The three rulers of decentralization are not included
Publish: 2021-04-20 21:53:41
1. A brief history of relativity http://www.newmind40.com/01_ 09 / SWH. HTM
relativity http://post..com/f?kz=11197627
relativity is the essence of physics. http://www.qglt.com/bbs/ReadFile?whichfile=304738&typeid=18
Einstein and relativity http://www.oursci.org/ency/physics/001.htm
special relativity is a theory based on the four-dimensional view of time and space, so we should understand the content of relativity, We should first have a general understanding of the concept of time and space of relativity. There are many kinds of multi-dimensional space in mathematics, but so far, the physical world we know is only four-dimensional, that is, three-dimensional space plus one-dimensional time. The high-dimensional space mentioned in modern microphysics has another meaning, only mathematical meaning, which is not discussed here
four dimensional space-time is the lowest dimension of the real world, and our world happens to be four-dimensional. As for the high-dimensional real space, at least now we can't perceive it. I mentioned an example in a post. When a ruler rotates in three-dimensional space (excluding time), its length does not change, but when it rotates, its coordinate values change, and the coordinates are related. The meaning of four-dimensional space-time is that time is the fourth dimensional coordinate, which is related to the spatial coordinate, that is to say, space-time is a unified and indivisible whole, and they are a relationship of "one ebb and the other"
four dimensional space-time is not limited to this. From the relationship between mass and energy, mass and energy are actually the same thing. Mass (or energy) is not independent, but related to the state of motion. For example, the greater the speed, the greater the mass. In the four-dimensional space-time, mass (or energy) is actually the fourth component of the four-dimensional momentum. Momentum is the quantity describing the motion of matter, so it is natural that mass is related to the state of motion. In four-dimensional space-time, momentum and energy are unified, which is called energy momentum four vectors. In addition, four-dimensional velocity, four-dimensional acceleration, four-dimensional force and four-dimensional form of electromagnetic field equations are defined in four-dimensional space-time. It is worth mentioning that the four-dimensional form of the electromagnetic field equations is more perfect, which completely unifies electricity and magnetism. The electric field and magnetic field are described by a unified electromagnetic field tensor. The physical laws of four-dimensional space-time are much more perfect than those of three-dimensional space-time, which shows that our world is indeed four-dimensional. It can be said that at least it is more perfect than Newtonian mechanics. At least because of its perfection, we can't doubt it
in the theory of relativity, time and space form an inseparable whole - four-dimensional space-time, and energy and momentum form an inseparable whole - four-dimensional momentum. This shows that there may be a deep connection between some seemingly unrelated quantities in nature. When we discuss general relativity in the future, we will see that there is also a profound connection between space-time and the four vectors of energy and momentum<
3 basic principle of special relativity
matter moves eternally in the interaction, there is no matter that does not move, and there is no matter that does not move. Because matter moves in the interaction, it is necessary to describe the motion in the mutual relationship of matter, and it is impossible to describe the motion in isolation. In other words, motion must have a reference, which is the frame of reference
Galileo once pointed out that the motion of a moving ship is indistinguishable from that of a stationary ship. That is to say, when you are in a closed cabin and completely isolated from the outside world, even if you have the most developed mind and the most advanced instruments, you can not perceive whether your ship is moving at a constant speed or stationary. There is no way to perceive the size of the speed, because there is no reference. For example, we don't know the overall motion of our universe, because the universe is closed. Einstein cited it as the first basic principle of special relativity: the principle of special relativity. Its content is: inertial frames are completely equivalent and indistinguishable
the famous Michelson Morey experiment completely negates the ether theory of light and concludes that light has nothing to do with reference system. That is to say, whether you are standing on the ground or on a speeding train, the measured speed of light is the same. This is the second basic principle of special relativity, the invariance of the speed of light
from these two basic principles, we can directly dece all the contents of special relativity, such as coordinate transformation, velocity transformation and so on. For example, the change of speed is contradictory to the traditional law, but it has been proved correct in practice. For example, the speed of a train is 10m / s, and the speed of a person on the train is 10m / s relative to the car. The speed of the person on the ground is not 20m / s, but (20-10 ^ (- 15)) m / s left and right. In general, this relativistic effect can be ignored, but it increases obviously when the speed of light is close to the speed of light. For example, the train speed is 0. 99 times the speed of light, the speed of man is 0. 99 times the speed of light, so the ground observer's conclusion is not 1. 98 times the speed of light, but zero. 999949 times the speed of light. The people in the car didn't slow down when they saw the light coming from behind. It was also the speed of light for him. Therefore, in this sense, the speed of light can not be surpassed, because no matter in that reference frame, the speed of light is constant. Velocity transformation has been proved to be impeccable by numerous experiments in particle physics. Because of this unique property of light, it is selected as the only ruler of four-dimensional space-time<
4 special relativity effect
according to the principle of special relativity, inertial frames are completely equivalent. Therefore, in the same inertial frame, there is a unified time, which is called simultaneity. Relativity proves that in different inertial frames, there is no unified simultaneity, that is, two events (time and space points) are in the same relevant frame at the same time, In another inertial frame, there may be different Simultaneities, which is the relativity of simultaneity. In the inertial frame, the time course of the same physical process is exactly the same. If the same physical process is used to measure time, a unified time can be obtained in the whole inertial frame. In the future general relativity, we can know that space-time is not uniform in non inertial frame, that is to say, there is no unified time in the same non inertial frame, so we can not establish a unified simultaneity< It is found that the time progress of moving inertial frame is slow, which is the so-called clock slow effect. It can be understood that a moving clock moves more slowly than a stationary clock. Moreover, the faster the moving speed is, the slower the clock moves. When it approaches the speed of light, the clock almost stops
the length of a ruler is in an inertial frame; At the same time & quot; The difference between the coordinate values of the two endpoints. Due to & quot; At the same time & quot; The length measured in different inertial frames is also different. The theory of relativity proves that the ruler moving in the direction of the length of the ruler is shorter than the ruler at rest. This is the so-called ruler shrinking effect. When the speed is close to the speed of light, the ruler shrinks to a point
5 special relativity effect 2
from the above statement, we can see that the principle of clock slowness and ruler contraction is that the time schele is relative. In other words, the time schele is related to the reference frame. This fundamentally negates Newton's view of absolute time and space. Relativity holds that absolute time does not exist, but time is still an objective quantity. For example, in the twin ideal experiment to be discussed in the next issue, the elder brother is 15 years old when he comes back from the spaceship, and the younger brother may be 45 years old, indicating that time is relative, but the elder brother does live for 15 years, and the younger brother also confirms that he has lived for 45 years, which has nothing to do with the reference system, and the time is & quot; Absolute & quot;. This shows that no matter what the state of motion of an object is, the time it experiences is an objective quantity and absolute, which is called fixed time. That is to say, no matter what form of exercise you take, you think that your coffee drinking speed is normal, and your life is not disturbed, but others may see that it took you 100 years to drink coffee, and it took only one second from putting down the cup to dying< After the birth of the theory of relativity, there was once a very interesting problem twin paradox. Twins a and B. A is on the earth. B goes to Star Trek by rocket and returns to earth after a long time. Einstein asserted by the theory of relativity that the two experienced different times. When they met again, B would be younger than a. Many people have doubts that a sees B in sports, and B sees a in sports. Why can't a be younger than B? Since the earth can be approximated as an inertial frame, B has to undergo the process of acceleration and deceleration, and it is a reference frame of variable acceleration motion, which is very complicated to discuss. Therefore, many people mistakenly think that the theory of relativity is a self contradictory theory. If we use the concept of space-time map and world line to discuss this problem, it will be much easier, just need to use a lot of mathematical knowledge and formulas. This is just a language to describe the simplest case. However, only language can not be more detailed details, interested please refer to some books on relativity. Our conclusion is that B is younger than a in any reference frame
in order to simplify the problem, only this case is discussed. The rocket accelerates to sub light speed in a very short time, and after flying for a period of time, it turns around in a very short time, then flies for a period of time, and decelerates in a very short time to meet the earth. The purpose of this treatment is to omit the effects of acceleration and deceleration. It is well discussed in the earth reference system that the rocket is always a moving clock, and B is younger than a when it meets again. In the rocket reference frame, the earth is a moving clock in the process of constant velocity, and the time process is slower than that in the rocket, but the most important thing is the process of the rocket turning around. In the process of U-turn, the earth from the far behind the rocket after a very short time across half a circle, to the far in front of the rocket. This is a & quot; Superluminal & quot; Process. But this superluminal speed is not contradictory to the theory of relativity; Superluminal & quot; It can't transmit any information. It's not really superluminal. Without this U-turn process, the rocket and the earth would not meet. Because there is no unified time for different reference frames, it is impossible to compare their ages, only when they meet. After the rocket turns around, B can't directly accept a's message, because it takes time for the message to be transmitted. The actual process that B saw was that in the process of turning around, the time schele of the earth speeded up sharply. In B's opinion, a reality is younger than B, and then it grows old quickly when turning around, and a grows old slower than itself when returning. When we met again, I was still younger than a. In other words, there is no logical contradiction in relativity<
7 Summary of special relativity
relativity requires the laws of physics to remain unchanged under coordinate transformation (Lorentz change). The classical electromagnetic theory can be incorporated into the framework of relativity without modification, while the situation of Newtonian mechanics only remains unchanged in the Galilean transformation, and the original concise form becomes extremely complex under the Lorentz transformation. Therefore, classical mechanics and mechanics need to be modified, and the situation of the modified mechanical system remains unchanged under Lorentz transformation, which is called relativistic mechanics
after the establishment of special relativity, it has played a great role in promoting physics. Moreover, it goes deep into the scope of quantum mechanics and becomes an indispensable theory for the study of high-speed particles. However, behind the success, there are two principles left behind
relativity http://post..com/f?kz=11197627
relativity is the essence of physics. http://www.qglt.com/bbs/ReadFile?whichfile=304738&typeid=18
Einstein and relativity http://www.oursci.org/ency/physics/001.htm
special relativity is a theory based on the four-dimensional view of time and space, so we should understand the content of relativity, We should first have a general understanding of the concept of time and space of relativity. There are many kinds of multi-dimensional space in mathematics, but so far, the physical world we know is only four-dimensional, that is, three-dimensional space plus one-dimensional time. The high-dimensional space mentioned in modern microphysics has another meaning, only mathematical meaning, which is not discussed here
four dimensional space-time is the lowest dimension of the real world, and our world happens to be four-dimensional. As for the high-dimensional real space, at least now we can't perceive it. I mentioned an example in a post. When a ruler rotates in three-dimensional space (excluding time), its length does not change, but when it rotates, its coordinate values change, and the coordinates are related. The meaning of four-dimensional space-time is that time is the fourth dimensional coordinate, which is related to the spatial coordinate, that is to say, space-time is a unified and indivisible whole, and they are a relationship of "one ebb and the other"
four dimensional space-time is not limited to this. From the relationship between mass and energy, mass and energy are actually the same thing. Mass (or energy) is not independent, but related to the state of motion. For example, the greater the speed, the greater the mass. In the four-dimensional space-time, mass (or energy) is actually the fourth component of the four-dimensional momentum. Momentum is the quantity describing the motion of matter, so it is natural that mass is related to the state of motion. In four-dimensional space-time, momentum and energy are unified, which is called energy momentum four vectors. In addition, four-dimensional velocity, four-dimensional acceleration, four-dimensional force and four-dimensional form of electromagnetic field equations are defined in four-dimensional space-time. It is worth mentioning that the four-dimensional form of the electromagnetic field equations is more perfect, which completely unifies electricity and magnetism. The electric field and magnetic field are described by a unified electromagnetic field tensor. The physical laws of four-dimensional space-time are much more perfect than those of three-dimensional space-time, which shows that our world is indeed four-dimensional. It can be said that at least it is more perfect than Newtonian mechanics. At least because of its perfection, we can't doubt it
in the theory of relativity, time and space form an inseparable whole - four-dimensional space-time, and energy and momentum form an inseparable whole - four-dimensional momentum. This shows that there may be a deep connection between some seemingly unrelated quantities in nature. When we discuss general relativity in the future, we will see that there is also a profound connection between space-time and the four vectors of energy and momentum<
3 basic principle of special relativity
matter moves eternally in the interaction, there is no matter that does not move, and there is no matter that does not move. Because matter moves in the interaction, it is necessary to describe the motion in the mutual relationship of matter, and it is impossible to describe the motion in isolation. In other words, motion must have a reference, which is the frame of reference
Galileo once pointed out that the motion of a moving ship is indistinguishable from that of a stationary ship. That is to say, when you are in a closed cabin and completely isolated from the outside world, even if you have the most developed mind and the most advanced instruments, you can not perceive whether your ship is moving at a constant speed or stationary. There is no way to perceive the size of the speed, because there is no reference. For example, we don't know the overall motion of our universe, because the universe is closed. Einstein cited it as the first basic principle of special relativity: the principle of special relativity. Its content is: inertial frames are completely equivalent and indistinguishable
the famous Michelson Morey experiment completely negates the ether theory of light and concludes that light has nothing to do with reference system. That is to say, whether you are standing on the ground or on a speeding train, the measured speed of light is the same. This is the second basic principle of special relativity, the invariance of the speed of light
from these two basic principles, we can directly dece all the contents of special relativity, such as coordinate transformation, velocity transformation and so on. For example, the change of speed is contradictory to the traditional law, but it has been proved correct in practice. For example, the speed of a train is 10m / s, and the speed of a person on the train is 10m / s relative to the car. The speed of the person on the ground is not 20m / s, but (20-10 ^ (- 15)) m / s left and right. In general, this relativistic effect can be ignored, but it increases obviously when the speed of light is close to the speed of light. For example, the train speed is 0. 99 times the speed of light, the speed of man is 0. 99 times the speed of light, so the ground observer's conclusion is not 1. 98 times the speed of light, but zero. 999949 times the speed of light. The people in the car didn't slow down when they saw the light coming from behind. It was also the speed of light for him. Therefore, in this sense, the speed of light can not be surpassed, because no matter in that reference frame, the speed of light is constant. Velocity transformation has been proved to be impeccable by numerous experiments in particle physics. Because of this unique property of light, it is selected as the only ruler of four-dimensional space-time<
4 special relativity effect
according to the principle of special relativity, inertial frames are completely equivalent. Therefore, in the same inertial frame, there is a unified time, which is called simultaneity. Relativity proves that in different inertial frames, there is no unified simultaneity, that is, two events (time and space points) are in the same relevant frame at the same time, In another inertial frame, there may be different Simultaneities, which is the relativity of simultaneity. In the inertial frame, the time course of the same physical process is exactly the same. If the same physical process is used to measure time, a unified time can be obtained in the whole inertial frame. In the future general relativity, we can know that space-time is not uniform in non inertial frame, that is to say, there is no unified time in the same non inertial frame, so we can not establish a unified simultaneity< It is found that the time progress of moving inertial frame is slow, which is the so-called clock slow effect. It can be understood that a moving clock moves more slowly than a stationary clock. Moreover, the faster the moving speed is, the slower the clock moves. When it approaches the speed of light, the clock almost stops
the length of a ruler is in an inertial frame; At the same time & quot; The difference between the coordinate values of the two endpoints. Due to & quot; At the same time & quot; The length measured in different inertial frames is also different. The theory of relativity proves that the ruler moving in the direction of the length of the ruler is shorter than the ruler at rest. This is the so-called ruler shrinking effect. When the speed is close to the speed of light, the ruler shrinks to a point
5 special relativity effect 2
from the above statement, we can see that the principle of clock slowness and ruler contraction is that the time schele is relative. In other words, the time schele is related to the reference frame. This fundamentally negates Newton's view of absolute time and space. Relativity holds that absolute time does not exist, but time is still an objective quantity. For example, in the twin ideal experiment to be discussed in the next issue, the elder brother is 15 years old when he comes back from the spaceship, and the younger brother may be 45 years old, indicating that time is relative, but the elder brother does live for 15 years, and the younger brother also confirms that he has lived for 45 years, which has nothing to do with the reference system, and the time is & quot; Absolute & quot;. This shows that no matter what the state of motion of an object is, the time it experiences is an objective quantity and absolute, which is called fixed time. That is to say, no matter what form of exercise you take, you think that your coffee drinking speed is normal, and your life is not disturbed, but others may see that it took you 100 years to drink coffee, and it took only one second from putting down the cup to dying< After the birth of the theory of relativity, there was once a very interesting problem twin paradox. Twins a and B. A is on the earth. B goes to Star Trek by rocket and returns to earth after a long time. Einstein asserted by the theory of relativity that the two experienced different times. When they met again, B would be younger than a. Many people have doubts that a sees B in sports, and B sees a in sports. Why can't a be younger than B? Since the earth can be approximated as an inertial frame, B has to undergo the process of acceleration and deceleration, and it is a reference frame of variable acceleration motion, which is very complicated to discuss. Therefore, many people mistakenly think that the theory of relativity is a self contradictory theory. If we use the concept of space-time map and world line to discuss this problem, it will be much easier, just need to use a lot of mathematical knowledge and formulas. This is just a language to describe the simplest case. However, only language can not be more detailed details, interested please refer to some books on relativity. Our conclusion is that B is younger than a in any reference frame
in order to simplify the problem, only this case is discussed. The rocket accelerates to sub light speed in a very short time, and after flying for a period of time, it turns around in a very short time, then flies for a period of time, and decelerates in a very short time to meet the earth. The purpose of this treatment is to omit the effects of acceleration and deceleration. It is well discussed in the earth reference system that the rocket is always a moving clock, and B is younger than a when it meets again. In the rocket reference frame, the earth is a moving clock in the process of constant velocity, and the time process is slower than that in the rocket, but the most important thing is the process of the rocket turning around. In the process of U-turn, the earth from the far behind the rocket after a very short time across half a circle, to the far in front of the rocket. This is a & quot; Superluminal & quot; Process. But this superluminal speed is not contradictory to the theory of relativity; Superluminal & quot; It can't transmit any information. It's not really superluminal. Without this U-turn process, the rocket and the earth would not meet. Because there is no unified time for different reference frames, it is impossible to compare their ages, only when they meet. After the rocket turns around, B can't directly accept a's message, because it takes time for the message to be transmitted. The actual process that B saw was that in the process of turning around, the time schele of the earth speeded up sharply. In B's opinion, a reality is younger than B, and then it grows old quickly when turning around, and a grows old slower than itself when returning. When we met again, I was still younger than a. In other words, there is no logical contradiction in relativity<
7 Summary of special relativity
relativity requires the laws of physics to remain unchanged under coordinate transformation (Lorentz change). The classical electromagnetic theory can be incorporated into the framework of relativity without modification, while the situation of Newtonian mechanics only remains unchanged in the Galilean transformation, and the original concise form becomes extremely complex under the Lorentz transformation. Therefore, classical mechanics and mechanics need to be modified, and the situation of the modified mechanical system remains unchanged under Lorentz transformation, which is called relativistic mechanics
after the establishment of special relativity, it has played a great role in promoting physics. Moreover, it goes deep into the scope of quantum mechanics and becomes an indispensable theory for the study of high-speed particles. However, behind the success, there are two principles left behind
2. The inner city of Lishu mountain, right? It's enough to find three yardsticks. Go to liaori's home in the inner city, use the items in the plot items column, and put the ruler on the table to open the door of Lishu mountain.
3. Unknown_Error
4. Slide rule, or slide rule, usually refers to logarithmic slide rule, is an analog computer, which is usually composed of three mutually locked calibrated strips and a sliding window (called cursor). It was widely used before the 1970s, and then it was replaced by electronic calculators and became obsolete technology
basic concepts
in its most basic form, the slide rule uses two logarithmic scales for multiplication and division, which are common operations that are time-consuming and error prone on paper. The user determines the position of the decimal point in the result by estimation. In calculations involving addition, subtraction, multiplication and division, addition and subtraction are performed on paper, not on a ruler
in fact, even the most basic students use the slide rule with far more than two scales. Most of the rulers are composed of three straight bars, which are parallel aligned and locked to each other, so that the middle bar can slide along the length direction relative to the other two. The outer two are fixed so that their relative positions remain unchanged. Some rulers (& quot; Double sided & quot; There are scales on both sides of the ruler and the slide bar. Some have scales on one side of the outer bar and on both sides of the slide bar. The others have scales on only one side; Single side & quot; Type). A slide mark has one or more vertical alignment lines for recording intermediate results on any scale, and can also be used to find corresponding points on non adjacent scales
more complex rulers can perform other calculations, such as square roots, exponents, logarithms, and trigonometric functions
generally, mathematical calculation is carried out by aligning the marks on the sliding rod with those on other fixed rods, and the results are read out by observing the relative positions of other marks on the rod
operation
multiplication
the figure below shows a simplified slide rule with two logarithmic scales. In other words, a number x is printed on the distance of each ruler; Index & quot Mark with the number 1) where the distance is proportional to log X
logarithm transforms multiplication and division operations into addition and subtraction, thanks to the two rules log (XY) = log (x) + log (y) and log (x / y) = log (x) - log (y). Slide the top scale to the right by the distance of log (x), and align each digit y (at the position of log (y) on the top scale) with the position of log (x) + log (y) on the bottom scale. Because log (x) + log (y) = log (XY), the position of the bottom scale is marked as XY, which is the proct of X and y
the figure below shows 2 times any other number. The index (1) of the upper scale is aligned with the index (2) of the lower scale. This shifts the whole upper scale to the right by log (2). The number on the upper scale (multiplier) corresponds to the proct on the lower scale. For example, 3.5 of the upper scale is aligned with the proct of 7 of the lower scale, while 4 is aligned with 8, and so on, as shown in the figure:
the operation may & quot; Out of range & quot;. For example, the figure above shows that 7 of the upper scale does not have any number alignment of the lower scale, so it does not give 2 and 1376; 7. In this case, the user can move the upper scale to the left by 0.2 instead of 2, as shown in the figure below:
here, the user of the slide rule must remember to adjust the decimal point to get the final answer. We need to find 2 and 1376; 7, but we actually calculated 0.2 and 1376; 7 = 1.4 So the real answer is 14 instead of 1.4...
division
the figure below shows the calculation of 5.5/2. 2 of the top scale is on top of 5.5 of the bottom scale. The top one is just above quotient 2.75
other operations
in addition to the logarithmic scale, some rulers have other mathematical functions recorded on the auxiliary scale. The most common ones are trigonometric functions, which usually have sine and tangent, logarithm (log10), natural logarithm (LN) and exponential function (Ex). Some rulers contain a Pythagorean scale for calculating the sides of triangles and a scale for calculating circles. Others have scales for calculating hyperbolic functions. On the ruler, the scales and their marks are highly standardized. The main changes lie in which scales are included and the order in which they appear
A, B double decadal logarithmic scale
C, D single decadal logarithmic scale
k triple decadal logarithmic scale
CF, DF from π Instead of C and D scales starting from 1,
Ci, Di, DIF reciprocal scales, from right to left,
s is used to find sine and cosine on D scale,
t is used to find tangent on D and di scale,
st is used to find sine and tangent on small angle,
L linear scale, It is used with C and D scales to find the logarithm of base 10 and the power of 10
LLN a set of logarithmic scales for finding the natural logarithm and exponent
the front and back sides of a K & E 4081-3 ruler< There are single ten (C and D), double ten (A and b), and three ten (k) scales. For example, to compute x 2, we can find x on D and read its square on a. If we reverse this process, we can calculate the square root, and we can also calculate the power of 3, 1 / 3, 2 / 3, and 3 / 2. You must be careful when looking for the bottom x on the scale. Sometimes there will be more than one place where x appears. For example, there are two nines on scale A. to find the square root of nine, we must use the first nine; The second nine gives the square root of 90
trigonometric function
for angles between 5.7 and 90 degrees, the sine can be found by comparing the s scale with C or D. The S-Scale has a second set of angles (sometimes in different colors), increasing in the opposite direction, which is used to calculate cosine. Tangent can be compared with t scale and C, D scale, or CI scale for angle greater than 45. The sine and tangent of an angle less than 5.7 degrees can be found using the st scale. Inverse trigonometric functions can be found by the opposite process
logarithm and exponent
the logarithm and exponent with the base of 10 can be found by L scale, which is linear. When the bottom is e, use ll scale
physical design
standard straight slide rule
the length of the slide rule is the length of the scale, not the length of the whole equipment. The most common high-end slide rule is a 10 inch plex ruler, while the student ruler is often a 10 inch simplex ruler. A pocket ruler is usually five inches long
usually, the separator marks the accuracy of two significant digits, and then the user estimates the third digit. Some high-end rulers have a vernier with a magnifying glass that doubles the accuracy, making the 10 inch ruler as easy to use as the 20 inch ruler
there are some tips that can be used to increase convenience. The triangle scale sometimes has two marks, one black and one red, indicating the complementary angle, which is called & quot; Darmstadt" Style. Duplex slide rule often copies some scales on the back. The scale is often & quot; Split & quot; In order to achieve higher accuracy
special slide rules are designed for different engineering, commercial and banking purposes. These commonly used calculations are often directly represented by special scales, such as loan calculation, optimal purchase quantity, or special engineering equations
there are two basic types of circular slide rule, one has two cursors, the other has a movable disc and a cursor. The basic advantage of the circular slide rule is that the longest dimension is reced to about three times (i.e π Times). For example, a 10 cm round ruler and a 30 cm ordinary ruler have the same accuracy. The circle ruler also eliminates & quot; Cross border & quot; Calculation, because the scale is designed as & quot; Surround & quot; Of; They never need to redirect when the result is close to 1.0 - the ruler is always in bounds
circular rulers are mechanically stronger, move more smoothly, and are more accurate than straight ones because they rely on only one central bearing. The central support is rarely detached. The bearing also avoids scratching the surface and vernier. Only the most expensive straight slide rule offers these features
the scale with the highest accuracy is placed in the outermost ring. High end circular slide rule does not need & quot; Split & quot; It uses spiral scale for difficult scale (such as double logarithmic scale). An eight inch high-level circular ruler can have a 50 inch double logarithmic scale
technically speaking, the real disadvantage of the circular slide rule is that the less important scale is closer to the center, so the accuracy is poor. Historically, the main drawback of circular rulers is that they are not standard. Most students learn how to use the ruler on the straight one, and then they don't find it necessary to change to the round one
E6B is still in daily use around the world today. This is a circular slide rule first made in 1930s, which is used to help aircraft pilots calculate dead reckoning. This is still available in all flying shops and is still widely used. While GPS reces the use of dead reckoning in aviation, E6B is still used as the first choice or used dead reckoning instrument, and most flight schools take its mastery as the learning requirement
in 1952, Breitling, a Swiss watch company, introced a pilot's watch with an integrated circular ruler for flight time calculation: Breitling navitimer. Navitimer is called & quot; Aeronautical computer;, It features flight speed, climb speed, flight time, distance, and fuel consumption functions, as well as kilometer mile and gallon liter fuel capacity conversion functions
materials
traditionally, the slide rule is made of hardwood, such as mahogany or boxwood, plus glass or metal chutes. In 1895, a Japanese company began to use bamboo to make rulers, which are less sensitive to temperature and humidity. These bamboo rulers were introced to Sweden in the autumn of 1933[ http://runeberg.org/tektid/1933a/0348.html ]Probably just a little earlier than the introction of Germany
the best early slide rule is made of bamboo, which is stable in size, firm and natural self-lubricating. They use celluloid or plastic scales. Some are made of mahogany. The original slide rule is made of plastic, or aluminum painted with plastic
all advanced slide rules are engraved with numbers and scales, and then filled with paint or other resin. The quality of the painted or branded slide rule is a little poor, because the scale is easy to wear off
early cursors were glass with metal frames. Later cursors were acrylic or polycarbonate sliding on teflon bearings
the vernier with magnifying glass can help engineers with poor eyesight, and can double the accuracy of the ruler
the advanced slide rule is equipped with a delicate hook to prevent the ruler from accidentally detaching, and a buffer to prevent the scale or cursor from slipping when the ruler is thrown on the table
the recommended cleaning method of engraving scale is to gently scrub with steel velvet. For the paint slide rule, the safe way is to use a commercial window cleaner and a soft cloth<
History
slide rule was invented in 1620-1630, shortly after John Napier's concept of logarithm was published. Edmund Gunter of Oxford invented a calculation tool using a single logarithmic scale, which can be used for multiplication and division when combined with other measuring tools. In 1630, William oughtred of Cambridge invented the circular slide rule. In 1632, he combined two Gantt slide rules and combined them by hand to form a device that can be regarded as a modern slide rule. And Newton of his time
basic concepts
in its most basic form, the slide rule uses two logarithmic scales for multiplication and division, which are common operations that are time-consuming and error prone on paper. The user determines the position of the decimal point in the result by estimation. In calculations involving addition, subtraction, multiplication and division, addition and subtraction are performed on paper, not on a ruler
in fact, even the most basic students use the slide rule with far more than two scales. Most of the rulers are composed of three straight bars, which are parallel aligned and locked to each other, so that the middle bar can slide along the length direction relative to the other two. The outer two are fixed so that their relative positions remain unchanged. Some rulers (& quot; Double sided & quot; There are scales on both sides of the ruler and the slide bar. Some have scales on one side of the outer bar and on both sides of the slide bar. The others have scales on only one side; Single side & quot; Type). A slide mark has one or more vertical alignment lines for recording intermediate results on any scale, and can also be used to find corresponding points on non adjacent scales
more complex rulers can perform other calculations, such as square roots, exponents, logarithms, and trigonometric functions
generally, mathematical calculation is carried out by aligning the marks on the sliding rod with those on other fixed rods, and the results are read out by observing the relative positions of other marks on the rod
operation
multiplication
the figure below shows a simplified slide rule with two logarithmic scales. In other words, a number x is printed on the distance of each ruler; Index & quot Mark with the number 1) where the distance is proportional to log X
logarithm transforms multiplication and division operations into addition and subtraction, thanks to the two rules log (XY) = log (x) + log (y) and log (x / y) = log (x) - log (y). Slide the top scale to the right by the distance of log (x), and align each digit y (at the position of log (y) on the top scale) with the position of log (x) + log (y) on the bottom scale. Because log (x) + log (y) = log (XY), the position of the bottom scale is marked as XY, which is the proct of X and y
the figure below shows 2 times any other number. The index (1) of the upper scale is aligned with the index (2) of the lower scale. This shifts the whole upper scale to the right by log (2). The number on the upper scale (multiplier) corresponds to the proct on the lower scale. For example, 3.5 of the upper scale is aligned with the proct of 7 of the lower scale, while 4 is aligned with 8, and so on, as shown in the figure:
the operation may & quot; Out of range & quot;. For example, the figure above shows that 7 of the upper scale does not have any number alignment of the lower scale, so it does not give 2 and 1376; 7. In this case, the user can move the upper scale to the left by 0.2 instead of 2, as shown in the figure below:
here, the user of the slide rule must remember to adjust the decimal point to get the final answer. We need to find 2 and 1376; 7, but we actually calculated 0.2 and 1376; 7 = 1.4 So the real answer is 14 instead of 1.4...
division
the figure below shows the calculation of 5.5/2. 2 of the top scale is on top of 5.5 of the bottom scale. The top one is just above quotient 2.75
other operations
in addition to the logarithmic scale, some rulers have other mathematical functions recorded on the auxiliary scale. The most common ones are trigonometric functions, which usually have sine and tangent, logarithm (log10), natural logarithm (LN) and exponential function (Ex). Some rulers contain a Pythagorean scale for calculating the sides of triangles and a scale for calculating circles. Others have scales for calculating hyperbolic functions. On the ruler, the scales and their marks are highly standardized. The main changes lie in which scales are included and the order in which they appear
A, B double decadal logarithmic scale
C, D single decadal logarithmic scale
k triple decadal logarithmic scale
CF, DF from π Instead of C and D scales starting from 1,
Ci, Di, DIF reciprocal scales, from right to left,
s is used to find sine and cosine on D scale,
t is used to find tangent on D and di scale,
st is used to find sine and tangent on small angle,
L linear scale, It is used with C and D scales to find the logarithm of base 10 and the power of 10
LLN a set of logarithmic scales for finding the natural logarithm and exponent
the front and back sides of a K & E 4081-3 ruler< There are single ten (C and D), double ten (A and b), and three ten (k) scales. For example, to compute x 2, we can find x on D and read its square on a. If we reverse this process, we can calculate the square root, and we can also calculate the power of 3, 1 / 3, 2 / 3, and 3 / 2. You must be careful when looking for the bottom x on the scale. Sometimes there will be more than one place where x appears. For example, there are two nines on scale A. to find the square root of nine, we must use the first nine; The second nine gives the square root of 90
trigonometric function
for angles between 5.7 and 90 degrees, the sine can be found by comparing the s scale with C or D. The S-Scale has a second set of angles (sometimes in different colors), increasing in the opposite direction, which is used to calculate cosine. Tangent can be compared with t scale and C, D scale, or CI scale for angle greater than 45. The sine and tangent of an angle less than 5.7 degrees can be found using the st scale. Inverse trigonometric functions can be found by the opposite process
logarithm and exponent
the logarithm and exponent with the base of 10 can be found by L scale, which is linear. When the bottom is e, use ll scale
physical design
standard straight slide rule
the length of the slide rule is the length of the scale, not the length of the whole equipment. The most common high-end slide rule is a 10 inch plex ruler, while the student ruler is often a 10 inch simplex ruler. A pocket ruler is usually five inches long
usually, the separator marks the accuracy of two significant digits, and then the user estimates the third digit. Some high-end rulers have a vernier with a magnifying glass that doubles the accuracy, making the 10 inch ruler as easy to use as the 20 inch ruler
there are some tips that can be used to increase convenience. The triangle scale sometimes has two marks, one black and one red, indicating the complementary angle, which is called & quot; Darmstadt" Style. Duplex slide rule often copies some scales on the back. The scale is often & quot; Split & quot; In order to achieve higher accuracy
special slide rules are designed for different engineering, commercial and banking purposes. These commonly used calculations are often directly represented by special scales, such as loan calculation, optimal purchase quantity, or special engineering equations
there are two basic types of circular slide rule, one has two cursors, the other has a movable disc and a cursor. The basic advantage of the circular slide rule is that the longest dimension is reced to about three times (i.e π Times). For example, a 10 cm round ruler and a 30 cm ordinary ruler have the same accuracy. The circle ruler also eliminates & quot; Cross border & quot; Calculation, because the scale is designed as & quot; Surround & quot; Of; They never need to redirect when the result is close to 1.0 - the ruler is always in bounds
circular rulers are mechanically stronger, move more smoothly, and are more accurate than straight ones because they rely on only one central bearing. The central support is rarely detached. The bearing also avoids scratching the surface and vernier. Only the most expensive straight slide rule offers these features
the scale with the highest accuracy is placed in the outermost ring. High end circular slide rule does not need & quot; Split & quot; It uses spiral scale for difficult scale (such as double logarithmic scale). An eight inch high-level circular ruler can have a 50 inch double logarithmic scale
technically speaking, the real disadvantage of the circular slide rule is that the less important scale is closer to the center, so the accuracy is poor. Historically, the main drawback of circular rulers is that they are not standard. Most students learn how to use the ruler on the straight one, and then they don't find it necessary to change to the round one
E6B is still in daily use around the world today. This is a circular slide rule first made in 1930s, which is used to help aircraft pilots calculate dead reckoning. This is still available in all flying shops and is still widely used. While GPS reces the use of dead reckoning in aviation, E6B is still used as the first choice or used dead reckoning instrument, and most flight schools take its mastery as the learning requirement
in 1952, Breitling, a Swiss watch company, introced a pilot's watch with an integrated circular ruler for flight time calculation: Breitling navitimer. Navitimer is called & quot; Aeronautical computer;, It features flight speed, climb speed, flight time, distance, and fuel consumption functions, as well as kilometer mile and gallon liter fuel capacity conversion functions
materials
traditionally, the slide rule is made of hardwood, such as mahogany or boxwood, plus glass or metal chutes. In 1895, a Japanese company began to use bamboo to make rulers, which are less sensitive to temperature and humidity. These bamboo rulers were introced to Sweden in the autumn of 1933[ http://runeberg.org/tektid/1933a/0348.html ]Probably just a little earlier than the introction of Germany
the best early slide rule is made of bamboo, which is stable in size, firm and natural self-lubricating. They use celluloid or plastic scales. Some are made of mahogany. The original slide rule is made of plastic, or aluminum painted with plastic
all advanced slide rules are engraved with numbers and scales, and then filled with paint or other resin. The quality of the painted or branded slide rule is a little poor, because the scale is easy to wear off
early cursors were glass with metal frames. Later cursors were acrylic or polycarbonate sliding on teflon bearings
the vernier with magnifying glass can help engineers with poor eyesight, and can double the accuracy of the ruler
the advanced slide rule is equipped with a delicate hook to prevent the ruler from accidentally detaching, and a buffer to prevent the scale or cursor from slipping when the ruler is thrown on the table
the recommended cleaning method of engraving scale is to gently scrub with steel velvet. For the paint slide rule, the safe way is to use a commercial window cleaner and a soft cloth<
History
slide rule was invented in 1620-1630, shortly after John Napier's concept of logarithm was published. Edmund Gunter of Oxford invented a calculation tool using a single logarithmic scale, which can be used for multiplication and division when combined with other measuring tools. In 1630, William oughtred of Cambridge invented the circular slide rule. In 1632, he combined two Gantt slide rules and combined them by hand to form a device that can be regarded as a modern slide rule. And Newton of his time
5. Unknown_Error
6. Some angles can be divided into three equal parts, and some angles can't be divided into three equal parts. For example, 90 degrees and 45 degrees can be divided into three equal parts, but 60 degrees can't be divided into three equal parts. In other words, the ruler can only make some specific angles. The following content is extracted from the Internet. The problem of dividing any angle into three equal parts may appear earlier than the other two geometric problems, and no relevant records can be found in history. But no doubt it's very natural that we can think of it now. Five or six hundred years before the epoch, Greek mathematicians had already thought of the method of bisecting any angle. Just as we learned in geometry textbooks or geometric paintings: take the vertex of a known angle as the center of the circle, use an appropriate radius to make the arc intersection angle, and get two intersections on both sides. Then take these two points as the center of the circle, and draw an arc with an appropriate length as the radius, The intersection of the two arcs is connected with the top of the angle, and the known angle is divided into two equal parts. Since it is so easy to bisect a given angle, it will naturally change the problem slightly: how about trisection? In this way, the problem arises naturally. It has been proved that there is no solution to this problem on the premise of drawing with ruler. History of Trisection: in the 4th century BC, Ptolemy I established the capital of Alexandria. By virtue of his superior geographical environment, he developed maritime trade and handicrafts, and awarded academic awards. He built a large-scale "palace of art gods" as an academic research and teaching center; He also built the famous Alexandria library with a collection of 750000 volumes. Ptolemy I deeply understood the importance of developing science and culture. He invited famous scholars to Alexandria, where many famous Greek mathematicians came at that time. There is a round villa on the outskirts of Alexandria, in which a princess lives. There is a river in the middle of the round villa, and the princess's room is just built at the center of the circle. Villa north and south walls each opened a door, a bridge was built on the river, the location of the bridge and the north and South doors just in a straight line. The articles given by the king every day were transported in from the north gate, first put into the warehouse at the south gate, and then the princess sent someone to take them back from the south gate. One day, the princess asked the attendant, "which section of the road is farther from the north gate to my bedroom or from the north gate to the bridge?" The attendant didn't know, so he quickly went to measure, and the result was that the two sections of the road were the same distance. After a few years, the little princess grew up, and the king wanted to build a villa for her. The little princess proposed that her villa should be built like her sister's, with rivers, bridges and North and South gates. The king promised that the little princess's villa would start soon. When the south gate was built and the location of the bridge and the north gate was determined, there was a problem: how to make the distance from the north gate to the bedroom as far as the distance from the north gate to the bridge? Set the location of the north gate as Q, the location of the South Gate as P, the bedroom (Center) as O, and the bridge as K. to determine the location of the north gate and the bridge, the key is to make ∠ OPQ, and set the angle between Po and the river as α From QK = Qo, ∠ qko = ∠ QoK, but ∠ qko= α+ Therefore, in △ qko, ∠ qko + ∠ QoK + ∠ oqk= α+ ∠KPO+ α+ ∠KPO+∠KPO =3∠KPO+2 α=π I.e. KPO= π- two α/ 3 as long as you can put 180-2 α If the angle is divided into three equal parts, the location of the bridge and the north gate can be determined. The key to solve the problem is how to divide a corner equally. The craftsmen tried to determine the position of the bridge by drawing with ruler, but it took them a long time to solve it. So they went to Archimedes. Archimedes solved the problem of dividing a corner into three equal parts by making fixed marks on the ruler, so as to determine the position of the north gate. When everyone praised Archimedes for his greatness, Archimedes said, "this method of determining the location of the north gate is feasible, but it is only a temporary measure. It has flaws." Archimedes's so-called flaw is to mark on the ruler, which is equal to making a scale, which is not allowed in the rule of ruler and gauge drawing. This story raised a mathematical problem: how to divide any given angle by ruler and ruler, which was not even solved by Archimedes.
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