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Finding work done when forces change A graph of force versus displacement lets you determine the work done by a force This graphical method can be used to solve problems in which the force is changing Figure 10-5a shows the work done by a constant force of 200 N that is exerted to lift an object a distance of 150 m The work done by this constant force is represented by W Fd (200 N)(150 m) 300 J The shaded area under the graph is equal to (200 N)(150 m), or 300 J The area under a force-displacement graph is equal to the work done by that force, even if the force changes Figure 10-5b shows the force exerted by a spring, which varies linearly from 00 to 200 N as it is compressed 150 m The work done by the force that compressed the spring is the area under the graph, which is the area of a triangle, 1 (base)(altitude), or 2 W 1 (200 N)(150 m) 150 J 2 Work done by many forces Newton s second law of motion relates the net force on an object to its acceleration In the same way, the work-energy theorem relates the net work done on a system to its energy change If several forces are exerted on a system, calculate the work done by each force, and then add the results

Each pixel of the animations or movies you watch, and each letter of the instant messages you send presents your computer with several hundred equations Each equation must be solved in a few billionths of a second if it takes a bit longer, you might complain that your computer is slow Early Computers The earliest computers

could solve very complex arrays of equations, just as yours can, but it took them a lot longer to do so There were several reasons for this First, the mathematics of algorithms (problemsolving strategies) still was new Computer scientists were only beginning to learn how to arrange a particular problem, such as the conversion of a picture into an easily-transmittable form, so that it could be solved by a machine

function PIPHP_PageUpdated($page, $datafile) { $contents = @file_get_contents($page); if (!$contents) return FALSE; $checksum = md5($contents); if (file_exists($datafile)) { $rawfile = file_get_contents($datafile); $data = explode("\n", rtrim($rawfile)); $left = array_map("PIPHP_PU_F1", $data); $right = array_map("PIPHP_PU_F2", $data); $exists = -1; for ($j = 0 ; $j < count($left) ; ++$j) { if ($left[$j] == $page) { $exists = $j; if ($right[$j] == $checksum) return 0; } } if ($exists > -1) { $rawfile = str_replace($right[$exists], $checksum, $rawfile); file_put_contents($datafile, $rawfile); return 1; } } else $rawfile = ""; file_put_contents($datafile, $rawfile "$page!1!$checksum\n");

Multiple-Hierarchy Dimension 2/2

Memory Third, electronic memories were extremely expensive You may know that a larger memory lets your computer work faster When one byte of memory required eight circuit boards, 1024 bytes (or 1 K) of memory was enormous Because memory was so precious, computer programs had to be written with great cleverness Astronants got to the Moon with 64 K of memory in Apollo s on-board computers

Machine Size Second, the machines were physically large Computers work by switching patterns of electric currents that represent binary numbers A 16-bit machine works with binary numbers that are 16 bits long If a 64-bit number must be dealt with, the machine must repeat the same operation four times A 32-bit machine would have to repeat the operation only twice, thus making it that much faster But a 32-bit machine is four times the size of a 16-bit machine; that is, it has four times as many wires and transistor switches, and even 8-bit machines were the size of the old UNIVAC shown above Moreover, current travels along wires at speeds no greater than about two-thirds the speed of light This is a long time if the computer wires are 15 m long and must move information in less than 10 9 s

When Gordon Moore and others invented the integrated circuit around 1960, the size and cost of computer circuitry dropped drastically Physically smaller, and thus faster, machines could be built and very large memories became possible Today, the transistors on a chip are now smaller than bacteria The cost and size of computers have dropped so much that your cell phone has far more computing power than most big office machines of the 1970s