.

Toppling condition

A body on a horizontal surface topples when the vertical line through its centre of mass falls outside its base. Inside the base it merely presses harder on one edge.

Resultant of two forces

For forces P and Q with angle θ between their lines of action, the resultant satisfies R² = P² + Q² + 2PQ·cos θ. Direction follows from the sine rule.

Equilibrium of a rigid body

The resultant force must be zero and the total moment about any one point must be zero. Because moments vanish about every point, three independent equations are available.

Centripetal acceleration

For motion in a circle of radius r at speed v, acceleration is v²/r or rω², directed toward the centre. Speed may be constant while velocity changes direction.

Vertical circle minimum speed

For a particle on a light string to complete a vertical circle, the speed at the lowest point must be at least √(5gr). Below this the string slackens.

Conical pendulum period

A bob moving in a horizontal circle with the string at angle θ to the vertical satisfies tan θ = ω²r/g, so the period depends only on the cone height.

Defining equation of SHM

Simple harmonic motion requires acceleration proportional to displacement from a fixed point and directed toward it: a = −ω²x. The period is then 2π/ω.

SHM velocity–displacement relation

In SHM, v = ±ω√(A² − x²), where A is amplitude and x is displacement. Speed is greatest at the centre and zero at the extremes.

Gravitational potential energy

For two masses separated by distance r, potential energy is −GMm/r, taken as zero at infinity. The negative sign means the force is attractive and energy must be supplied to escape.

Kepler's third law form

For a circular orbit, equating gravitational and centripetal force gives T² = 4π²r³/(GM). The period squared is proportional to the orbital radius cubed.

Differentiation from first principles

The derivative is the limit of [f(x + h) − f(x)]/h as h tends to zero. Use this method whenever a question explicitly demands first principles.

Quotient rule

For y = u/v, the derivative is (u′v − uv′)/v². The order of subtraction matters, so the denominator squared is always positive.

Chain rule

For a composite function, dy/dx = dy/du · du/dx. Identify the inner function, differentiate the outer with respect to it, then multiply by the inner derivative.

Integral of 1/x

The integral of 1/x with respect to x is ln|x| + C, not x⁰/0. The modulus is needed because the logarithm is undefined for negative arguments.

Second derivative test

At a stationary point, d²y/dx² negative means a maximum, positive means a minimum, and zero is inconclusive, requiring a sign-change check of the first derivative.

Volume of revolution

Rotating y = f(x) about the x-axis between limits gives volume π∫y² dx. Each thin slice is a disc of radius y and thickness dx.

Trapezoidal rule

For n equal strips of width h, the integral is approximately (h/2) times the first plus last ordinates plus twice all interior ordinates. Error falls as h².

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. It rescales the joint probability by the probability of the conditioning event.

Independent events

Events A and B are independent exactly when P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A). Knowing one occurs does not alter the other's probability.

Binomial distribution mean and variance

For X ~ B(n, p), the mean is np and the variance is np(1 − p). The model needs a fixed number of independent trials with constant success probability.

Standardising a normal variable

Convert X ~ N(μ, σ²) to z = (x − μ)/σ, then read probabilities from standard normal tables. For a sample mean, replace σ by σ/√n.

Variance of aX + b

Adding a constant b leaves variance unchanged, while multiplying by a multiplies variance by a². The mean becomes aE(X) + b.

Bayes' theorem

P(A|B) = P(A)·P(B|A)/P(B), where P(B) is found by the law of total probability. It reverses the direction of a conditional probability.