Physical vector quantities examples. Which quantity is vector and which is scalar? Just something complicated
(tensors of rank 0), on the other hand, tensor quantities (strictly speaking, tensors of rank 2 or more). It can also be contrasted with certain objects of a completely different mathematical nature.
In most cases, the term vector is used in physics to denote a vector in the so-called “physical space”, that is, in the usual three-dimensional space of classical physics or in four-dimensional space-time in modern physics(in the latter case, the concept of a vector and a vector quantity coincides with the concept of a 4-vector and a 4-vector quantity).
The use of the phrase “vector quantity” is practically exhausted by this. As for the use of the term “vector”, it, despite the default inclination to the same field of applicability, in large quantities cases still go very far beyond such limits. See below for details.
Use of terms vector And vector quantity in physics
In general, in physics the concept of a vector almost completely coincides with that in mathematics. However, there is a terminological specificity associated with the fact that in modern mathematics this concept is somewhat overly abstract (in relation to the needs of physics).
In mathematics, when pronouncing “vector,” they mean rather a vector in general, that is, any vector of any abstract linear space of any dimension and nature, which, unless special efforts are made, can even lead to confusion (not so much, of course, in essence, as for ease of use). If it is necessary to be more specific, in the mathematical style one has to either speak at quite a length (“vector of such and such a space”), or keep in mind what is implied by the explicitly described context.
In physics, we are almost always talking not about mathematical objects (possessing certain formal properties) in general, but about their specific (“physical”) connection. Taking these considerations of specificity into account with considerations of brevity and convenience, it can be understood that terminological practice in physics differs markedly from that of mathematics. However, it is not in obvious contradiction with the latter. This can be achieved with a few simple “tricks”. First of all, these include the agreement on the use of the term by default (when the context is not specifically specified). Thus, in physics, unlike mathematics, the word vector without additional clarification usually means not “some vector of any linear space in general,” but primarily a vector associated with “ordinary physical space” (the three-dimensional space of classical physics or the four-dimensional space -time of relativistic physics). For vectors of spaces that are not directly and directly related to “physical space” or “space-time”, special names are used (sometimes including the word “vector”, but with clarification). If a vector of some space that is not directly and directly related to “physical space” or “space-time” (and which is difficult to immediately characterize in any specific way) is introduced into theory, it is often specifically described as an “abstract vector”.
Everything that has been said applies to the term “vector quantity” even more than to the term “vector”. The silence in this case even more strictly implies a binding to “ordinary space” or space-time, and the use of abstract vector spaces in relation to elements is almost never encountered, at least such a use seems to be the rarest exception (if not a reservation at all).
In physics, vectors most often, and vector quantities - almost always - are called vectors of two classes that are similar to each other:
Examples of vector physical quantities: speed, force, heat flow.
Genesis of vector quantities
How are physical “vector quantities” related to space? First of all, what is striking is that the dimension of vector quantities (in the usual sense of using this term, which is explained above) coincides with the dimension of the same “physical” (and “geometric”) space, for example, space is three-dimensional and vector electric field three-dimensional. Intuitively, one can also notice that any vector physical quantity, no matter what vague connection it has with ordinary spatial extension, nevertheless it has a very definite direction precisely in this ordinary space.
However, it turns out that much more can be achieved by directly “reducing” the entire set of vector quantities of physics to the simplest “geometric” vectors, or rather even to one vector - the vector of elementary displacement, and it would be more correct to say - by deriving them all from it.
This procedure has two different (although essentially repeating each other in detail) implementations for the three-dimensional case of classical physics and for the four-dimensional space-time formulation common to modern physics.
Classic 3D case
We will start from the usual three-dimensional “geometric” space in which we live and can move.
Let's take the vector of infinitesimal displacement as the initial and reference vector. It's pretty obvious that this is a regular "geometric" vector (just like a finite displacement vector).
Let us now immediately note that multiplying a vector by a scalar always gives a new vector. The same can be said about the sum and difference of vectors. In this chapter we will not make a difference between polar and axial vectors, so we note that the vector product of two vectors also gives a new vector.
Also, the new vector gives the differentiation of the vector with respect to the scalar (since such a derivative is the limit of the ratio of the difference of vectors to the scalar). This can be said further about derivatives of all higher orders. The same is true for integration over scalars (time, volume).
Now note that, based on the radius vector r or from elementary displacement d r, we easily understand that vectors are (since time is a scalar) such kinematic quantities as
From speed and acceleration, multiplied by a scalar (mass), we get
Since we are now interested in pseudovectors, we note that
- Using the Lorentz force formula, the electric field strength and the magnetic induction vector are tied to the force and velocity vectors.
Continuing this procedure, we discover that all vector quantities known to us are now not only intuitively, but also formally, tied to the original space. Namely, all of them, in a sense, are its elements, since they are expressed essentially as linear combinations of other vectors (with scalar factors, perhaps dimensional, but scalar, and therefore formally quite legal).
In physics courses, we often encounter quantities for which it is enough to know only numerical values to describe them. For example, mass, time, length.
Quantities that are characterized only numerical value, are called scalar or scalars.
In addition to scalar quantities, quantities are used that have both a numerical value and a direction. For example, speed, acceleration, force.
Quantities that are characterized by numerical value and direction are called vector or vectors.
Vector quantities are indicated by the corresponding letters with an arrow at the top or in bold. For example, the force vector is denoted by \(\vec F\) or F . The numerical value of a vector quantity is called the modulus or length of the vector. The value of the force vector is denoted by F or \(\left|\vec F \right|\).
Vector image
Vectors are represented by directed segments. The beginning of the vector is the point from which the directed segment begins (point A in Fig. 1), the end of the vector is the point at which the arrow ends (point B in Fig. 1).
Rice. 1.
The two vectors are called equal, if they have the same length and are directed in the same direction. Such vectors are represented by directed segments having the same lengths and directions. For example, in Fig. 2 shows the vectors \(\vec F_1 =\vec F_2\).
Rice. 2. When two or more vectors are depicted in one drawing, the segments are constructed on a pre-selected scale. For example, in Fig. Figure 3 shows vectors whose lengths are \(\upsilon_1\) = 2 m/s, \(\upsilon_2\) = 3 m/s.
Rice. 3. Method for specifying a vector
On a plane, a vector can be specified in several ways:
1. Specify the coordinates of the beginning and end of the vector. For example, the vector \(\Delta\vec r\) in Fig. 4 is given by the coordinates of the beginning of the vector – (2, 4) (m), the end – (6, 8) (m).
Rice. 4. 2. Indicate the magnitude of the vector (its value) and the angle between the direction of the vector and some pre-selected direction on the plane. Often for such a direction in positive side axis 0 X. Angles measured from this direction counterclockwise are considered positive. In Fig. 5 vector \(\Delta\vec r\) is given by two numbers b and \(\alpha\) , indicating the length and direction of the vector.
Rice. 5. Vector- a purely mathematical concept that is only used in physics or other applied sciences and which allows one to simplify the solution of some complex problems.
Vector− directed straight segment.
In the know elementary physics we have to operate with two categories of quantities − scalar and vector.
Scalar quantities (scalars) are quantities characterized by a numerical value and sign. The scalars are length − l, mass − m, path − s, time − t, temperature − T, electric charge − q, energy − W, coordinates, etc.
All algebraic operations (addition, subtraction, multiplication, etc.) apply to scalar quantities.
Example 1.
Determine the total charge of the system, consisting of the charges included in it, if q 1 = 2 nC, q 2 = −7 nC, q 3 = 3 nC.
Full system charge
q = q 1 + q 2 + q 3 = (2 − 7 + 3) nC = −2 nC = −2 × 10 −9 C.
Example 2.
For quadratic equation type
ax 2 + bx + c = 0;
x 1,2 = (1/(2a)) × (−b ± √(b 2 − 4ac)).
Vector Quantities (vectors) are quantities, to determine which it is necessary to indicate, in addition to the numerical value, the direction. Vectors − speed v, strength F, impulse p, electric field strength E, magnetic induction B etc.
The numerical value of a vector (modulus) is denoted by a letter without a vector symbol or the vector is enclosed between vertical bars r = |r|.
Graphically, the vector is represented by an arrow (Fig. 1),
The length of which on a given scale is equal to its magnitude, and the direction coincides with the direction of the vector.
Two vectors are equal if their magnitudes and directions coincide.
Vector quantities are added geometrically (according to the rule of vector algebra).
Finding a vector sum from given component vectors is called vector addition.
The addition of two vectors is carried out according to the parallelogram or triangle rule. Sum vector
c = a + b
equal to the diagonal of a parallelogram built on vectors a And b. Module it
с = √(a 2 + b 2 − 2abcosα) (Fig. 2). 
At α = 90°, c = √(a 2 + b 2 ) is the Pythagorean theorem.
The same vector c can be obtained using the triangle rule if from the end of the vector a set aside vector b. Trailing vector c (connecting the beginning of the vector a and the end of the vector b) is the vector sum of terms (component vectors a And b).
The resulting vector is found as the trailing line of the broken line whose links are the component vectors (Fig. 3). 
Example 3.
Add two forces F 1 = 3 N and F 2 = 4 N, vectors F 1 And F 2 make angles α 1 = 10° and α 2 = 40° with the horizon, respectively
F = F 1 + F 2(Fig. 4). 
The result of the addition of these two forces is a force called the resultant. Vector F directed along the diagonal of a parallelogram built on vectors F 1 And F 2, both sides, and is equal in modulus to its length.
Vector module F find by the cosine theorem
F = √(F 1 2 + F 2 2 + 2F 1 F 2 cos(α 2 − α 1)),
F = √(3 2 + 4 2 + 2 × 3 × 4 × cos(40° − 10°)) ≈ 6.8 H.
If
(α 2 − α 1) = 90°, then F = √(F 1 2 + F 2 2 ).
Angle which is vector F is equal to the Ox axis, we find it using the formula
α = arctan((F 1 sinα 1 + F 2 sinα 2)/(F 1 cosα 1 + F 2 cosα 2)),
α = arctan((3.0.17 + 4.0.64)/(3.0.98 + 4.0.77)) = arctan0.51, α ≈ 0.47 rad.
Projection of vector a onto the Ox (Oy) axis − scalar quantity, depending on the angle α between the direction of the vector a and Ox (Oy) axis. (Fig. 5) 
Vector projections a on the Ox and Oy axes of the rectangular coordinate system. (Fig. 6) 
To avoid mistakes when determining the sign of the projection of a vector onto an axis, it is useful to remember the following rule: if the direction of the component coincides with the direction of the axis, then the projection of the vector onto this axis is positive, but if the direction of the component is opposite to the direction of the axis, then the projection of the vector is negative. (Fig. 7) 
Subtraction of vectors is an addition in which a vector is added to the first vector, numerically equal to the second, in the opposite direction
a − b = a + (−b) = d(Fig. 8). 
Let it be necessary from the vector a subtract vector b, their difference − d. To find the difference between two vectors, you need to go to the vector a add vector ( −b), that is, a vector d = a − b will be a vector directed from the beginning of the vector a to the end of the vector ( −b) (Fig. 9). 
In a parallelogram built on vectors a And b both sides, one diagonal c has the meaning of the sum, and the other d− vector differences a And b(Fig. 9).
Product of a vector a by scalar k equals vector b= k a, the modulus of which is k times greater than the modulus of the vector a, and the direction coincides with the direction a for positive k and the opposite for negative k.
Example 4.
Determine the momentum of a body weighing 2 kg moving at a speed of 5 m/s. (Fig. 10) 
Body impulse p= m v; p = 2 kg.m/s = 10 kg.m/s and directed towards the speed v.
Example 5.
Charge q = −7.5 nC placed in electric field with voltage E = 400 V/m. Find the magnitude and direction of the force acting on the charge.
The force is F= q E. Since the charge is negative, the force vector is directed in the direction opposite to the vector E. (Fig. 11) 
Division vector a by a scalar k is equivalent to multiplying a by 1/k.
Dot product vectors a And b called the scalar “c”, equal to the product of the moduli of these vectors and the cosine of the angle between them
(a.b) = (b.a) = c,
с = ab.cosα (Fig. 12) 
Example 6.
Find the work done by a constant force F = 20 N, if the displacement is S = 7.5 m, and the angle α between the force and the displacement is α = 120°.
The work done by a force is equal, by definition, to the scalar product of force and displacement
A = (F.S) = FScosα = 20 H × 7.5 m × cos120° = −150 × 1/2 = −75 J.
Vector artwork vectors a And b called a vector c, numerically equal to the product of the absolute values of vectors a and b multiplied by the sine of the angle between them:
c = a × b = ,
с = ab × sinα.
Vector c perpendicular to the plane in which the vectors lie a And b, and its direction is related to the direction of the vectors a And b the rule of the right screw (Fig. 13). 
Example 7.
Determine the force acting on a conductor 0.2 m long, placed in a magnetic field, the induction of which is 5 T, if the current strength in the conductor is 10 A and it forms an angle α = 30° with the direction of the field.
Ampere power
dF = I = Idl × B or F = I(l)∫(dl × B),
F = IlBsinα = 5 T × 10 A × 0.2 m × 1/2 = 5 N.
Consider problem solving.
1. How are two vectors directed, the moduli of which are identical and equal to a, if the modulus of their sum is equal to: a) 0; b) 2a; c) a; d) a√(2); e) a√(3)?
Solution.
a) Two vectors are directed along one straight line in opposite directions. The sum of these vectors is zero. 
b) Two vectors are directed along one straight line in the same direction. The sum of these vectors is 2a. 
c) Two vectors are directed at an angle of 120° to each other. The sum of the vectors is a. The resulting vector is found using the cosine theorem: 
a 2 + a 2 + 2aacosα = a 2 ,
cosα = −1/2 and α = 120°.
d) Two vectors are directed at an angle of 90° to each other. The modulus of the sum is equal to
a 2 + a 2 + 2aacosα = 2a 2 ,
cosα = 0 and α = 90°. 
e) Two vectors are directed at an angle of 60° to each other. The modulus of the sum is equal to
a 2 + a 2 + 2aacosα = 3a 2 ,
cosα = 1/2 and α = 60°.
Answer: The angle α between the vectors is equal to: a) 180°; b) 0; c) 120°; d) 90°; e) 60°.
2. If a = a 1 + a 2 orientation of vectors, what can be said about the mutual orientation of vectors a 1 And a 2, if: a) a = a 1 + a 2 ; b) a 2 = a 1 2 + a 2 2 ; c) a 1 + a 2 = a 1 − a 2?
Solution.
a) If the sum of vectors is found as the sum of the modules of these vectors, then the vectors are directed along one straight line, parallel to each other a 1 ||a 2.
b) If the vectors are directed at an angle to each other, then their sum is found using the cosine theorem for a parallelogram
a 1 2 + a 2 2 + 2a 1 a 2 cosα = a 2 ,
cosα = 0 and α = 90°.
vectors are perpendicular to each other a 1 ⊥ a 2.
c) Condition a 1 + a 2 = a 1 − a 2 can be executed if a 2− zero vector, then a 1 + a 2 = a 1 .
Answers. A) a 1 ||a 2; b) a 1 ⊥ a 2; V) a 2− zero vector.
3. Two forces of 1.42 N each are applied to one point of the body at an angle of 60° to each other. At what angle should two forces of 1.75 N each be applied to the same point on the body so that their action balances the action of the first two forces?
Solution.
According to the conditions of the problem, two forces of 1.75 N each balance two forces of 1.42 N each. This is possible if the modules of the resulting vectors of force pairs are equal. We determine the resulting vector using the cosine theorem for a parallelogram. For the first pair of forces:
F 1 2 + F 1 2 + 2F 1 F 1 cosα = F 2 ,
for the second pair of forces, respectively
F 2 2 + F 2 2 + 2F 2 F 2 cosβ = F 2 .
Equating the left sides of the equations
F 1 2 + F 1 2 + 2F 1 F 1 cosα = F 2 2 + F 2 2 + 2F 2 F 2 cosβ.
Let's find the required angle β between the vectors
cosβ = (F 1 2 + F 1 2 + 2F 1 F 1 cosα − F 2 2 − F 2 2)/(2F 2 F 2).
After calculations,
cosβ = (2.1.422 + 2.1.422.cos60° − 2.1.752)/(2.1.752) = −0.0124,
β ≈ 90.7°.
Second solution.
Let's consider the projection of vectors onto the coordinate axis OX (Fig.). 
Using the relationship between the parties in right triangle, we get
2F 1 cos(α/2) = 2F 2 cos(β/2),
where
cos(β/2) = (F 1 /F 2)cos(α/2) = (1.42/1.75) × cos(60/2) and β ≈ 90.7°.
4. Vector a = 3i − 4j. What must be the scalar quantity c for |c a| = 7,5?
Solution.
c a= c( 3i − 4j) = 7,5
Vector module a will be equal
a 2 = 3 2 + 4 2 , and a = ±5,
then from
c.(±5) = 7.5,
let's find that
c = ±1.5.
5. Vectors a 1 And a 2 exit from the origin and have Cartesian end coordinates (6, 0) and (1, 4), respectively. Find the vector a 3 such that: a) a 1 + a 2 + a 3= 0; b) a 1 − a 2 + a 3 = 0.
Solution.
Let's depict the vectors in the Cartesian coordinate system (Fig.) 
a) The resulting vector along the Ox axis is
a x = 6 + 1 = 7.
The resulting vector along the Oy axis is
a y = 4 + 0 = 4.
For the sum of vectors to be equal to zero, it is necessary that the condition be satisfied
a 1 + a 2 = −a 3.
Vector a 3 modulo will be equal to the total vector a 1 + a 2, but directed in the opposite direction. Vector end coordinate a 3 is equal to (−7, −4), and the modulus
a 3 = √(7 2 + 4 2) = 8.1.
B) The resulting vector along the Ox axis is equal to
a x = 6 − 1 = 5,
and the resulting vector along the Oy axis
a y = 4 − 0 = 4.
When the condition is met
a 1 − a 2 = −a 3,
vector a 3 will have the coordinates of the end of the vector a x = –5 and a y = −4, and its modulus is equal to
a 3 = √(5 2 + 4 2) = 6.4.
6. A messenger walks 30 m to the north, 25 m to the east, 12 m to the south, and then takes an elevator to a height of 36 m in a building. What is the distance L traveled by him and the displacement S?
Solution.
Let us depict the situation described in the problem on a plane on an arbitrary scale (Fig.). 
End of vector O.A. has coordinates 25 m to the east, 18 m to the north and 36 up (25; 18; 36). The distance traveled by a person is equal to
L = 30 m + 25 m + 12 m +36 m = 103 m.
We find the magnitude of the displacement vector using the formula
S = √((x − x o) 2 + (y − y o) 2 + (z − z o) 2 ),
where x o = 0, y o = 0, z o = 0.
S = √(25 2 + 18 2 + 36 2) = 47.4 (m).
Answer: L = 103 m, S = 47.4 m.
7. Angle α between two vectors a And b equals 60°. Determine the length of the vector c = a + b and angle β between vectors a And c. The magnitudes of the vectors are a = 3.0 and b = 2.0.
Solution.
The length of the vector equal to the sum of the vectors a And b Let's determine using the cosine theorem for a parallelogram (Fig.). 
с = √(a 2 + b 2 + 2abcosα).
After substitution
c = √(3 2 + 2 2 + 2.3.2.cos60°) = 4.4.
To determine the angle β, we use the sine theorem for triangle ABC:
b/sinβ = a/sin(α − β).
At the same time, you should know that
sin(α − β) = sinαcosβ − cosαsinβ.
Solving a simple one trigonometric equation, we arrive at the expression
tgβ = bsinα/(a + bcosα),
hence,
β = arctan(bsinα/(a + bcosα)),
β = arctan(2.sin60/(3 + 2.cos60)) ≈ 23°.
Let's check using the cosine theorem for a triangle:
a 2 + c 2 − 2ac.cosβ = b 2 ,
where
cosβ = (a 2 + c 2 − b 2)/(2ac)
And
β = arccos((a 2 + c 2 − b 2)/(2ac)) = arccos((3 2 + 4.4 2 − 2 2)/(2.3.4.4)) = 23°.
Answer: c ≈ 4.4; β ≈ 23°.
Solve problems.
8. For vectors a And b defined in Example 7, find the length of the vector d = a − b corner γ
between a And d.
9. Find the projection of the vector a = 4.0i + 7.0j to a straight line, the direction of which makes an angle α = 30° with the Ox axis. Vector a and the straight line lie in the xOy plane.
10. Vector a makes an angle α = 30° with straight line AB, a = 3.0. At what angle β to straight line AB should the vector be directed? b(b = √(3)) so that the vector c = a + b was parallel to AB? Find the length of the vector c.
11. Three vectors are given: a = 3i + 2j − k; b = 2i − j + k; с = i + 3j. Find a) a+b; b) a+c; V) (a, b); G) (a, c)b − (a, b)c.
12. Angle between vectors a And b is equal to α = 60°, a = 2.0, b = 1.0. Find the lengths of the vectors c = (a, b)a + b And d = 2b − a/2.
13. Prove that the vectors a And b are perpendicular if a = (2, 1, −5) and b = (5, −5, 1).
14. Find the angle α between the vectors a And b, if a = (1, 2, 3), b = (3, 2, 1).
15. Vector a makes an angle α = 30° with the Ox axis, the projection of this vector onto the Oy axis is equal to a y = 2.0. Vector b perpendicular to the vector a and b = 3.0 (see figure). 
Vector c = a + b. Find: a) projections of the vector b on the Ox and Oy axis; b) the value of c and the angle β between the vector c and the Ox axis; c) (a, b); d) (a, c).
Answers:
9. a 1 = a x cosα + a y sinα ≈ 7.0.
10. β = 300°; c = 3.5.
11. a) 5i + j; b) i + 3j − 2k; c) 15i − 18j + 9 k.
12. c = 2.6; d = 1.7.
14. α = 44.4°.
15. a) b x = −1.5; b y = 2.6; b) c = 5; β ≈ 67°; c) 0; d) 16.0.
By studying physics, you have great opportunities to continue your education at a technical university. This will require a parallel deepening of knowledge in mathematics, chemistry, language, and less often other subjects. The winner of the Republican Olympiad, Savich Egor, graduates from one of the faculties of MIPT, where great demands are placed on knowledge in chemistry. If you need help at the State Academy of Sciences in chemistry, then contact the professionals; you will definitely receive qualified and timely assistance.
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 about the movement 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 values, it would seem, does 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.
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.
Science also uses speed in 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 chemical reaction, group speed, connection speed, angular speed, etc. Mathematically characterized by the derivative of the function.
Acceleration(usually denoted in theoretical mechanics), the derivative of speed with respect to time is a vector quantity that shows 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.
A branch of mechanics that studies motion in three-dimensional Euclidean space, its recording, as well as the recording of velocities and accelerations in various systems reference 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. This characteristic feature convenient to use 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 moves forward sewing machine, piston in cylinder steam engine or engine internal combustion, car body (but not wheels!) 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. Absolutely solid- 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 -
This topic belongs to the section:
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..
If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:
What will we do with the received material:
If this material was useful to you, you can save it to your page on social networks:
| Tweet |
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 movement - view mechanical movement. 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 more complex look movement than a rectilinear one, 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, geometric point, the position of which 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
Galileo's principle of relativity
· Inertial reference system Galileo's inertial reference system
Plastic deformation
Let's bend the steel plate (for example, a hacksaw) a little, and then after a while let it go. 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
energy 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 force
· Moment of force. 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
