1.
Muscle contraction
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Muscle contraction is the activation of tension-generating sites within muscle fibers. The termination of muscle contraction is followed by muscle relaxation, which is a return of the fibers to their low tension-generating state. Muscle contractions can be described based on two variables, length and tension, a muscle contraction is described as isometric if the muscle tension changes but the muscle length remains the same. In contrast, a muscle contraction is isotonic if muscle length changes, if the muscle length shortens, the contraction is concentric, if the muscle length lengthens, the contraction is eccentric. In natural movements that underlie locomotor activity, muscle contractions are multifaceted as they are able to produce changes in length, therefore, neither length nor tension is likely to remain the same in muscles that contract during locomotor activity. In vertebrates, skeletal muscle contractions are neurogenic as they require synaptic input from neurons to produce muscle contractions. A single motor neuron is able to innervate multiple muscle fibers, once innervated, the protein filaments within each skeletal muscle fiber slide past each other to produce a contraction, which is explained by the sliding filament theory. The contraction produced can be described as a twitch, summation, or tetanus, in skeletal muscles, muscle tension is at its greatest when the muscle is stretched to an intermediate length as described by the length-tension relationship. Unlike skeletal muscle, the contractions of smooth and cardiac muscles are myogenic, the mechanisms of contraction in these muscle tissues are similar to those in skeletal muscle tissues. Muscle contractions can be described based on two variables, force and length, force itself can be differentiated as either tension or load. Muscle tension is the force exerted by the muscle on an object whereas a load is the force exerted by an object on the muscle, when muscle tension changes without any corresponding changes in muscle length, the muscle contraction is described as isometric. If the muscle length changes while muscle tension remains the same, in an isotonic contraction, the muscle length can either shorten to produce a concentric contraction or lengthen to produce an eccentric contraction. Furthermore, if the muscle length shortens, the contraction is concentric, but if the muscle length lengthens, then the contraction is eccentric. In natural movements that underlie locomotor activity, muscle contractions are multifaceted as they are able to produce changes in length, therefore, neither length nor tension is likely to remain constant when the muscle is active during locomotor activity. An isometric contraction of a muscle generates tension without changing length, an example can be found when the muscles of the hand and forearm grip an object, the joints of the hand do not move, but muscles generate sufficient force to prevent the object from being dropped. In isotonic contraction, the tension in the muscle remains constant despite a change in muscle length and this occurs when a muscles force of contraction matches the total load on the muscle. In concentric contraction, muscle tension is sufficient to overcome the load, and this occurs when the force generated by the muscle exceeds the load opposing its contraction. During a concentric contraction, a muscle is stimulated to contract according to the sliding filament theory and this occurs throughout the length of the muscle, generating a force at the origin and insertion, causing the muscle to shorten and changing the angle of the joint
2.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
4.
Capillary wave
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A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension. Capillary waves are common in nature, and are referred to as ripples. The wavelength of capillary waves in water is less than a few centimeters. A higher wavelength on an interface will result in gravity–capillary waves which are influenced by both the effects of surface tension and gravity, as well as by fluid inertia. Ordinary gravity waves have a higher wavelength. When generated by wind in open water, a nautical name for them is cats paw waves. Light breezes which stir up such small ripples are also referred to as cats paws. On the open ocean, much larger ocean surface waves may result from coalescence of smaller wind-caused ripple-waves, the wavelength is λ =2 π k. For the boundary between fluid and vacuum, the dispersion relation reduces to ω2 = σ ρ | k |3, in general, waves are also affected by gravity and are then called gravity–capillary waves. Notice the factor / in the first term is the Atwood number, for large wavelengths, only the first term is relevant and one has gravity waves. Phase velocity is two thirds of group velocity in this limit, between these two limits, an interesting and common situation occurs when the dispersion caused by gravity cancels out the dispersion due to the capillary effect. At a certain wavelength, the group velocity equals the phase velocity, at precisely this same wavelength, the phase velocity of gravity–capillary waves as a function of wavelength has a minimum. Waves with wavelengths smaller than this critical wavelength λm are dominated by surface tension. The value of this wavelength and the minimum phase speed cm are. For the air–water interface, λm is found to be 1.7 cm, if one drops a small stone or droplet into liquid, the waves then propagate outside an expanding circle of fluid at rest, this circle is a caustics which corresponds to the minimal group velocity. The derivation of the dispersion relation is therefore quite involved. Therefore, first the assumptions involved are pointed out, there are three contributions to the energy, due to gravity, to surface tension, and to hydrodynamics. The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of g and σ, for gravity, an assumption is made of the density of the fluids being constant, and likewise g
