The rules by which vectors are added. Laws of addition of forces in mechanics Vector sum of forces acting on a moving
When several forces are simultaneously applied to one body, the body begins to move with acceleration, which is the vector sum of the accelerations that would arise under the influence of each force separately. The rule of vector addition is applied to forces acting on a body and applied to one point.
Definition 1
The vector sum of all forces simultaneously acting on a body is the force resultant, which is determined by the rule of vector addition of forces:
R → = F 1 → + F 2 → + F 3 → + . . . + F n → = ∑ i = 1 n F i → .
Equals effective force acts on the body in the same way as the sum of all forces acting on it.
Definition 2To add 2 forces use rule parallelogram(picture 1).
Picture 1 . Addition of 2 forces according to the parallelogram rule
Let us derive the formula for the modulus of the resultant force using the cosine theorem:
R → = F 1 → 2 + F 2 → 2 + 2 F 1 → 2 F 2 → 2 cos α
Definition 3
If it is necessary to add more than 2 forces, use polygon rule: from the end
The 1st force must draw a vector equal and parallel to the 2nd force; from the end of the 2nd force it is necessary to draw a vector equal and parallel to the 3rd force, etc.
Figure 2. Addition of forces using the polygon rule
The final vector drawn from the point of application of forces to the end of the last force is equal in magnitude and direction to the resultant force. Figure 2 clearly illustrates an example of finding the resultant forces from 4 forces: F 1 →, F 2 →, F 3 →, F 4 →. Moreover, the summed vectors do not necessarily have to be in the same plane.
The result of the force acting on a material point will depend only on its modulus and direction. U solid There are certain sizes. Therefore, forces with the same magnitudes and directions cause different movements of a rigid body depending on the point of application.
Definition 4
Line of action of force called a straight line passing through the force vector.
Figure 3. Addition of forces applied to different points of the body
If forces are applied to different points of the body and do not act parallel to each other, then the resultant is applied to the point of intersection of the lines of action of the forces (Figure 3 ). A point will be in equilibrium if the vector sum of all forces acting on it is equal to 0: ∑ i = 1 n F i → = 0 → . IN in this case equals 0 and the sum of the projections of these forces onto any coordinate axis.
Definition 5Decomposition of forces into two components- this is the replacement of one force by 2, applied at the same point and producing the same effect on the body as this one force. The decomposition of forces is carried out, like addition, by the parallelogram rule.
The problem of decomposing one force (the modulus and direction of which are given) into 2, applied at one point and acting at an angle to each other, has a unique solution in the following cases when the following are known:
- directions of 2 component forces;
- module and direction of one of the component forces;
- modules of 2 component forces.
It is necessary to decompose the force F into 2 components located in the same plane with F and directed along straight lines a and b (Figure 4 ). Then it is enough to draw 2 straight lines from the end of the vector F, parallel to straight lines a and b. The segment F A and the segment F B represent the required forces.
Figure 4. Decomposition of the force vector in directions
Example 2
The second version of this problem is to find one of the projections of the force vector using the given force vectors and the 2nd projection (Figure 5 a).
Figure 5. Finding the projection of the force vector from given vectors
In the second version of the problem, it is necessary to construct a parallelogram along the diagonal and one of the sides, as in planimetry. Figure 5 b shows such a parallelogram and indicates the desired component F 2 → force F → .
So, the 2nd solution: add to the force a force equal to - F 1 → (Figure 5 c). As a result, we obtain the desired force F →.
Example 3
Three forces F 1 → = 1 N; F 2 → = 2 N; F 3 → = 3 N are applied to one point, are in the same plane (Figure 6 a) and make angles with the horizontal α = 0 °; β = 60°; γ = 30° respectively. It is necessary to find the resultant force.
Solution
Figure 6. Finding the resultant force from given vectors
Let's draw mutually perpendicular axes O X and O Y so that the O X axis coincides with the horizontal along which the force F 1 → is directed. Let's make a projection of these forces onto the coordinate axes (Figure 6 b). The projections F 2 y and F 2 x are negative. The sum of the projections of forces onto the coordinate axis O X is equal to the projection onto this axis of the resultant: F 1 + F 2 cos β - F 3 cos γ = F x = 4 - 3 3 2 ≈ - 0.6 N.
Similarly, for projections onto the O Y axis: - F 2 sin β + F 3 sin γ = F y = 3 - 2 3 2 ≈ - 0.2 N.
We determine the modulus of the resultant using the Pythagorean theorem:
F = F x 2 + F y 2 = 0.36 + 0.04 ≈ 0.64 N.
We find the direction of the resultant using the angle between the resultant and the axis (Figure 6 c):
t g φ = F y F x = 3 - 2 3 4 - 3 3 ≈ 0.4.
Example 4
A force F = 1 kN is applied at point B of the bracket and is directed vertically downward (Figure 7 a). It is necessary to find the components of this force in the directions of the bracket rods. All necessary data is shown in the figure.
Solution
Figure 7. Finding the components of force F in the directions of the bracket rods
Given:
F = 1 k N = 1000 N
Let the rods be screwed to the wall at points A and C. Figure 7 b shows the decomposition of the force F → into components along the directions A B and B C. From here it is clear that
F 1 → = F t g β ≈ 577 N;
F 2 → = F cos β ≈ 1155 N.
Answer: F 1 → = 557 N; F 2 → = 1155 N.
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According to Newton's first law, in inertial frames of reference, a body can change its speed only if other bodies act on it. The mutual action of bodies on each other is expressed quantitatively using the following physical quantity, as force (). A force can change the speed of a body, both in magnitude and in direction. Force is a vector quantity; it has a modulus (magnitude) and a direction. The direction of the resultant force determines the direction of the acceleration vector of the body on which the force in question acts.
The basic law by which the direction and magnitude of the resultant force is determined is Newton’s second law:
where m is the mass of the body on which the force acts; - the acceleration that the force imparts to the body in question. The essence of Newton's second law is that the forces that act on a body determine the change in the speed of the body, and not just its speed. It must be remembered that Newton's second law works for inertial frames of reference.
If several forces act on a body, then their combined action is characterized by the resultant force. Let us assume that several forces act on the body simultaneously, and the body moves with an acceleration equal to the vector sum of the accelerations that would appear under the influence of each of the forces separately. The forces acting on the body and applied to one point must be added according to the rule of vector addition. The vector sum of all forces acting on a body at one moment in time is called the resultant force ():
When several forces act on a body, Newton's second law is written as:
The resultant of all forces acting on the body can be equal to zero if there is mutual compensation of the forces applied to the body. In this case, the body moves with constant speed or is at rest.
When depicting forces acting on a body in a drawing, in the case of uniformly accelerated movement of the body, the resultant force directed along the acceleration should be depicted longer than the oppositely directed force (sum of forces). When uniform motion(or rest) the dynamics of the vectors of forces directed in opposite directions are the same.
To find the resultant force, you should depict in the drawing all the forces that must be taken into account in the problem acting on the body. Forces should be added according to the rules of vector addition.
Examples of solving problems on the topic “Resultant force”
EXAMPLE 1
| Exercise | A small ball hangs on a thread, it is at rest. What forces act on this ball, depict them in the drawing. What is the resultant force applied to the body? |
| Solution | Let's make a drawing. Let's consider the reference system associated with the Earth. In our case, this reference system can be considered inertial. A ball suspended on a thread is acted upon by two forces: the force of gravity directed vertically downward () and the reaction force of the thread (tension force of the thread): . Since the ball is at rest, the force of gravity is balanced by the tension force of the thread: Expression (1.1) corresponds to Newton’s first law: the resultant force applied to a body at rest in an inertial frame of reference is zero. |
| Answer | The resultant force applied to the ball is zero. |
EXAMPLE 2
| Exercise | Two forces act on the body and and , where - constants. . What is the resultant force applied to the body? |
| Solution | Let's make a drawing. Since the vectors of force and are perpendicular to each other, therefore, we find the length of the resultant as: |
How vector addition occurs is not always clear to students. Children have no idea what is hidden behind them. You just have to remember the rules, and not think about the essence. Therefore, it is about the principles of addition and subtraction of vector quantities that a lot of knowledge is required.
The addition of two or more vectors always results in one more. Moreover, it will always be the same, regardless of how it is found.
Most often in school course geometry considers the addition of two vectors. It can be performed according to the triangle or parallelogram rule. These drawings look different, but the result of the action is the same.
How does addition occur using the triangle rule?
It is used when the vectors are non-collinear. That is, they do not lie on the same straight line or on parallel ones.
In this case, the first vector must be plotted from some arbitrary point. From its end it is required to draw parallel and equal to the second. The result will be a vector starting from the beginning of the first and ending at the end of the second. The pattern resembles a triangle. Hence the name of the rule.
If the vectors are collinear, then this rule can also be applied. Only the drawing will be located along one line.
How is addition performed using the parallelogram rule?
Yet again? applies only to non-collinear vectors. The construction is carried out according to a different principle. Although the beginning is the same. We need to set aside the first vector. And from its beginning - the second. Based on them, complete the parallelogram and draw a diagonal from the beginning of both vectors. This will be the result. This is how vector addition is performed according to the parallelogram rule.
So far there have been two. But what if there are 3 or 10 of them? Use the following technique.
How and when does the polygon rule apply?
If you need to perform addition of vectors, the number of which is more than two, do not be afraid. It is enough to put them all aside sequentially and connect the beginning of the chain with its end. This vector will be the required amount.
What properties are valid for operations with vectors?
About the zero vector. Which states that when added to it, the original is obtained.
About the opposite vector. That is, about one that has the opposite direction and equal magnitude. Their sum will be zero.
On the commutativity of addition. What has been known since primary school. Changing the positions of the terms does not change the result. In other words, it doesn't matter which vector to put off first. The answer will still be correct and unique.
On the associativity of addition. This law allows you to add any vectors from a triple in pairs and add a third one to them. If you write this using symbols, you get the following:
first + (second + third) = second + (first + third) = third + (first + second).
What is known about vector difference?
There is no separate subtraction operation. This is due to the fact that it is essentially addition. Only the second of them is given the opposite direction. And then everything is done as if adding vectors were considered. Therefore, there is practically no talk about their difference.
In order to simplify the work with their subtraction, the triangle rule is modified. Now (when subtracting) the second vector must be set aside from the beginning of the first. The answer will be the one that connects the end point of the minuend with the same one as the subtrahend. Although you can postpone it as described earlier, simply by changing the direction of the second.
How to find the sum and difference of vectors in coordinates?
The problem gives the coordinates of the vectors and requires finding out their values for the final result. In this case, there is no need to perform constructions. That is, you can use simple formulas that describe the rule for adding vectors. They look like this:
a (x, y, z) + b (k, l, m) = c (x + k, y + l, z + m);
a (x, y, z) -b (k, l, m) = c (x-k, y-l, z-m).
It is easy to see that the coordinates simply need to be added or subtracted depending on the specific task.
First example with solution
Condition. Given a rectangle ABCD. Its sides are equal to 6 and 8 cm. The intersection point of the diagonals is designated by the letter O. It is required to calculate the difference between the vectors AO and VO.
Solution. First you need to draw these vectors. They are directed from the vertices of the rectangle to the point of intersection of the diagonals.
If you look closely at the drawing, you can see that the vectors are already combined so that the second of them is in contact with the end of the first. It's just that his direction is wrong. It should start from this point. This is if the vectors are adding, but the problem involves subtraction. Stop. This action means that you need to add the oppositely directed vector. This means that VO needs to be replaced with OV. And it turns out that the two vectors have already formed a pair of sides from the triangle rule. Therefore, the result of their addition, that is, the desired difference, is the vector AB.
And it coincides with the side of the rectangle. To write down your numerical answer, you will need the following. Draw a rectangle lengthwise so that the larger side is horizontal. Start numbering the vertices from the bottom left and go counterclockwise. Then the length of vector AB will be 8 cm.
Answer. The difference between AO and VO is 8 cm.
Second example and its detailed solution
Condition. The diagonals of the rhombus ABCD are 12 and 16 cm. The point of their intersection is designated by the letter O. Calculate the length of the vector formed by the difference between the vectors AO and VO.
Solution. Let the designation of the vertices of the rhombus be the same as in the previous problem. Similar to the solution to the first example, it turns out that the required difference is equal to the vector AB. And its length is unknown. Solving the problem came down to calculating one of the sides of the rhombus.
For this purpose, you will need to consider the triangle ABO. It is rectangular because the diagonals of a rhombus intersect at an angle of 90 degrees. And its legs are equal to half the diagonals. That is, 6 and 8 cm. The side sought in the problem coincides with the hypotenuse in this triangle.
To find it you will need the Pythagorean theorem. The square of the hypotenuse will be equal to the sum numbers 6 2 and 8 2. After squaring, the values obtained are: 36 and 64. Their sum is 100. It follows that the hypotenuse is equal to 10 cm.
Answer. The difference between the vectors AO and VO is 10 cm.
Third example with detailed solution
Condition. Calculate the difference and sum of two vectors. Their coordinates are known: the first one has 1 and 2, the second one has 4 and 8.
Solution. To find the sum you will need to add the first and second coordinates in pairs. The result will be the numbers 5 and 10. The answer will be a vector with coordinates (5; 10).
For the difference, you need to subtract the coordinates. After performing this action, the numbers -3 and -6 will be obtained. They will be the coordinates of the desired vector.
Answer. The sum of the vectors is (5; 10), their difference is (-3; -6).
Fourth example
Condition. The length of the vector AB is 6 cm, BC is 8 cm. The second is laid off from the end of the first at an angle of 90 degrees. Calculate: a) the difference between the modules of the vectors VA and BC and the module of the difference between VA and BC; b) the sum of the same modules and the module of the sum.
Solution: a) The lengths of the vectors are already given in the problem. Therefore, calculating their difference is not difficult. 6 - 8 = -2. The situation with the difference module is somewhat more complicated. First you need to find out which vector will be the result of the subtraction. For this purpose, one should set aside the vector VA, which is directed towards the opposite side AB. Then draw the vector BC from its end, directing it in the direction opposite to the original one. The result of subtraction is the vector CA. Its modulus can be calculated using the Pythagorean theorem. Simple calculations lead to a value of 10 cm.
b) The sum of the moduli of the vectors is equal to 14 cm. To find the second answer, some transformation will be required. Vector BA is oppositely directed to that given - AB. Both vectors are directed from the same point. In this situation, you can use the parallelogram rule. The result of the addition will be a diagonal, and not just a parallelogram, but a rectangle. Its diagonals are equal, which means that the modulus of the sum is the same as in the previous paragraph.
Answer: a) -2 and 10 cm; b) 14 and 10 cm.
This is the vector sum of all forces acting on the body.
The cyclist leans towards the turn. The force of gravity and the reaction force of the support from the side of the earth give a resultant force that imparts centripetal acceleration required for circular motion
Relationship with Newton's second law
Let's remember Newton's law: 
The resultant force can be equal to zero in the case when one force is compensated by another, the same force, but opposite in direction. In this case, the body is at rest or moving uniformly.
If the resultant force is NOT zero, then the body moves with uniform acceleration. Actually, it is this force that is the reason uneven movement. Direction of resultant force Always coincides in direction with the acceleration vector.
When it is necessary to depict the forces acting on a body, while the body moves with uniform acceleration, it means that in the direction of acceleration the acting force is longer than the opposite one. If the body moves uniformly or is at rest, the length of the force vectors is the same.
Finding the resultant force
In order to find the resultant force, it is necessary: firstly, to correctly designate all the forces acting on the body; then draw coordinate axes, select their directions; in the third step it is necessary to determine the projections of the vectors on the axes; write down the equations. Briefly: 1) identify the forces; 2) select the axes and their directions; 3) find the projections of forces on the axis; 4) write down the equations.
How to write equations? If in a certain direction the body moves uniformly or is at rest, then the algebraic sum (taking into account signs) of the projections of forces is equal to zero. If a body moves uniformly accelerated in a certain direction, then the algebraic sum of the projections of forces is equal to the product of mass and acceleration, according to Newton’s second law.
Examples
A body moving uniformly on a horizontal surface is subject to the force of gravity, the reaction force of the support, the force of friction and the force under which the body moves.
Let us denote the forces, choose the coordinate axes
Let's find the projections
Writing down the equations
A body that is pressed against a vertical wall moves downward with uniform acceleration. The body is acted upon by the force of gravity, the force of friction, the reaction of the support and the force with which the body is pressed. The acceleration vector is directed vertically downwards. The resultant force is directed vertically downwards.
The body moves uniformly along a wedge whose slope is alpha. The body is acted upon by the force of gravity, the reaction force of the support, and the force of friction.
The main thing to remember
1) If the body is at rest or moving uniformly, then the resultant force is zero and the acceleration is zero;
2) If the body moves uniformly accelerated, then the resultant force is not zero;
3) The direction of the resultant force vector always coincides with the direction of acceleration;
4) Be able to write equations of projections of forces acting on a body
A block is a mechanical device, a wheel that rotates around its axis. Blocks can be mobile And motionless.
Fixed block used only to change the direction of force.
Bodies connected by an inextensible thread have equal accelerations.
Movable block designed to change the amount of effort applied. If the ends of the rope clasping the block make equal angles with the horizon, then lifting the load will require a force half as much as the weight of the load. The force acting on a load is related to its weight as the radius of a block is to the chord of an arc encircled by a rope.
The acceleration of body A is half the acceleration of body B.
In fact, any block is lever arm, in the case of a fixed block - equal arms, in the case of a movable one - with a ratio of shoulders of 1 to 2. As for any other lever, the following rule applies to the block: the number of times we win in effort, the same number of times we lose in distance
A system consisting of a combination of several movable and fixed blocks is also used. This system is called a polyspast.
When several forces act simultaneously on one body, the body moves with acceleration, which is the vector sum of the accelerations that would arise under the action of each force separately. The forces acting on a body and applied to one point are added according to the rule of vector addition.
The vector sum of all forces simultaneously acting on a body is called the resultant force and is determined by the rule of vector addition of forces: $\overrightarrow(R)=(\overrightarrow(F))_1+(\overrightarrow(F))_2+(\overrightarrow(F)) _3+\dots +(\overrightarrow(F))_n=\sum^n_(i=1)((\overrightarrow(F))_i)$.
The resultant force has the same effect on a body as the sum of all forces applied to it.
To add two forces, the parallelogram rule is used (Fig. 1):
Figure 1. Addition of two forces according to the parallelogram rule
In this case, we find the modulus of the sum of two forces using the cosine theorem:
\[\left|\overrightarrow(R)\right|=\sqrt((\left|(\overrightarrow(F))_1\right|)^2+(\left|(\overrightarrow(F))_2\right |)^2+2(\left|(\overrightarrow(F))_1\right|)^2(\left|(\overrightarrow(F))_2\right|)^2(cos \alpha \ ))\ ]
If you need to add more than two forces applied at one point, then use the polygon rule: ~ from the end of the first force draw a vector equal and parallel to the second force; from the end of the second force - a vector equal and parallel to the third force, and so on.
Figure 2. Addition of forces according to the polygon rule
The closing vector drawn from the point of application of forces to the end of the last force is equal in magnitude and direction to the resultant. In Fig. 2 this rule is illustrated by the example of finding the resultant of four forces $(\overrightarrow(F))_1,\ (\overrightarrow(F))_2,(\overrightarrow(F))_3,(\overrightarrow(F) )_4$. Note that the vectors being added do not necessarily belong to the same plane.
The result of a force acting on a material point depends only on its modulus and direction. A solid body has certain dimensions. Therefore, forces of equal magnitude and direction cause different movements of a rigid body depending on the point of application. The straight line passing through the force vector is called the line of action of the force.
Figure 3. Addition of forces applied to different points body
If forces are applied to different points of the body and do not act parallel to each other, then the resultant is applied to the point of intersection of the lines of action of the forces (Fig. 3).
A point is in equilibrium if the vector sum of all forces acting on it is equal to zero: $\sum^n_(i=1)((\overrightarrow(F))_i)=\overrightarrow(0)$. In this case, the sum of the projections of these forces onto any coordinate axis is also zero.
The replacement of one force by two, applied at the same point and producing the same effect on the body as this one force, is called the decomposition of forces. The decomposition of forces is carried out, as is their addition, according to the parallelogram rule.
The problem of decomposing one force (the modulus and direction of which are known) into two, applied at one point and acting at an angle to each other, has a unique solution in the following cases, if known:
- directions of both components of forces;
- module and direction of one of the component forces;
- modules of both component forces.
Let, for example, we want to decompose the force $F$ into two components lying in the same plane with F and directed along straight lines a and b (Fig. 4). To do this, it is enough to draw two lines parallel to a and b from the end of the vector representing F. The segments $F_A$ and $F_B$ will depict the required forces.
Figure 4. Decomposition of the force vector by directions
Another version of this problem is to find one of the projections of the force vector given the force vectors and the second projection. (Fig. 5 a).
Figure 5. Finding the projection of the force vector using given vectors
The problem comes down to constructing a parallelogram along the diagonal and one of the sides, known from planimetry. In Fig. 5b such a parallelogram is constructed and the required component $(\overrightarrow(F))_2$ of the force $(\overrightarrow(F))$ is indicated.
The second solution is to add to the force a force equal to - $(\overrightarrow(F))_1$ (Fig. 5c). As a result, we obtain the desired force $(\overrightarrow(F))_2$.
Three forces~$(\overrightarrow(F))_1=1\ N;;\ (\overrightarrow(F))_2=2\ N;;\ (\overrightarrow(F))_3=3\ N$ applied to one point, lie in the same plane (Fig. 6 a) and make angles~ with the horizontal $\alpha =0()^\circ ;;\beta =60()^\circ ;;\gamma =30()^\ circ $respectively. Find the resultant of these forces.
Let us draw two mutually perpendicular axes OX and OY so that the OX axis coincides with the horizontal along which the force $(\overrightarrow(F))_1$ is directed. Let's project these forces onto the coordinate axes (Fig. 6 b). The projections $F_(2y)$ and $F_(2x)$ are negative. The sum of the projections of forces onto the OX axis is equal to the projection onto this axis of the resultant: $F_1+F_2(cos \beta \ )-F_3(cos \gamma \ )=F_x=\frac(4-3\sqrt(3))(2)\ approx -0.6\ H$. Similarly, for projections onto the OY axis: $-F_2(sin \beta \ )+F_3(sin \gamma =F_y=\ )\frac(3-2\sqrt(3))(2)\approx -0.2\ H$ . The modulus of the resultant is determined by the Pythagorean theorem: $F=\sqrt(F^2_x+F^2_y)=\sqrt(0.36+0.04)\approx 0.64\ Н$. The direction of the resultant is determined using the angle between the resultant and the axis (Fig. 6 c): $tg\varphi =\frac(F_y)(F_x)=\ \frac(3-2\sqrt(3))(4-3\sqrt (3))\approx 0.4$
The force $F = 1kH$ is applied at point B of the bracket and is directed vertically downwards (Fig. 7a). Find the components of this force in the directions of the bracket rods. The required data is shown in the figure.
F = 1 kN = 1000N
$(\mathbf \beta )$ = $30^(\circ)$
$(\overrightarrow(F))_1,\ (\overrightarrow(F))_2$ - ?
Let the rods be attached to the wall at points A and C. The decomposition of the force $(\overrightarrow(F))$ into components along the directions AB and BC is shown in Fig. 7b. This shows that $\left|(\overrightarrow(F))_1\right|=Ftg\beta \approx 577\ H;\ \ $
\[\left|(\overrightarrow(F))_2\right|=F(cos \beta \ )\approx 1155\ H. \]
Answer: $\left|(\overrightarrow(F))_1\right|$=577 N; $\left|(\overrightarrow(F))_2\right|=1155\ Н$