General equation of a straight line - theory, examples, problem solving. Write the equation of motion of a rigid body around a fixed axis General equation of a straight line - basic information
This article is part of the topic equation of a straight line in a plane. Here we will analyze from all sides: we will start with the proof of a theorem that defines the form of the general equation of a straight line, then we will consider an incomplete general equation of a straight line, we will give examples of incomplete equations of a straight line with graphic illustrations, in conclusion we will dwell on the transition from the general equation of a straight line to other types of equations of this straight line and we will give detailed solutions to typical problems on the compilation of the general equation of a straight line.
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General equation of a straight line - basic information.
Let's analyze this algorithm when solving an example.
Example.
Write the parametric equations of the straight line, which is given by the general equation of the straight line 
.
Solution.
First, we reduce the original general equation of a straight line to the canonical equation of a straight line:
Now we take the left and right parts of the resulting equation equal to the parameter . We have 
Answer:
From the general equation of a straight line, it is possible to obtain an equation of a straight line with a slope coefficient only when . What do you need to do to switch? Firstly, in the left side of the general equation of the straight line, only the term should be left, the remaining terms must be transferred to the right side with the opposite sign: 
. Secondly, divide both parts of the resulting equality by the number B , which is different from zero, 
. And that's all.
Example.
The line in the rectangular coordinate system Oxy is given by the general equation of the line. Get the equation of this line with the slope.
Solution.
Let's take the necessary steps:
Answer:
When a straight line is given by a complete general equation of a straight line, it is easy to obtain an equation of a straight line in segments of the form . To do this, we transfer the number C to the right side of the equality with the opposite sign, divide both parts of the resulting equality by -C, and in conclusion we transfer the coefficients for the variables x and y to the denominators: 
DETERMINING THE SPEED OF A MOUNTING CHUCK USING A BALLISTIC TORSIONAL PENDULUM
Goal of the work: study of conservation laws on the example of a ballistic torsion pendulum.
Instruments and accessories: ballistic torsion pendulum, a set of mounting cartridges, a millisecond watch block.
Description of the experimental setup
General form ballistic pendulum is shown in the figure. Base 1 equipped with adjustable legs 2 to level the instrument. Column fixed at the base 3 , on which the upper 4 , bottom 5 and middle 6 brackets. A firing device is attached to the middle bracket 7 , as well as a transparent screen with an angular scale printed on it 8 and photoelectric sensor 9 . brackets 4 And 5 have clamps for attaching steel wire 10 , on which a pendulum is suspended, consisting of two bowls filled with plasticine 11 , two transportable goods 12 , two rods 13 , walkers 14 .
Work order
1. Having removed the transparent screen, set the weights at a distance r1 from the axis of rotation.
3. Insert the chuck into the spring device.
4. Push the cartridge out of the spring device.
6. Turn on the time counter (on the panel, the indicators of the meter show "0").
7. Deviate the pendulum at an angle φ1, and then let it go.
8. Press the "STOP" button, when the counter shows nine oscillations, record the time of ten full oscillations t1. Calculate the oscillation period T1. Enter the data in table No. 1, repeat points 7.8 four more times.
9. Install weights at distance r2. Follow steps 2-8 for distances r2.
10. Calculate the formula for the speed for five measurements:
11. Estimate the absolute error in calculating the speed by analyzing five speed values (Table No. 1).
r \u003d 0.12 m, m \u003d 3.5 g., M \u003d 0.193 kg.
Table #1
| experience number | r1 = 0.09 m | r2 = 0.02 m | |||||||
| φ1 | t1 | T1 | φ2 | t2 | T2 | V | |||
| deg. | glad. | With | deg. | glad. | With | m/s | |||
| 1. | |||||||||
| 2. | |||||||||
| 3. | |||||||||
| 4. | |||||||||
| 5. |
Settlement part
Control questions
Formulate the law of conservation of angular momentum.
The angular momentum of the "chuck-pendulum" system relative to the axis is conserved:
Formulate the law of conservation of energy.
When the pendulum oscillates, the kinetic energy of the rotational motion of the system is converted into the potential energy of the elastically deformed wire during torsion:
Write the equation of motion of a rigid body around a fixed axis
4. What is a torsion pendulum and how is the period of its oscillation determined?
A torsion pendulum is a massive steel rod rigidly attached to a vertical wire. At the ends of the rod, bowls with plasticine are fixed, which allows the cartridge to “stick” to the pendulum. Also on the rod there are two identical weights that can move along the rod relative to its axis of rotation. This makes it possible to change the moment of inertia of the pendulum. A “walker” is rigidly fixed to the pendulum, allowing photoelectric sensors to count the number of its full oscillations. Torsional vibrations are caused by elastic forces arising in the wire during its torsion. In this case, the period of oscillation of the pendulum:
5. How else can you determine the speed of the mounting chuck in this work?
1.AB=2j-3j.1)Find the coordinates of point A if B(-1;4).2)Find the coordinates of the midpoint of the segment AB.3)Write the equation of the straight line AB.2.The points are givenA (-3; 4), B (2; 1), C (-1; a). It is known that AB \u003d BC. Find a.3. The radius of the circle is 6. The center of the circle belongs to the Ox axis and has a positive abscissa. The circle passes through the point (5; 0). Write the equation of the circle. 4. The vector a is co-directed with the vector b (-1; 2) and has the length of the vector c (-3; 4).
vector a (5; - 9). The answer should be 2x - 3y = 38.
2. With parallel transfer, point A (4:3) goes to point A1 (5;4). Write the equation of the curve into which the parabola y \u003d x ^ 2 (meaning x squared) - 3x + 1 passes with such a movement. The answer should be: x^2 - 5x +6.
Help Please with questions on geometry (Grade 9)! 1) Formulate and prove a lemma about collinear vectors. 2) What does it mean to decompose a vector in twogiven vectors. 3) Formulate and prove a theorem on the expansion of a vector in two non-collinear vectors. 4) Explain how a rectangular coordinate system is introduced. 5) What are coordinate vectors? 6) Formulate and prove the statement about the decomposition of an arbitrary vector in coordinate vectors. 7) What are vector coordinates? 8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector by a number according to the given coordinates of the vectors. 9) What is the radius vector of a point? Prove that the coordinates of a point are equal to the corresponding coordinates of the vectors. 10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end. 11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends. 12) Derive a formula for calculating the length of a vector by its coordinates. 13) Derive a formula for calculating the distance between two points by their coordinates. 14) Give an example of solving a geometric problem using the coordinate method. 15) What equation is called the equation of this line? Give an example. 16) Derive the equation of a circle of a given radius centered at a given point. 17) Write the equation for a circle of given radius centered at the origin. 18) Derive the equation of this line in a rectangular coordinate system. 19) Write the equation of the lines passing through the given point M0 (X0: Y0) and parallel to the coordinate axes. 20) Write the equation of the coordinate axes. 21) Give examples of using the equations of a circle and a straight line in solving geometric problems.
1) Formulate and prove a lemma about collinear vectors.2) What does it mean to decompose a vector in two given vectors.
3) Formulate and prove a theorem on the expansion of a vector in two non-collinear vectors.
4) Explain how a rectangular coordinate system is introduced.
5) What are coordinate vectors?
6) Formulate and prove the statement about the decomposition of an arbitrary vector in coordinate vectors.
7) What are vector coordinates?
8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector by a number according to the given coordinates of the vectors.
9) What is the radius vector of a point? Prove that the coordinates of the point are equal to the corresponding coordinates of the vectors.
10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end.
11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends.
12) Derive a formula for calculating the length of a vector by its coordinates.
13) Derive a formula for calculating the distance between two points by their coordinates.
14) Give an example of solving a geometric problem using the coordinate method.
15) What equation is called the equation of this line? Give an example.
16) Derive the equation of a circle of a given radius centered at a given point.
17) Write the equation for a circle of given radius centered at the origin.
18) Derive the equation of this line in a rectangular coordinate system.
19) Write the equation of the lines passing through the given point M0 (X0: Y0) and parallel to the coordinate axes.
20) Write the equation of the coordinate axes.
21) Give examples of using the equations of a circle and a straight line in solving geometric problems.
Please, it's very necessary! Preferably with drawings (where necessary)!
