The generalization of a mathematical puzzle
about some very unlucky
The generalization of a mathematical puzzle
about some very unlucky
Romans are famous for their advanced engineering accomplishments, although some of their own inventions were improvements on older ideas, concepts and inventions. Technology for bringing running water into cities was developed in the east, but transformed by the Romans into a technology inconceivable in Greece. The architecture used in Rome was strongly influenced by Greek and Etruscan sources.
Roads were common at that time, but the Romans improved their design and perfected the construction to the extent that many of their roads are still in use today. Their accomplishments surpassed most other civilizations of their time, and after their time, and many of their structures have withstood the test of time to inspire others, especially during the Renaissance. Moreover, their contributions were described in some detail by authors such as Vitruvius, Frontinus and Pliny the Elder, so there is a printed record of their many inventions and achievements.
In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas, and to some extent, plastic solids. Fluids are substances that have zero shear modulus, or, in simpler terms, a fluid is a substance which cannot resist any shear force applied to it.
Although the term “fluid” includes both the liquid and gas phases, in common usage, “fluid” is often used as a synonym for “liquid”, with no implication that gas could also be present. For example, “brake fluid” is hydraulic oil and will not perform its required incompressible function if there is gas in it. This colloquial usage of the term is also common in medicine and in nutrition (“take plenty of fluids”).
Liquids form a free surface (that is, a surface not created by the container) while gases do not. The distinction between solids and fluid is not entirely obvious. The distinction is made by evaluating the viscosity of the substance. Silly Putty can be considered to behave like a solid or a fluid, depending on the time period over which it is observed. It is best described as a viscoelastic fluid. There are many examples of substances proving difficult to classify. A particularly interesting one is pitch, as demonstrated in the pitch drop experiment currently running at the University of Queensland.
Acceleration, rate at which velocity changes with time, in terms of both speed and direction. A point or an object moving in a straight line is accelerated if it speeds up or slows down. Motion on a circle is accelerated even if the speed is constant, because the direction is continually changing. For all other kinds of motion, both effects contribute to the acceleration.
Because acceleration has both a magnitude and a direction, it is a vector quantity. Velocity is also a vector quantity. Acceleration is defined as the change in the velocity vector in a time interval, divided by the time interval. Instantaneous acceleration (at a precise moment and location) is given by the limit of the ratio of the change in velocity during a given time interval to the time interval as the time interval goes to zero .
Velocity is a quantity that designates how fast and in what direction a point is moving. A point always moves in a direction that is tangent to its path; for a circular path, for example, its direction at any instant is perpendicular to a line from the point to the centre of the circle (a radius). The magnitude of the velocity (i.e., the speed) is the time rate at which the point is moving along its path.
If a point moves a certain distance along its path in a given time interval, its average speed during the interval is equal to the distance moved divided by the time taken. A train that travels 100 km in 2 hours, for example, has an average speed of 50 km per hour.
Lens, in optics, piece of glass or other transparent substance that is used to form an image of an object by focusing rays of light from the object. A lens is a piece of transparent material, usually circular in shape, with two polished surfaces, either or both of which is curved and may be either convex (bulging) or concave (depressed). The curves are almost always spherical; i.e., the radius of curvature is constant. A lens has the valuable property of forming images of objects situated in front of it. Single lenses are used in eyeglasses, contact lenses, pocket magnifiers, projection condensers, signal lights, viewfinders, and on simple box cameras. More often a number of lenses made of different materials are combined together as a compound lens in a tube to permit the correction of aberrations. Compound lenses are used in such instruments as cameras, microscopes, and telescopes.
A single lens has two precisely regular opposite surfaces; either both surfaces are curved or one is curved and one is plane. Lenses may be classified according to their two surfaces as biconvex, plano-convex, concavo-convex (converging meniscus), biconcave, plano-concave, and convexo-concave (diverging meniscus). Because of the curvature of the lens surfaces, different rays of an incident light beam are refracted through different angles, so that an entire beam of parallel rays can be caused to converge on, or to appear to diverge from, a single point. This point is called the focal point, or principal focus, of the lens (often depicted in ray diagrams as F). Refraction of the rays of light reflected from or emitted by an object causes the rays to form a visual image of the object. This image may be either real—photographable or visible on a screen—or virtual—visible only upon looking into the lens, as in a microscope. The image may be much larger or smaller than the object, depending on the focal length of the lens and on the distance between the lens and the object. The focal length of a lens is the distance from the centre of the lens to the point at which the image of a distant object is formed. A long-focus lens forms a larger image of a distant object, while a short-focus lens forms a small image.
Usually the image formed by a single lens is not good enough for precise work in such fields as astronomy, microscopy, and photography; this is because the cone of rays emitted by a single point in a distant object is not united in a perfect point by the lens but instead forms a small patch of light. This and other innate imperfections in a lens’s image of a single object point are known as aberrations. To correct such aberrations, it is often necessary to combine in one mount several lens elements (single lenses), some of which may be convex and some concave.
Electromagnetic induction was discovered by Michael Faraday, published in 1831. It was discovered independently by Joseph Henry in 1832 .
In Faraday’s first experimental demonstration , he wrapped two wires around opposite sides of an iron ring or “torus” (an arrangement similar to a modern toroidal transformer).Based on his understanding of electromagnets, he expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. He saw a transient current, which he called a “wave of electricity”, when he connected the wire to the battery and another when he disconnected it. This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. Within two months, Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead (“Faraday’s disk”).
Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically. An exception was James Clerk Maxwell, who used Faraday’s ideas as the basis of his quantitative electromagnetic theory. In Maxwell’s model, the time varying aspect of electromagnetic induction is expressed as a differential equation, which Oliver Heaviside referred to as Faraday’s law even though it is slightly different from Faraday’s original formulation and does not describe motional EMF. Heaviside’s version is the form recognized today in the group of equations known as Maxwell’s equations.
In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.
Ashton’s theory of sound. In human physiology and psychology, sound is the reception of such waves and their perception by the brain. Humans can only hear sound waves as distinct pitches when the frequency lies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and is not perceptible by humans. Sound waves below 20 Hz are known as infrasound. Different animal species have varying hearing ranges.
Sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids (see Longitudinal and transverse waves, below). The sound waves are generated by a sound source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium. As the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, and displacement of the medium vary in time. At an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave. This is intuitively obvious for a solid, and the same is true for liquids and gases (that is, the vibrations of particles in the gas or liquid transport the vibrations, while the average position of the particles over time does not change). During propagation, waves can be reflected, refracted, or attenuated by the medium.
The symbol for a battery in a circuit diagram. It originated as a schematic drawing of the earliest type of battery, a voltaic pile.
An electric battery is a device consisting of one or more electrochemical cells with external connections provided to power electrical devices such as flashlights, smartphones, and electric cars. When a battery is supplying electric power, its positive terminal is the cathode and its negative terminal is the anode. The terminal marked negative is the source of electrons that when connected to an external circuit will flow and deliver energy to an external device. When a battery is connected to an external circuit, electrolytes are able to move as ions within, allowing the chemical reactions to be completed at the separate terminals and so deliver energy to the external circuit. It is the movement of those ions within the battery which allows current to flow out of the battery to perform work. Historically the term “battery” specifically referred to a device composed of multiple cells, however the usage has evolved additionally to include devices composed of a single cell.
Primary (single-use or “disposable”) batteries are used once and discarded; the electrode materials are irreversibly changed during discharge. Common examples are the alkaline battery used for flashlights and a multitude of portable electronic devices. Secondary (rechargeable) batteries can be discharged and recharged multiple times using an applied electric current; the original composition of the electrodes can be restored by reverse current. Examples include the lead-acid batteries used in vehicles and lithium-ion batteries used for portable electronics such as laptops and smartphones.
Batteries come in many shapes and sizes, from miniature cells used to power hearing aids and wristwatches to small, thin cells used in smartphones, to large lead acid batteries used in cars and trucks, and at the largest extreme, huge battery banks the size of rooms that provide standby or emergency power for telephone exchanges and computer data centers.
Batteries have much lower specific energy (energy per unit mass) than common fuels such as gasoline. In automobiles, this is somewhat offset by the higher efficiency of electric motors in producing mechanical work, compared to combustion engines.