Vector− clean mathematical concept, which is only used in physics or other applied sciences and which allows you to simplify the solution of some complex problems.
Vector− directed straight segment.
I 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 apply to scalar quantities algebraic operations(addition, subtraction, multiplication, etc.).
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 kind
ax 2 + bx + c = 0;
x 1,2 = (1/(2a)) × (−b ± √(b 2 − 4ac)).
Vector quantities (vectors) are quantities, for the determination of which it is necessary to indicate, in addition to numerical value so is the direction. Vectors − speed v, force F, impulse p, tension electric field E, magnetic induction B and 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.
The projection of vector a onto the Ox (Oy) axis is a 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 axis rectangular system coordinates (Fig. 6) 
To avoid mistakes when determining the sign of the vector projection onto the axis, it is useful to remember next 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 of 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.
A charge q = −7.5 nC is placed in an electric field with a strength of 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 towards 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 moduli of these vectors by the cosine of the angle between them
(a.b) = (b.a) = c,
с = ab.cosα (Fig. 12) 
Example 6.
Find a job 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 right screw rule (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 sides. 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.
The magnitude of the displacement vector can be found 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.
Vector length, equal to the amount 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 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 continue your education in 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.
All quantities that we encounter in physics and, in particular, in one of its branches of mechanics, can be divided into two types:
a) scalar, which are determined by one real positive or negative number. Examples of such quantities include time, temperature;
b) vector, which are determined by a directed spatial segment of a line (or three scalar quantities) and have the properties given below.
Example vector quantities serve as force, speed, acceleration.
Cartesian coordinate system
When talking about directed segments, you should indicate the object in relation to which this direction is determined. The Cartesian coordinate system, the components of which are the axes, is taken as such an object.
An axis is a straight line on which direction is indicated. Three mutually perpendicular to the axis, intersecting at point O, named accordingly, form a rectangular Cartesian coordinate system. Cartesian system coordinates can be right (Fig. 1) or left (Fig. 2). These systems are mirror images of each other and cannot be combined by any movement.
In all subsequent presentation, a right-handed coordinate system is adopted throughout. In the right coordinate system, the positive direction of reference for all angles is taken counterclockwise.
This corresponds to the direction in which the x and y axes align when viewed from the positive direction of the axis
Free Vectors
A vector characterized only by length and direction in given system coordinates is called free. Free vector represented by a line segment given length and a direction, the beginning of which is located at any point in space. In the drawing, the vector is represented by an arrow (Fig. 3).
Vectors are designated by one bold letter or two letters corresponding to the beginning and end of an arrow with a dash above them or
The magnitude of a vector is called its modulus and is denoted in one of the following ways
Equality of vectors
Since the main characteristics of a vector are its length and direction, vectors are called equal if their directions and magnitudes coincide. In a particular case, equal vectors can be directed along one straight line. Equality of vectors, for example a and b (Fig. 4), is written as:
If the vectors (a and b) are equal in magnitude, but diametrically opposite in direction (Fig. 5), then this is written in the form:
Vectors that have the same or diametrically opposite directions are called collinear.
Multiplying a vector by a scalar
The product of vector a and scalar K is called a vector in modulus, equal in direction to vector a if K is positive, and diametrically opposite to it if K is negative.
Unit vector
A vector whose modulus equal to one and the direction coincides with the given vector a, is called the unit vector given vector or its ortom. Ort is denoted by . Any vector can be represented through its unit vector as
Unit vectors located along the positive directions of the coordinate axes are designated accordingly (Fig. 6).
Vector addition
The rule for adding vectors is postulated (the justification for this postulate is observations on real objects vector nature). This postulate is that two vectors
They are transferred to some point in space so that their origins coincide (Fig. 7). The directed diagonal of a parallelogram built on these vectors (Fig. 7) is called the sum of vectors; the addition of vectors is written in the form
and is called addition according to the parallelogram rule.
The specified rule for adding vectors can also be implemented in the following way: at any point in space, a vector is plotted further, a vector is plotted from the end of the vector (Fig. 8). A vector a, the beginning of which coincides with the beginning of the vector and the end of which coincides with the end of the vector, will be the sum of vectors
Final Rule Vector addition is convenient if you need to add more than two vectors. Indeed, if you need to add several vectors, then using specified rule, you should construct a polyline whose sides are the given vectors, and the beginning of any vector coincides with the end of the previous vector. The sum of these vectors will be a vector whose beginning coincides with the beginning of the first vector, and the end coincides with the end of the last vector (Fig. 9). If the given vectors form closed polygon, then we say that the sum of the vectors is zero.
From the rule for constructing the sum of vectors it follows that their sum does not depend on the order in which the terms are taken, or the addition of vectors is commutative. For two vectors, the latter can be written as:
Vector subtraction
Subtracting a vector from a vector is carried out according to the following rule: a vector is constructed and a vector - is laid off from its end (Fig. 10). Vector a, the beginning of which coincides with the beginning
vector and the end - with the end of the vector is equal to the difference between the vectors and The operation performed can be written in the form:
Vector decomposition into components
To decompose a given vector means to represent it as the sum of several vectors, which are called its components.
Let us consider the problem of decomposing the vector a, if it is specified that its components should be directed along three coordinate axes. To do this, we will construct a parallelepiped, the diagonal of which is the vector a and the edges are parallel to the coordinate axes (Fig. 11). Then, as is obvious from the drawing, the sum of the vectors located along the edges of this parallelepiped gives vector a:
Projection of a vector onto an axis
The projection of a vector onto an axis is the size of a directed segment, which is bounded by planes perpendicular to the axis, passing through the beginning and end of the vector (Fig. 12). The points of intersection of these planes with the axis (A and B) are called the projection of the beginning and end of the vector, respectively.
The projection of a vector has a plus sign if its directions, counting from the projection of the beginning of the vector to the projection of its end, coincide with the direction of the axis. If these directions do not coincide, then the projection has a minus sign.
The projections of vector a on the coordinate axes are designated accordingly
Vector coordinates
The components of vector a, located parallel to the coordinate axes through the projections of the vector and unit vectors can be written as:
Hence:
where they completely define the vector and are called its coordinates.
Denoting through the angles that vector a makes with the coordinate axes, the projections of vector a on the axes can be written in the form:
Hence for the modulus of vector a we have the expression:
Since the definition of a vector by its projections is unique, two equal vectors will have equal coordinates.
Addition of vectors through their coordinates
As follows from Fig. 13, the projection of the sum of vectors onto the axis is equal to algebraic sum their projections. Therefore, from the vector equality:
the following three scalar equalities follow:
or the coordinates of the total vector are equal to the algebraic sum of the coordinates of the component vectors.
Dot product of two vectors
The scalar product of two vectors is denoted a b and is determined by the product of their modules and the cosine of the angle between them:
The dot product of two vectors can also be defined as the product of the modulus of one of the vectors and the projection of the other vector onto the direction of the first vector.
From the definition of the scalar product it follows that
i.e., the commutative law takes place.
Relative to addition scalar product has the distributive property:
which directly follows from the property that the projection of the sum of vectors is equal to the algebraic sum of their projections.
The scalar product through projections of vectors can be written as:
Cross product of two vectors
The cross product of two vectors is denoted axb. This is a vector c whose modulus equal to the product moduli of the vectors being multiplied by the sine of the angle between them:
Vector c is directed perpendicular to the plane defined by vectors a and b so that if viewed from the end of vector c, then in order to align vector a with vector b as quickly as possible, the first vector had to be rotated in the positive direction (counterclockwise; Fig. 14). A vector representing vector product two vectors is called an axial vector (or pseudovector). Its direction depends on the choice of coordinate system or the condition on the positive direction of the angles. Direction indicated vector c corresponds to the right system of Cartesian coordinate axes, the choice of which was agreed upon earlier.
Vector quantity (vector)- This physical quantity, which has two characteristics - module 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 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) - vector physical quantity characterizing the speed of movement and direction of movement 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 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.
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 of a sewing machine, the piston in the cylinder of a steam engine or engine moves progressively 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 axes rotation.
Absolutely solid body- the second supporting object of mechanics along with the material point.
There are several definitions:
1. Absolutely rigid body - model concept 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 movement.
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 the apparatus with which physical laws can be precisely formulated.. theory Greek consideration.. standard method testing theories directly experimental verification experiment is a 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 acceleration. 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 movement
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 curvilinear movement body, we see that its speed is different moments different. 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 the two
Galileo's principle of relativity
· Inertial system reference system Galilean 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 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 kinetic energies all n material
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
Scalar and vector quantities
- Vector calculus (for example, displacement (s), force (F), acceleration (a), velocity (V) energy (E)).
scalar quantities that are completely determined by specifying them numerical values(length (L), area (S), volume (V), time (t), mass (m), etc.) ;
- Scalar quantities: temperature, volume, density, electric potential, potential energy of a body (for example, in a gravity field). Also the modulus of any vector (for example, those listed below).
Vector quantities: radius vector, speed, acceleration, electric field strength, intensity magnetic field. And many others :)
- vector quantity has numerical expression and direction: speed, acceleration, force, electromagnetic induction, displacement, etc., and the scalar is only a numerical expression: volume, density, length, width, height, mass (not to be confused with weight) temperature
- vector, for example, speed (v), force (F), displacement (s), impulse (p), energy (E). An arrow-vector is placed above each of these letters. that's why they are vector. and scalar ones are mass (m), volume (V), area (S), time (t), height (h)
- Vector movements are linear, tangential movements.
Scalar motions are closed motions that screen vector motions.
Vector movements are transmitted through scalar ones, as through intermediaries, just as current is transmitted from atom to atom through a conductor. - Scalar quantities: temperature, volume, density, electric potential, potential energy of a body (for example, in a gravity field). Also the modulus of any vector (for example, those listed below).
Vector quantities: radius vector, speed, acceleration, electric field strength, magnetic field strength. And many others:-
- A scalar quantity (scalar) is a physical quantity that has only one characteristic: a numerical value.
A scalar quantity can be positive or negative.
Examples scalar quantities: mass, temperature, path, work, time, period, frequency, density, energy, volume, electrical capacity, voltage, current, etc.
Mathematical operations with scalar quantities are algebraic operations.
Vector quantity
A vector quantity (vector) is a physical quantity that has two characteristics: module and direction in space.
Examples of vector quantities: speed, force, acceleration, tension, etc.
Geometrically, a vector is depicted as a directed segment of a straight line, the length of which is scaled to the modulus of the vector.
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.
Vector quantity except numerical value, 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're talking about 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 presents the main vector physical quantities.
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 one is linear speed. It is introduced when considering a rectilinear uniform motion. At the same time, she turns out equal to the ratio the distance traveled by the body to the time of movement.
The same formula can be used when 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 obtain visual representation about 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 forces acting on the body mechanical forces. 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. It turns out that it is not her who is more important appearance, and 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 customary to designate it Latin letter r.
Here the following question may arise: “Is the path a vector quantity?” IN general case this statement is not true. Path equal to length trajectory and has no specific direction. An exception is the situation when rectilinear movement in one direction is considered. 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, the acceleration can be both positive and negative meaning. At straight motion it is directed towards higher speed. If the movement occurs along curvilinear trajectory, 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 value acceleration. 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 important role has the law of conservation of momentum, 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. The masses of the platform and the car are 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 designations: the speed of the car before the impact is v1, the speed of the car with the platform after coupling is v, the mass of the car is m1, the mass of the platform is m2. According to the conditions of the problem, it is necessary to find out the value of the speed v.
Solution Rules similar 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 support reaction 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 equal to zero. Only the car moved, its momentum is the product of m1 and v1.
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: m1 * v1 = (m1 + m2) * 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 required speed: v = m1 * v1 / (m1 + m2).
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 number 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 m1 and m2. Their speeds will be v1 and v2 respectively. starting speed grenades - v. The problem requires calculating the value of v2.
In order for the larger fragment to continue to move in the same direction as the entire grenade, the second one must fly in reverse side. 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 down the law of conservation of momentum of a body in projection onto the OX axis, it will look like this: (m1 + m2) * v = m1 * v1 - m2 * v2. From it it is easy to express the required speed. It will be determined by the formula: v2 = ((m1 + m2) * v - m1 * v1) / m2. 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 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 general view, 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 the water flow in the river v1 and the boat's own speed v2 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 opposite bank 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 shows two similar to a triangle. 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 = v1 / v2. After the transformation, the formula for the desired value is obtained: s = l * (v1 / v2).
2). In this version of the problem, the total velocity vector is perpendicular to the shores. It is equal vector sum v1 and v2. The sine of the angle by which the natural velocity vector must deviate is equal to the ratio of the modules v1 and v2. 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 = √(v22 – v12), then t = l / (√(v22 – v12)).
Answer. 1). s = l * (v1 / v2), 2). sin α = v1 / v2, t = l / (√(v22 – v12)).