Vectors are a powerful tool for mathematics and physics. The basic laws of mechanics and electrodynamics are formulated in the language of vectors. To understand physics, you need to learn how to work with vectors.
This chapter contains a detailed presentation of the material necessary to begin studying mechanics:
! Vector addition
! Multiplying a scalar by a vector
! Angle between vectors
! Projection of a vector onto an axis
! Vectors and coordinates on the plane
! Vectors and coordinates in space
! Dot product of vectors
It will be useful to return to the text of this application in the first year when studying analytical geometry and linear algebra in order to understand, for example, where the axioms of linear and Euclidean space come from.
7.1 Scalar and vector quantities
In the process of studying physics, we encounter two types of quantities: scalar and vector.
Definition. A scalar quantity, or scalar, is a physical quantity for which (in suitable units of measurement) one number is sufficient.
There are a lot of scalars in physics. The body weight is 3 kg, the air temperature is 10 C, the network voltage is 220 V. . . In all these cases, the quantity we are interested in is given by a single number. Therefore, mass, temperature and electrical voltage are scalars.
But a scalar in physics is not just a number. A scalar is a number equipped with dimension 1. So, when specifying the mass, we cannot write m = 3; You must specify the unit of measurement, for example, m = 3 kg. And if in mathematics we can add the numbers 3 and 220, then in physics we cannot add 3 kilograms and 220 volts: we have the right to add only those scalars that have the same dimension (mass with mass, voltage with voltage, etc.) .
Definition. A vector quantity, or vector, is a physical quantity characterized by: 1) a non-negative scalar; 2) direction in space. In this case, the scalar is called the modulus of the vector, or its absolute value.
Let's assume that the car is moving at a speed of 60 km/h. But this is incomplete information about the movement, isn’t it? It may also be important where the car is going, in what direction. Therefore, it is important to know not only the modulus (absolute value) of the car’s speed, in this case it is 60 km/h, but also its direction in space. This means that speed is a vector.
Another example. Let's say there is a brick weighing 1 kg on the floor. A force of 100 N acts on the brick (this is the modulus of the force, or its absolute value). How will the brick move? The question is meaningless until the direction of the force is specified. If the force acts upward, then the brick will move upward. If the force acts horizontally, then the brick will move horizontally. And if the force acts vertically downward, then the brick will not move at all; it will only be pressed into the floor. We see, therefore, that force is also a vector.
A vector quantity in physics also has dimension. The dimension of a vector is the dimension of its modulus.
We will denote vectors by letters with an arrow. Thus, the velocity vector can be denoted
through ~v, and the force vector through F. Actually, a vector is an arrow or, as they also say, a directed segment (Fig. 7.1).
Rice. 7.1. Vector ~v
The starting point of the arrow is called the beginning of the vector, and the end point (tip) of the arrow
end of the vector. In mathematics, a vector starting at point A and ending at point B is denoted
also AB; We will also sometimes need such a notation.
A vector whose beginning and end coincide is called a zero vector (or zero) and
denoted by ~. The zero vector is simply a point; it has no definite direction.
The length of the zero vector is, of course, zero.
1 There are also dimensionless scalars: friction coefficient, efficiency, refractive index of the medium. . . Thus, the refractive index of water is 1.33; this is comprehensive information; this number does not have any dimension.
Drawing arrows completely solves the problem of graphically representing vector quantities. The direction of the arrow indicates the direction of a given vector, and the length of the arrow on a suitable scale is the magnitude of that vector.
Suppose, for example, that two cars are moving towards each other at speeds u = 30 km/h and v = 60 km/h. Then the vectors ~u and ~v of the car speeds will have opposite directions, and the length of the vector ~v is twice as large (Fig. 7.2).
Rice. 7.2. Vector ~v is twice as long
As you already understood, a letter without an arrow (for example, u or v in the previous paragraph) indicates the magnitude of the corresponding vector. In mathematics, the modulus of the vector ~v is usually denoted j~vj, but physicists, if the situation allows, will prefer the letter v without the arrow.
Vectors are called collinear if they are located on the same line or on parallel lines.
Let there be two collinear vectors. If their directions coincide, then the vectors are called codirectional; if their directions are different, then the vectors are called oppositely directed. So, above in Fig. 7.2 vectors ~u and ~v are oppositely directed.
Two vectors are called equal if they are codirectional and have equal modules (Fig. 7.3).
Rice. 7.3. Vectors ~a and b are equal: ~a = b
Thus, the equality of vectors does not necessarily mean that their beginnings and ends coincide: we can move a vector parallel to itself, and this will result in a vector equal to the original one. This transfer is constantly used in cases where it is desirable to reduce the beginnings of vectors to one point, for example, when finding the sum or difference of vectors. We now move on to consider operations on vectors.
The two words that frighten schoolchildren - vector and scalar - are not actually scary. If you approach the topic with interest, then everything can be understood. In this article we will consider which quantity is vector and which is scalar. More precisely, we will give examples. Every student probably noticed that in physics some quantities are denoted not only by a symbol, but also by an arrow on top. What do they mean? This will be discussed below. Let's try to figure out how it differs from scalar.
Examples of vectors. How are they designated?
What is meant by vector? That which characterizes movement. It doesn't matter whether in space or on a plane. What quantity is a vector quantity in general? For example, an airplane flies at a certain speed at a certain altitude, has a specific mass, and began moving from the airport with the required acceleration. What is the motion of an airplane? What made him fly? Of course, acceleration, speed. Vector quantities from the physics course are clear examples. To put it bluntly, a vector quantity is associated with motion, displacement.
Water also moves at a certain speed from the height of the mountain. Do you see? Movement is carried out not by volume or mass, but by speed. A tennis player allows the ball to move with the help of a racket. It sets the acceleration. By the way, the force applied in this case is also a vector quantity. Because it is obtained as a result of given speeds and accelerations. Power can also change and carry out specific actions. The wind that moves the leaves on the trees can also be considered an example. Because there is speed.
Positive and negative quantities
A vector quantity is a quantity that has a direction in the surrounding space and a magnitude. The scary word appeared again, this time module. Imagine that you need to solve a problem where a negative acceleration value will be recorded. In nature, negative meanings, it would seem, do not exist. How can speed be negative?
A vector has such a concept. This applies, for example, to forces that are applied to the body, but have different directions. Remember the third where action is equal to reaction. The guys are playing tug of war. One team wears blue T-shirts, the other team wears yellow T-shirts. The latter turn out to be stronger. Let us assume that their force vector is directed positively. At the same time, the first ones cannot pull the rope, but they try. An opposing force arises.
Vector or scalar quantity?
Let's talk about how a vector quantity differs from a scalar quantity. Which parameter has no direction, but has its own meaning? Let's list some scalar quantities below:
Do they all have a direction? No. Which quantity is vector and which is scalar can only be shown with visual examples. In physics there are such concepts not only in the section “Mechanics, dynamics and kinematics”, but also in the paragraph “Electricity and magnetism”. The Lorentz force is also a vector quantity.
Vector and scalar in formulas
Physics textbooks often contain formulas that have an arrow at the top. Remember Newton's second law. Force ("F" with an arrow on top) is equal to the product of mass ("m") and acceleration ("a" with an arrow on top). As mentioned above, force and acceleration are vector quantities, but mass is scalar.
Unfortunately, not all publications have the designation of these quantities. This was probably done to simplify things so that schoolchildren would not be misled. It is best to buy those books and reference books that indicate vectors in formulas.
The illustration will show which quantity is a vector one. It is recommended to pay attention to pictures and diagrams in physics lessons. Vector quantities have a direction. Where is it directed? Of course, down. This means that the arrow will be shown in the same direction.
Physics is studied in depth at technical universities. In many disciplines, teachers talk about what quantities are scalar and vector. Such knowledge is required in the following areas: construction, transport, natural sciences.
Physics and mathematics cannot do without the concept of “vector quantity”. You need to know and recognize it, and also be able to operate with it. You should definitely learn this so as not to get confused and make stupid mistakes.
How to distinguish a scalar quantity from a vector quantity?
The first one always has only one characteristic. This is its numerical value. Most scalar quantities can take on both positive and negative values. Examples of these are electric charge, work, or temperature. But there are scalars that cannot be negative, for example, length and mass.
A vector quantity, in addition to a numerical quantity, which is always taken modulo, is also characterized by direction. Therefore, it can be depicted graphically, that is, in the form of an arrow, the length of which is equal to the absolute value directed in a certain direction.
When writing, each vector quantity is indicated by an arrow sign on the letter. If we are talking about a numerical value, then the arrow is not written or it is taken modulo.
What actions are most often performed with vectors?
First, a comparison. They may or may not be equal. In the first case, their modules are the same. But this is not the only condition. They must also have the same or opposite directions. In the first case, they should be called equal vectors. In the second they turn out to be opposite. If at least one of the specified conditions is not met, then the vectors are not equal.
Then comes addition. It can be made according to two rules: a triangle or a parallelogram. The first prescribes to first lay off one vector, then from its end the second. The result of the addition will be the one that needs to be drawn from the beginning of the first to the end of the second.
The parallelogram rule can be used when adding vector quantities in physics. Unlike the first rule, here they should be postponed from one point. Then build them up to a parallelogram. The result of the action should be considered the diagonal of the parallelogram drawn from the same point.
If a vector quantity is subtracted from another, then they are again plotted from one point. Only the result will be a vector that coincides with what is plotted from the end of the second to the end of the first.
What vectors are studied in physics?
There are as many of them as there are scalars. You can simply remember what vector quantities exist in physics. Or know the signs by which they can be calculated. For those who prefer the first option, this table will be useful. It contains the main vector
Now a little more about some of these quantities.
The first quantity is speed
It’s worth starting with examples of vector quantities. This is due to the fact that it is among the first to be studied.
Speed is defined as a characteristic of the movement of a body in space. It sets the numerical value and direction. Therefore, speed is a vector quantity. In addition, it is customary to divide it into types. The first is linear speed. It is introduced when considering rectilinear uniform motion. In this case, it turns out to be equal to the ratio of the path traveled by the body to the time of movement.
The same formula can be used for uneven movement. Only then will it be average. Moreover, the time interval that must be selected must be as short as possible. As the time interval tends to zero, the speed value is already instantaneous.
If arbitrary movement is considered, then speed is always a vector quantity. After all, it has to be decomposed into components directed along each vector directing the coordinate lines. In addition, it is defined as the derivative of the radius vector taken with respect to time.
The second quantity is strength
It determines the measure of the intensity of the impact that is exerted on the body by other bodies or fields. Since force is a vector quantity, it necessarily has its own magnitude and direction. Since it acts on the body, the point to which the force is applied is also important. To get a visual representation of force vectors, you can refer to the following table.
Also another vector quantity is the resultant force. It is defined as the sum of all mechanical forces acting on the body. To determine it, it is necessary to perform addition according to the principle of the triangle rule. You just need to lay off the vectors one by one from the end of the previous one. The result will be the one that connects the beginning of the first to the end of the last.
The third quantity is displacement
During movement, the body describes a certain line. It's called a trajectory. This line can be completely different. What is more important is not its appearance, but the starting and ending points of the movement. They are connected by a segment called a translation. This is also a vector quantity. Moreover, it is always directed from the beginning of the movement to the point where the movement was stopped. It is usually denoted by the Latin letter r.
Here the following question may arise: “Is the path a vector quantity?” In general, this statement is not true. The path is equal to the length of the trajectory and does not have a specific direction. The exception is the situation when it is considered in one direction. Then the magnitude of the displacement vector coincides in value with the path, and their direction turns out to be the same. Therefore, when considering motion along a straight line without changing the direction of movement, the path can be included in examples of vector quantities.
The fourth quantity is acceleration
It is a characteristic of the speed of change of speed. Moreover, acceleration can have both positive and negative values. When moving in a straight line, it is directed towards higher speed. If the movement occurs along a curved path, then its acceleration vector is decomposed into two components, one of which is directed towards the center of curvature along the radius.
The average and instantaneous acceleration values are distinguished. The first should be calculated as the ratio of the change in speed over a certain period of time to this time. When the time interval under consideration tends to zero, we speak of instantaneous acceleration.
Fifth value - momentum
In another way it is also called quantity of motion. Momentum is a vector quantity because it is directly related to the speed and force applied to the body. Both of them have a direction and give it to the impulse.
By definition, the latter is equal to the product of body mass and speed. Using the concept of momentum of a body, we can write Newton’s well-known law differently. It turns out that the change in momentum is equal to the product of force and a period of time.
In physics, the law of conservation of momentum plays an important role, which states that in a closed system of bodies its total momentum is constant.
We have very briefly listed which quantities (vector) are studied in the physics course.
Inelastic Impact Problem
Condition. There is a stationary platform on the rails. A carriage is approaching it at a speed of 4 m/s. and wagon - 10 and 40 tons, respectively. The car hits the platform and automatic coupling occurs. It is necessary to calculate the speed of the "car-platform" system after the impact.
Solution. First, you need to enter the following notations: the speed of the car before the impact is v 1, the speed of the car with the platform after coupling is v, the mass of the car is m 1, the mass of the platform is m 2. According to the conditions of the problem, it is necessary to find out the value of the speed v.
The rules for solving such tasks require a schematic representation of the system before and after interaction. It is reasonable to direct the OX axis along the rails in the direction where the car is moving.
Under these conditions, the car system can be considered closed. This is determined by the fact that external forces can be neglected. Gravity and are balanced, and friction on the rails is not taken into account.
According to the law of conservation of momentum, their vector sum before the interaction of the car and the platform is equal to the total for the coupling after the impact. At first the platform did not move, so its momentum was zero. Only the car was moving, its momentum was the product of m 1 and v 1 .
Since the impact was inelastic, that is, the car connected with the platform, and then they began to roll together in the same direction, the impulse of the system did not change direction. But its meaning has changed. Namely, the product of the sum of the mass of the car with the platform and the desired speed.
You can write the following equality: m 1 * v 1 = (m 1 + m 2) * v. It will be true for the projection of impulse vectors onto the selected axis. From it it is easy to derive the equality that will be needed to calculate the desired speed: v = m 1 * v 1 / (m 1 + m 2).
According to the rules, the values for mass should be converted from tons to kilograms. Therefore, when substituting them into the formula, you must first multiply the known quantities by a thousand. Simple calculations give a figure of 0.75 m/s.
Answer. The speed of the car with the platform is 0.75 m/s.
Problem with dividing the body into parts
Condition. The speed of a flying grenade is 20 m/s. It breaks into two pieces. The weight of the first is 1.8 kg. It continues to move in the direction in which the grenade was flying at a speed of 50 m/s. The second fragment has a mass of 1.2 kg. What is its speed?
Solution. Let the masses of the fragments be denoted by the letters m 1 and m 2. Their speeds will be v 1 and v 2 respectively. The initial speed of the grenade is v. The problem requires calculating the value of v 2 .
In order for the larger fragment to continue to move in the same direction as the entire grenade, the second one must fly in the opposite direction. If you choose the direction of the axis to be the one that was at the initial impulse, then after the break the large fragment flies along the axis, and the small one flies against the axis.
In this problem, it is allowed to use the law of conservation of momentum due to the fact that the grenade explodes instantly. Therefore, despite the fact that gravity acts on the grenade and its parts, it does not have time to act and change the direction of the impulse vector with its absolute value.
The sum of the vector magnitudes of the impulse after the grenade explosion is equal to that which was before it. If we write the conservation law in projection onto the OX axis, it will look like this: (m 1 + m 2) * v = m 1 * v 1 - m 2 * v 2 . From it it is easy to express the required speed. It will be determined by the formula: v 2 = ((m 1 + m 2) * v - m 1 * v 1) / m 2. After substituting numerical values and calculations, we get 25 m/s.
Answer. The speed of the small fragment is 25 m/s.
Problem about shooting at an angle
Condition. A gun is mounted on a platform of mass M. It fires a projectile of mass m. It flies out at an angle α to the horizon with a speed v (given relative to the ground). You need to know the speed of the platform after the shot.
Solution. In this problem, you can use the law of conservation of momentum in projection onto the OX axis. But only in the case when the projection of external resultant forces is equal to zero.
For the direction of the OX axis, you need to select the side where the projectile will fly, and parallel to the horizontal line. In this case, the projections of gravity forces and the reaction of the support on OX will be equal to zero.
The problem will be solved in a general form, since there is no specific data for known quantities. The answer is a formula.
The system's momentum before the shot was zero, since the platform and the projectile were stationary. Let the desired platform speed be denoted by the Latin letter u. Then its momentum after the shot will be determined as the product of the mass and the projection of the velocity. Since the platform will roll back (against the direction of the OX axis), the impulse value will have a minus sign.
The momentum of a projectile is the product of its mass and the projection of velocity onto the OX axis. Due to the fact that the velocity is directed at an angle to the horizon, its projection is equal to the velocity multiplied by the cosine of the angle. In literal equality it will look like this: 0 = - Mu + mv * cos α. From it, through simple transformations, the answer formula is obtained: u = (mv * cos α) / M.
Answer. The platform speed is determined by the formula u = (mv * cos α) / M.
River crossing problem
Condition. The width of the river along its entire length is the same and equal to l, its banks are parallel. The speed of water flow in the river v 1 and the boat’s own speed v 2 are known. 1). When crossing, the bow of the boat is directed strictly towards the opposite shore. How far s will it be carried downstream? 2). At what angle α should the bow of the boat be directed so that it reaches the opposite shore strictly perpendicular to the point of departure? How long will it take t for such a crossing?
Solution. 1). The total speed of the boat is the vector sum of two quantities. The first of these is the flow of the river, which is directed along the banks. The second is the boat’s own speed, perpendicular to the shores. The drawing produces two similar triangles. The first is formed by the width of the river and the distance over which the boat drifts. The second is by velocity vectors.
From them follows the following entry: s / l = v 1 / v 2. After the transformation, the formula for the desired value is obtained: s = l * (v 1 / v 2).
2). In this version of the problem, the total velocity vector is perpendicular to the shores. It is equal to the vector sum of v 1 and v 2. The sine of the angle by which the natural velocity vector must deviate is equal to the ratio of the modules v 1 and v 2. To calculate the travel time, you will need to divide the width of the river by the calculated full speed. The value of the latter is calculated using the Pythagorean theorem.
v = √(v 2 2 - v 1 2), then t = l / (√(v 2 2 - v 1 2)).
Answer. 1). s = l * (v 1 / v 2), 2). sin α = v 1 / v 2, t = l / (√(v 2 2 - v 1 2)).
Vector quantity (vector) is a physical quantity that has two characteristics - modulus and direction in space.
Examples of vector quantities: speed (), force (), acceleration (), etc.
Geometrically, a vector is depicted as a directed segment of a straight line, the length of which on a scale is the absolute value of the vector.
Radius vector(usually denoted or simply) - a vector that specifies the position of a point in space relative to some pre-fixed point, called the origin.
For an arbitrary point in space, the radius vector is the vector going from the origin to that point.
The length of the radius vector, or its modulus, determines the distance at which the point is located from the origin, and the arrow indicates the direction to this point in space.
On a plane, the angle of the radius vector is the angle by which the radius vector is rotated relative to the x-axis in a counterclockwise direction.
the line along which a body moves is called trajectory of movement. Depending on the shape of the trajectory, all movements can be divided into rectilinear and curvilinear.
The description of movement begins with an answer to the question: how has the position of the body in space changed over a certain period of time? How is a change in the position of a body in space determined?
Moving- a directed segment (vector) connecting the initial and final position of the body.
Speed(often denoted , from English. velocity or fr. vitesse) is a vector physical quantity that characterizes the speed of movement and direction of movement of a material point in space relative to the selected reference system (for example, angular velocity). The same word can be used to refer to a scalar quantity, or more precisely, the modulus of the derivative of the radius vector.
In science, speed is also used in a broad sense, as the speed of change of some quantity (not necessarily the radius vector) depending on another (usually changes in time, but also in space or any other). For example, they talk about the rate of temperature change, the rate of a chemical reaction, the group velocity, the rate of connection, the angular velocity, etc. The derivative of a function is characterized mathematically.
Acceleration(usually denoted in theoretical mechanics), the derivative of speed with respect to time is a vector quantity showing how much the speed vector of a point (body) changes as it moves per unit time (i.e. acceleration takes into account not only the change in the magnitude of the speed, but also its direction ).
For example, near the Earth, a body falling on the Earth, in the case where air resistance can be neglected, increases its speed by approximately 9.8 m/s every second, that is, its acceleration is equal to 9.8 m/s².
The branch of mechanics that studies motion in three-dimensional Euclidean space, its recording, as well as the recording of velocities and accelerations in various reference systems, is called kinematics.
The unit of acceleration is meters per second per second ( m/s 2, m/s 2), there is also a non-system unit Gal (Gal), used in gravimetry and equal to 1 cm/s 2.
Derivative of acceleration with respect to time i.e. the quantity characterizing the rate of change of acceleration over time is called jerk.
The simplest movement of a body is one in which all points of the body move equally, describing the same trajectories. This movement is called progressive. We obtain this type of motion by moving the splinter so that it remains parallel to itself at all times. During forward motion, trajectories can be either straight (Fig. 7, a) or curved (Fig. 7, b) lines.
It can be proven that during translational motion, any straight line drawn in the body remains parallel to itself. It is convenient to use this characteristic feature to answer the question of whether a given body movement is translational. For example, when a cylinder rolls along a plane, straight lines intersecting the axis do not remain parallel to themselves: rolling is not a translational motion. When the crossbar and square move along the drawing board, any straight line drawn in them remains parallel to itself, which means they move forward (Fig. 8). The needle of a sewing machine, the piston in the cylinder of a steam engine or internal combustion engine, the body of a car (but not the wheels!) move forward when driving on a straight road, etc.
Another simple type of movement is rotational movement body, or rotation. During rotational motion, all points of the body move in circles whose centers lie on a straight line. This straight line is called the axis of rotation (straight line 00" in Fig. 9). The circles lie in parallel planes perpendicular to the axis of rotation. The points of the body lying on the axis of rotation remain stationary. Rotation is not a translational movement: when the axis rotates OO" . The trajectories shown remain parallel only straight lines parallel to the axis of rotation.
Absolutely solid body- the second supporting object of mechanics along with the material point.
There are several definitions:
1. An absolutely rigid body is a model concept of classical mechanics, denoting a set of material points, the distances between which are maintained during any movements performed by this body. In other words, an absolutely solid body not only does not change its shape, but also maintains the distribution of mass inside unchanged.
2. An absolutely rigid body is a mechanical system that has only translational and rotational degrees of freedom. “Hardness” means that the body cannot be deformed, that is, no other energy can be transferred to the body other than the kinetic energy of translational or rotational motion.
3. An absolutely rigid body is a body (system), the relative position of any points of which does not change, no matter what processes it participates in.
In three-dimensional space and in the absence of connections, an absolutely rigid body has 6 degrees of freedom: three translational and three rotational. The exception is a diatomic molecule or, in the language of classical mechanics, a solid rod of zero thickness. Such a system has only two rotational degrees of freedom.
End of work -
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An unproven and unrefuted hypothesis is called an open problem.
Physics is closely related to mathematics; mathematics provides an apparatus with the help of which physical laws can be precisely formulated.. theory Greek consideration.. standard method of testing theories direct experimental verification experiment criterion of truth however often..
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All topics in this section:
The principle of relativity in mechanics
Inertial reference systems and the principle of relativity.
Galileo's transformations. Transformation invariants. Absolute and relative speeds and accelerations. Postulates of special technology
Rotational motion of a material point.
Rotational motion of a material point is the movement of a material point in a circle.
Rotational motion is a type of mechanical motion. At
Relationship between the vectors of linear and angular velocities, linear and angular accelerations.
A measure of rotational motion: the angle φ through which the radius vector of a point rotates in a plane normal to the axis of rotation.
Uniform rotational motion
Speed and acceleration during curved motion.
Curvilinear movement is a more complex type of movement than rectilinear movement, since even if the movement occurs on a plane, two coordinates that characterize the position of the body change. Speed and
Acceleration during curved motion.
Considering the curvilinear movement of a body, we see that its speed is different at different moments. Even in the case when the magnitude of the speed does not change, there is still a change in the direction of the speed
Newton's equation of motion
(1) where the force F in the general case
Center of mass
center of inertia, a geometric point whose position characterizes the distribution of masses in a body or mechanical system. The coordinates of the central mass are determined by the formulas
Law of motion of the center of mass.
Using the law of momentum change, we obtain the law of motion of the center of mass: dP/dt = M∙dVc/dt = ΣFi The center of mass of the system moves in the same way as the two
Bend the steel plate (for example, a hacksaw) a little, and then release it after a while. We will see that the hacksaw will completely (at least at first glance) restore its shape. If we take
EXTERNAL AND INTERNAL FORCES
. In mechanics, external forces in relation to a given system of material points (i.e., such a set of material points in which the movement of each point depends on the positions or movements of all axes
Kinetic energy
the energy of a mechanical system, depending on the speed of movement of its points. K. e. T of a material point is measured by half the product of the mass m of this point by the square of its speed
Kinetic energy.
Kinetic energy is the energy of a moving body. (From the Greek word kinema - movement). By definition, the kinetic energy of something at rest in a given frame of reference
A value equal to half the product of a body's mass and the square of its speed.
=J.
Kinetic energy is a relative quantity, depending on the choice of CO, because the speed of the body depends on the choice of CO.
That.
Moment of power
· Moment of power. Rice. Moment of power. Rice. Moment of force, quantities
Kinetic energy of a rotating body
Kinetic energy is an additive quantity. Therefore, the kinetic energy of a body moving in an arbitrary manner is equal to the sum of the kinetic energies of all n materials
Work and power during rotation of a rigid body.
Work and power during rotation of a rigid body.
Let's find an expression for work at temp
Basic equation for the dynamics of rotational motion
According to equation (5.8), Newton’s second law for rotational motion P In physics, there are several categories of quantities: vector and scalar. What is a vector quantity?
A vector quantity has two main characteristics:
If we consider a vector quantity regardless of direction, then such a segment can be measured. But the resulting result will reflect only partial characteristics of the quantity. To fully measure it, the value should be supplemented with other parameters of the directional segment.
In vector algebra there is a concept zero vector. This concept means a point. As for the direction of the zero vector, it is considered uncertain. To denote the zero vector, the arithmetic zero is used, typed in bold.
If we analyze all of the above, we can conclude that all directed segments define vectors. Two segments will define one vector only if they are equal. When comparing vectors, the same rule applies as when comparing scalar quantities. Equality means complete agreement in all respects.
What is a scalar quantity?
Unlike a vector, a scalar quantity has only one parameter - this its numerical value. It is worth noting that the analyzed value can have both a positive numerical value and a negative one.
Examples include mass, voltage, frequency or temperature. With such quantities you can perform various arithmetic operations: addition, division, subtraction, multiplication. A scalar quantity does not have such a characteristic as direction.
A scalar quantity is measured with a numerical value, so it can be displayed on a coordinate axis. For example, very often the axis of the distance traveled, temperature or time is constructed.
Main differences between scalar and vector quantities
From the descriptions given above, it is clear that the main difference between vector quantities and scalar quantities is their characteristics. A vector quantity has a direction and magnitude, while a scalar quantity has only a numerical value. Of course, a vector quantity, like a scalar quantity, can be measured, but such a characteristic will not be complete, since there is no direction.
In order to more clearly imagine the difference between a scalar quantity and a vector quantity, an example should be given. To do this, let’s take such an area of knowledge as climatology. If we say that the wind blows at a speed of 8 meters per second, then a scalar quantity will be introduced. But if we say that the north wind blows at a speed of 8 meters per second, then we are talking about a vector value.
Vectors play a huge role in modern mathematics, as well as in many areas of mechanics and physics. Most physical quantities can be represented as vectors. This allows us to generalize and significantly simplify the formulas and results used. Often vector values and vectors are identified with each other. For example, in physics you may hear that speed or force is a vector.