Cameroon GCE A-Level Pure Math with Mechanics key terms

The single force that has the same effect on a body as two or more forces acting together. Found by vector addition, often by resolving components or drawing a triangle of forces.

Splitting a force into two perpendicular components, usually horizontal and vertical, using cosine for the adjacent direction and sine for the opposite direction. This converts awkward directions into manageable scalar equations.

The state in which the vector sum of all forces on a particle is zero, so it has no acceleration. In component form, the sums of horizontal forces and of vertical forces each equal zero.

The borderline state where a body is just about to slide, so friction has reached its maximum value and Newton's laws are still satisfied with acceleration zero. Used to find the coefficient of friction.

The perpendicular contact force a surface exerts on a body resting on or pressing against it. On a horizontal surface it equals weight only when no other vertical forces act.

A contact force opposing relative motion or the tendency to move, acting along the surface. In the standard model its maximum value is the coefficient of friction times the normal reaction.

The dimensionless constant relating limiting friction to the normal reaction between two surfaces. It has no units and is usually denoted by the Greek letter mu.

A flat surface tilted at an angle to the horizontal. Weight is resolved parallel and perpendicular to the slope, giving components of magnitude mg sin(theta) and mg cos(theta).

The net force on a body equals the rate of change of its momentum; for constant mass this is force equals mass times acceleration. It links dynamics to kinematics through acceleration.

A dynamical equation formed by applying Newton's second law along a chosen direction, for example T minus mg equals ma for a particle moving vertically. Solving it gives acceleration or an unknown force.

The pulling force transmitted along a string, rope, or rod. In an inextensible string over a smooth pulley the tension is the same throughout, which is a key modelling assumption.

Two or more bodies joined by a string or in contact, moving with a common acceleration. Each body is treated separately with its own equation of motion, then the equations are solved together.

A measure of the turning effect of a force about a point, equal to the force times the perpendicular distance from the point to its line of action. Measured in newton-metres.

For a body in rotational equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that same point. Used to locate centres of mass and find unknown forces.

A pair of equal, opposite, and parallel forces whose lines of action do not coincide. A couple produces rotation only, with moment equal to one force times the perpendicular separation.

The point at which the whole mass of a body may be considered concentrated for the purpose of analysing translational motion and weight. For a uniform lamina it often lies on an axis of symmetry.

Rotation of a body about an edge when the vertical through its centre of mass falls outside the base. The condition for stability is that this vertical line passes through the base.

The vector giving the change in position of a particle from a fixed origin, with both magnitude and direction. Unlike distance, it depends only on the starting and finishing points.

The rate of change of displacement with respect to time, a vector quantity. Its sign or direction matters: a particle moving in the negative direction has negative velocity.

The rate of change of velocity with respect to time, a vector measured in metres per second squared. It is the second derivative of displacement with respect to time.

The five constant-acceleration formulae linking displacement, initial velocity, final velocity, acceleration, and time. They apply only when acceleration is uniform, such as free fall under gravity.

Motion of a body projected into the air and subject only to gravity. Horizontal and vertical motions are treated independently, with constant horizontal velocity and constant vertical acceleration g.

The velocity of one body as observed from another, found by vector subtraction of the observer's velocity from the body's velocity. Often solved with a velocity triangle or diagram.

A quantity having both magnitude and direction, such as force or velocity. In two dimensions it is written as a column of components or as a multiple of unit vectors i and j.

A vector of magnitude one, used to specify direction. Any vector can be written as its magnitude multiplied by a unit vector in its direction.

A vector giving the position of a point relative to a fixed origin, usually written in terms of i and j. The vector from A to B equals the position vector of B minus that of A.

A product of two vectors giving a scalar, equal to the product of their magnitudes and the cosine of the angle between them. It is zero when the vectors are perpendicular.

The instantaneous rate of change of a function, interpreted geometrically as the gradient of the tangent to its curve. Written as dy/dx, it is found from first principles or by standard rules.

The rule for differentiating a composite function: differentiate the outer function, then multiply by the derivative of the inner function. Written as dy/dx equals dy/du times du/dx.

A rule for differentiating a product of two functions: the derivative of the first times the second, plus the first times the derivative of the second. Often remembered as u dv plus v du.

A rule for differentiating one function divided by another, giving denominator times derivative of numerator minus numerator times derivative of denominator, all over the denominator squared.

A point on a curve where the gradient is zero, so the tangent is horizontal. It may be a maximum, a minimum, or a point of inflection.

The derivative of the derivative, measuring how the gradient itself changes. Its sign at a stationary point distinguishes a maximum from a minimum.

The integral of a function between two limits, evaluated as the difference of an antiderivative at the upper and lower limits. It represents the signed area between the curve and the axis.

The region between a graph and the horizontal axis, found by integrating the function between the relevant limits. Parts below the axis contribute negatively, so care with signs is needed.

A technique for evaluating integrals by changing the variable to simplify the integrand, remembering to change the differential and the limits accordingly. It reverses the chain rule.

A technique based on reversing the product rule, integrating the product of two functions by choosing one part to differentiate and the other to integrate. Useful for products of algebraic and exponential or trigonometric functions.

The volume swept out when a curve is rotated about an axis, found by integrating pi times the square of the function between limits. The axis of rotation determines which variable is used.

A sequence in which each term after the first differs from the previous one by a fixed constant. The nth term is a plus n minus one times d, and the sum has a standard formula.

A sequence in which each term is a fixed multiple of the previous one. The nth term is a times r to the power n minus one, and convergence depends on the size of the ratio.

The limiting value of the sum of a geometric series as the number of terms grows without bound, existing only when the common ratio is strictly between minus one and one. It equals a divided by one minus r.

The expansion of a power of a sum of two terms as a series, using binomial coefficients. For a positive integer power the series terminates; for other powers it is infinite and valid only in a range.

The decomposition of a rational expression into a sum of simpler fractions with linear or quadratic denominators. It makes integration and series expansion of such expressions possible.

The inverse of an exponential function: the logarithm of a number to a base is the power to which the base must be raised to give that number. Laws of logarithms turn products into sums.

A logarithm taken to base e, written ln x. Its derivative is one over x, which makes it essential for integrating reciprocals and solving exponential equations.

A function of the form a to the power x, in which the variable is in the exponent. Its derivative is proportional to the function itself, which models growth and decay.

A unit of angle measure defined so that one full turn is two pi radians. In radians the derivative of sine is cosine and arc length equals radius times angle.

An equation true for all angles, such as the Pythagorean identity linking sine squared and cosine squared to one. Identities are used to simplify expressions and solve equations.

Formulae relating sides and angles of any triangle, used when the triangle is not right-angled. The sine rule pairs sides with opposite angles; the cosine rule links three sides to one angle.

The distance measured along a circular arc, equal to the radius multiplied by the angle in radians. It is the curved counterpart of a straight-line segment.