In mathematics, physics and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as in here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a ray (a line segment with a definite direction), or graphically as an arrow connecting an initial point A with a terminal point B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.

## Basic Mathematics

Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

## Units and Dimensions

Dimensions and units are commonly confused, even though the solution to all engineering problems must include units. Dimensions are physical quantities that can be measured, whereas units are arbitrary names that correlate to particular dimensions to make it relative (e.g., a dimension is length, whereas a meter is a relative unit that describes length). All units for the same dimension are related to each other through a conversion factor (e.g., 2.54 cm is exactly equal to 1 in). There are seven base dimensions that can be combined to describe all of the other dimensions of interest in engineering and physics, among other disciplines. In fluid mechanics, we generally pick length, mass, time, and temperature as base dimensions.

## Kinematics

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the “geometry of motion” and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined.

## Chapter 14 – Oscillations SHM

In mechanics and physics, simple harmonic motion is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object’s displacement magnitude and acts towards the object’s equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely. Simple harmonic motion can serve as a mathematical model for a variety of motions but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke’s law. Simple harmonic motion can also be used to model molecular vibration as well.

## Wave Motion

A mechanical wave is a disturbance in matter that transfers energy through the matter. A mechanical wave starts when matter is disturbed. A source of energy is needed to disturb matter and start a mechanical wave.

Q: Where does the energy come from in the water wave pictured above?

A: The energy comes from the falling droplets of water, which have kinetic energy because of their motion.

The Medium

The energy of a mechanical wave can travel only through matter. The matter through which the wave travels is called the medium (plural, media). The medium in the water wave pictured above is water, a liquid. But the medium of a mechanical wave can be any state of matter, even a solid.

Q: How do the particles of the medium move when a wave passes through them?

A: The particles of the medium just vibrate in place. As they vibrate, they pass the energy of the disturbance to the particles next to them, which pass the energy to the particles next to them, and so on. Particles of the medium don’t actually travel along with the wave. Only the energy of the wave travels through the medium.