Gimlet Rule - a simplified visual demonstration using one hand to correctly multiply two vectors. The geometry of the school course implies pupils' awareness of the scalar product. In physics, the vector is often found.
The Concept of a Vector
We believe that there is no sense in interpreting the driller rule with no knowledge of the definition of a vector. It is required to open a bottle - the knowledge of the correct actions will help. A vector is a mathematical abstraction that does not really exist, showing these signs:
- A directional segment, indicated by an arrow.
- The starting point is the point of action of the force described by the vector.
- The length of the vector is equal to the modulus of the force, the field, the other described quantities.
Not always affect force. The vectors describe the field. The simplest example is shown to schoolchildren by teachers of physics. We imply lines of magnetic field intensity. Vectors are usually drawn along a tangent along. In the illustrations of the action on the conductor with a current you will see straight lines.
Vector values are often devoid of application space, action centers are selected by agreement. The moment of force emanates from the axis of the shoulder. Required to simplify addition. Suppose that levers of different lengths are affected by different forces applied to the shoulders with a common axis. By simple addition, subtraction of moments, we find the result.
Vectors help to solve many everyday problems and, although they act as mathematical abstractions, they really work. On the basis of a number of regularities, it is possible to predict the future behavior of an object along with scalar values: the population of the population, the ambient temperature. Environmentalists are interested in directions, the speed of flight of birds. Displacement is a vector quantity.
The Gimlet Rule helps to find the vector product of vectors. This is not a tautology. Just the result of the action will also be a vector. The gimlet rule describes the direction to which the arrow will point. As for the module, you need to apply the formula. The gimlet rule is a simplified purely qualitative abstraction of a complex mathematical operation.
Analytical geometry in space
Everyone knows the problem: standing on one side of the river, determine the width of the channel. It seems to the mind incomprehensible, solved in two ways by the methods of simplest geometry, which students learn. Let's do a number of simple actions:
- Detect a prominent landmark on the opposite bank, an imaginary point: a tree trunk, the mouth of a stream that flows into a stream.
- At the right angle of the opposite bank line, make a notch on this side of the channel.
- Find a place from which the landmark is visible at an angle of 45 degrees to the shore.
- The width of the river is equal to the distance of the end point from the notch.
We use the tangent of an angle. Not necessarily equal to 45 degrees. Need more accuracy - the angle is better to take sharp. Just the tangent of 45 degrees is one, the solution of the problem is simplified.
Similarly, it is possible to find answers to burning questions. Even in the electron-controlled microcosm. We can definitely say one thing: to the uninitiated the rule of a gimlet, the vector product of vectors seems to be boring, boring. A handy tool that helps in understanding many processes. Most will be interested in the principle of operation of the electric motor( regardless of the design).Can easily be explained using the left-hand rule.
In many branches of science, two rules follow side by side: the left, the right hand. Vector product can sometimes be described in one way or another. Sounds vague, we suggest to immediately consider an example:
- Suppose an electron moves. A negatively charged particle plows a constant magnetic field. Obviously, the trajectory will be bent due to the Lorentz force.skeptics will argue, according to some scientists, the electron is not a particle, but rather a superposition of fields. But the principle of uncertainty Heisenberg consider another time. So, the electron moves:
By placing the right hand so that the vector of the magnetic field perpendicularly enters the palm, the extended fingers indicate the direction of the particle's flight, bent 90 degrees to the side, the thumb will stretch in the direction of the force. The right-hand rule is another expression of the gimlet rule. Synonyms. It sounds different, in fact - one.
- We give the phrase Wikipedia, giving a weirdness. When reflected in a mirror, the right three of vectors becomes left, then you need to apply the rule of the left hand instead of the right. The electron flew in one direction, according to the methods adopted in physics, the current moves in the opposite direction. As if reflected in the mirror, therefore the Lorentz force is already determined by the left hand rule:
If you place your left hand so that the magnetic field vector perpendicularly enters the palm, the extended fingers indicate the direction of the current flow, bent 90 degrees to the side of the thumb, stretching, indicating the action vectorstrength
You see, situations are similar, the rules are simple. How to remember which one to apply? The main principle of uncertainty of physics. The vector product is calculated in many cases, with one rule being applied.
What is the rule to apply
The words are synonyms: arm, screw, gimlet
First we analyze the word-synonyms, many began to ask themselves: if the story here should affect the gimlet, why does the text constantly touch the hands. We introduce the concept of the right three, the right coordinate system. Total, 5 words-synonyms.
It was necessary to find out the vector product of vectors, it turned out: this does not work in school. Clarify the situation inquisitive schoolchildren.
School graphics on the board are drawn in the Cartesian coordinate system X-Y.The horizontal axis( positive part) is directed to the right - we hope, the vertical axis points up. We take one step, getting the right three. Imagine: from the beginning of the counting, the Z axis looks to the class. Now the schoolchildren know the definition of the right three vectors.
In Wikipedia it is written: it is permissible to take left triples, right, when calculating a vector product, they disagree. Usmanov is categorical in this respect. With the permission of Alexander Evgenievich, we give an exact definition: a vector product is a vector that satisfies three conditions:
- The product module is equal to the product of the modules of the original vectors and the sine of the angle between them.
- The result vector is perpendicular to the original( together they form a plane).
- The triple vectors( in order of context) the right.
The right three know. So, if the X axis is the first vector, Y is the second, Z will be the result. Why was called the right three? Apparently, it is connected with screws, gimlets. If the imaginary gimlet is twisted along the shortest path, the first vector is the second vector, the translational axis of the cutting tool will begin to move in the direction of the resulting vector:
- The gimlet rule applies to the product of two vectors.
- The driller rule qualitatively indicates the direction of the resultant vector of this action. Quantitatively, the length is the expression mentioned( the product of the modules of the vectors and the sine of the angle between them).
Now everyone understands: the Lorentz force is found according to the rule of a left-handed thread. The vectors are collected by the left triple, if mutually orthogonal( perpendicular to one another), the left coordinate system is formed. On the board, the Z-axis would look in the direction of the view( from the audience behind the wall).
Simple techniques for memorizing the rules of the gimlet
People forget that it is easier to determine the Lorentz force by the rule of the gimlet with left-handed thread. One who wants to understand the principle of operation of an electric motor should double-click like nuts. Depending on the design, the number of rotor coils is significant, or the circuit degenerates, becoming a squirrel cage. Knowledge seekers are helped by the Lorentz rule, which describes the magnetic field where copper conductors move.
To memorize, let's present the physics of the process. Suppose an electron moves in a field. The rule of the right hand is applied to find the direction of the force. It is proved: the particle carries a negative charge. The direction of the force on the conductor is the left-hand rule, remember: physicists completely from the left resources took that electric current flows in the opposite direction to where the electrons went. And this is wrong. Therefore it is necessary to apply the rule of the left hand.
Do not always go such a wilds. It would seem that the rules are more confusing, not quite. The right-hand rule is often used to calculate the angular velocity, which is the geometric product of the acceleration radius: V = ω x r. Many people will be helped by visual memory:
- The vector of the radius of the circular path is directed from the center to the circle.
- If the acceleration vector is directed upwards, the body moves counterclockwise.
Look, the right-hand rule is again here: if you position the palm so that the acceleration vector enters perpendicularly into the palm, extend your fingers in the direction of the radius, bent by 90 degrees, the thumb indicates the direction of movement of the object. It is enough to draw once on paper, remembering at least half a life. The picture is really simple. More on the physics lesson will not have to wrestle with a simple question - the direction of the vector of angular acceleration.
Similarly, the moment of force is determined. It comes perpendicularly from the axis of the shoulder, coincides with the direction with angular acceleration in the figure described above. Many will ask: what is needed? Why is the moment of force not a scalar quantity? Why the direction? In complex systems is not easy to trace the interaction. If there are a lot of axes, forces, vector addition of moments helps. You can greatly simplify the calculations.