The resultant of two forces. What is the resultant of the forces F1 and F2 acting on the cart what? What are the forces f1 and f2 equal to

To answer this question, it is necessary to draw some conclusions from the problem conditions:

1. The direction of these forces;
2. Modular value of forces F1 and F2;
3. Can these forces create such a resultant force to move the cart from its place?

Direction of forces

In order to determine the main characteristics of the movement of a cart under the influence of two forces, it is necessary to know their direction. For example, if a cart is pulled to the right by a force equal to 5 N and the same force is pulling the cart to the left, then it is logical to assume that the cart will stand still. If the forces are codirectional, to find the resultant force it is only necessary to find their sum. If any force is directed at an angle to the plane of motion of the cart, then the value of this force must be multiplied by the cosine of the angle between the direction of the force and the plane. Mathematically it would look like this:

F = F1 * cosa; Where

F – force directed parallel to the surface of motion.

The cosine theorem for finding the resulting vector of forces

If two forces have their origin at one point and there is a certain angle between their direction, then it is necessary to complete the triangle with the resulting vector (that is, the one that connects the ends of the vectors F1 and F2). Let's find the resulting force using the cosine theorem, which states that the square of any side of a triangle equal to the sum the squares of the other two sides of the triangle minus twice the product of these sides and the cosine of the angle between them. Let's write this in mathematical form:

F = F 1 2 + F 2 2 - 2 * F 1 * F 2 * cosa.

By substituting all known quantities, you can determine the magnitude of the resulting force.

Resultant. You already know that two forces balance each other when they are equal in magnitude and directed in opposite directions. Such, for example, are the force of gravity and the force of normal reaction acting on a book lying on a table. In this case, the resultant of the two forces is said to be zero. In general, the resultant of two or more forces is a force that produces the same effect on a body as the simultaneous action of these forces.

Let us consider experimentally how to find the resultant of two forces directed along one straight line.

Let's put experience

Let's place a light block on a smooth horizontal surface of the table (so that the friction between the block and the table surface can be neglected). We will pull the block to the right using one dynamometer, and to the left using two dynamometers, as shown in Fig. 16.3. Please note that the dynamometers on the left are attached to the block so that the tension forces of the springs of these dynamometers are different.

Rice. 16.3. How can you find the resultant of two forces?

We will see that the block is at rest if the magnitude of the force that pulls it to the right is equal to the sum of the magnitudes of the forces pulling the block to the left. The diagram of this experiment is shown in Fig. 16.4.

Rice. 16.4. Schematic representation of the forces acting on the block

The force F 3 balances the resultant of the forces F 1 and F 2, that is, it is equal to it in magnitude and opposite in direction. This means that the resultant of the forces F 1 and F 2 is directed to the left (like these forces), and its module is equal to F 1 + F 2. Thus, if two forces are directed in the same way, their resultant is directed in the same way as these forces, and the modulus of the resultant is equal to the sum of the moduli of the component forces.

Let's consider the force F 1. It balances the resultant forces F 2 and F 3, directed in opposite directions. This means that the resultant of the forces F 2 and F 3 is directed to the right (that is, towards the larger of these forces), and its module is equal to F 3 - F 2. Thus, if two forces that are not equal in magnitude are directed oppositely, their resultant is directed as the larger of these forces, and the module of the resultant is equal to the difference between the modules of the larger and smaller force.

Finding the resultant of several forces is called the addition of these forces.

Two forces are directed along one straight line. The modulus of one force is equal to 1 N, and the modulus of the other force is equal to 2 N. Can the modulus of the resultant of these forces be equal to: a) zero; b) 1 N; c) 2 N; d) 3 N?

The content of the article

STATICS, branch of mechanics, the subject of which is material bodies, which are at rest when exposed to external forces. In the broadest sense of the word, statics is the theory of equilibrium of any body - solid, liquid or gaseous. In a narrower sense, this term refers to the study of the equilibrium of solid bodies, as well as non-stretchable flexible bodies - cables, belts and chains. The equilibrium of deforming solids is considered in the theory of elasticity, and the equilibrium of liquids and gases is considered in hydroaeromechanics.
Cm. HYDROAEROMECHANICS.

Historical reference.

Statics is the oldest section of mechanics; some of its principles were already known to the ancient Egyptians and Babylonians, as evidenced by the pyramids and temples they built. Among the first creators of theoretical statics was Archimedes (c. 287–212 BC), who developed the theory of the lever and formulated the fundamental law of hydrostatics. The founder of modern statics was the Dutchman S. Stevin (1548–1620), who in 1586 formulated the law of addition of forces, or the parallelogram rule, and applied it to solve a number of problems.

Basic laws.

The laws of statics follow from general laws speakers like special case, when the velocities of solid bodies tend to zero, but historical reasons and pedagogical considerations, statics is often presented independently of dynamics, building it on the following postulated laws and principles: a) the law of addition of forces, b) the principle of equilibrium and c) the principle of action and reaction. In the case of solids (more precisely, ideally solid bodies that do not deform under the influence of forces), another principle is introduced, based on the definition of a rigid body. This is the principle of force transfer: the state of a solid body does not change when the point of application of force moves along the line of its action.

Force as a vector.

In statics, force can be considered as a pulling or pushing force that has a certain direction, magnitude and point of application. From a mathematical point of view, it is a vector, and therefore it can be represented by a directed segment of a straight line, the length of which is proportional to the magnitude of the force. (Vector quantities, unlike other quantities that do not have a direction, are denoted by bold letters.)

Parallelogram of forces.

Let's consider the body (Fig. 1, A), which is acted upon by forces F 1 and F 2 applied at point O and represented in the figure by directed segments O.A. And O.B.. As experience shows, the action of forces F 1 and F 2 is equivalent to one force R, represented by the segment O.C.. Magnitude of force R equal to the length of the diagonal of a parallelogram built on vectors O.A. And O.B. like its sides; its direction is shown in Fig. 1, A. Force R called the resultant force F 1 and F 2. Mathematically this is written as R = F 1 + F 2, where addition is understood in geometrically words mentioned above. This is the first law of statics, called the rule of parallelogram of forces.

Resultant force.

Instead of constructing a parallelogram OACB, to determine the direction and magnitude of the resultant R you can construct triangle OAC by moving the vector F 2 parallel to itself until its starting point (former point O) coincides with the end (point A) of the vector O.A.. The trailing side of triangle OAC will obviously have the same magnitude and the same direction as the vector R(Fig. 1, b). This method of finding the resultant can be generalized to a system of many forces F 1 , F 2 ,..., F n applied at the same point O of the body under consideration. So, if the system consists of four forces (Fig. 1, V), then we can find the resultant force F 1 and F 2, fold it with force F 3, then add the new resultant with force F 4 and as a result obtain the full resultant R. Resultant R, found by such a graphical construction, is represented by the closing side of the polygon of forces OABCD (Fig. 1, G).

The above definition of the resultant can be generalized to a system of forces F 1 , F 2 ,..., F n applied at points O 1, O 2,..., O n of the solid body. A point O, called the reduction point, is selected, and a system of parallel transferred forces equal in magnitude and direction to the forces is built at it F 1 , F 2 ,..., F n. Resultant R of these parallel transferred vectors, i.e. the vector represented by the closing side of the force polygon is called the resultant of the forces acting on the body (Fig. 2). It is clear that the vector R does not depend on the selected reference point. If the vector magnitude R(segment ON) is not equal to zero, then the body cannot be at rest: in accordance with Newton’s law, any body on which a force acts must move with acceleration. Thus, a body can be in a state of equilibrium only if the resultant of all forces applied to it is equal to zero. However, this necessary condition cannot be considered sufficient - a body can move when the resultant of all forces applied to it is equal to zero.

As a simple but important example to explain this, consider a thin rigid rod of length l, the weight of which is negligible compared to the magnitude of the forces applied to it. Let two forces act on the rod F And -F, applied to its ends, equal in magnitude, but oppositely directed, as shown in Fig. 3, A. In this case, the resultant R equal to F – F= 0, but the rod will not be in equilibrium; obviously it will rotate around its midpoint O. A system of two equal but oppositely directed forces acting in more than one straight line is a “force couple”, which can be characterized by the product of the magnitude of the force F on the shoulder" l. The significance of such a product can be shown by the following reasoning, which illustrates the rule of leverage derived by Archimedes and leads to the conclusion about the condition of rotational equilibrium. Let us consider a light homogeneous rigid rod capable of rotating around an axis at point O, which is acted upon by a force F 1 applied at a distance l 1 from the axis, as shown in Fig. 3, b. Under force F 1 rod will rotate around point O. As you can easily see from experience, rotation of such a rod can be prevented by applying some force F 2 at this distance l 2 so that the equality holds F 2 l 2 = F 1 l 1 .

Thus, rotation can be prevented in countless ways. It is only important to choose the force and the point of its application so that the product of the force by the shoulder is equal to F 1 l 1 . This is the rule of leverage.

It is not difficult to derive the equilibrium conditions for the system. Action of forces F 1 and F 2 on the axis causes counteraction in the form of a reaction force R, applied at point O and directed opposite to the forces F 1 and F 2. According to the law of mechanics about action and reaction, the magnitude of the reaction R equal to the sum of forces F 1 + F 2. Therefore, the resultant of all forces acting on the system is equal to F 1 + F 2 + R= 0, so the necessary equilibrium condition noted above is satisfied. Force F 1 creates a torque acting clockwise, i.e. moment of power F 1 l 1 relative to point O, which is balanced by a counterclockwise torque F 2 l 2 powers F 2. Obviously, the condition for equilibrium of a body is equality to zero algebraic sum moments, eliminating the possibility of rotation. If strength F acts on the rod at an angle q, as shown in Fig. 4, A, then this force can be represented as the sum of two components, one of which ( F p), value F cos q, acts parallel to the rod and is balanced by the reaction of the support - F p , and the other ( F n), size F sin q, directed at right angles to the lever. In this case, the torque is equal to Fl sin q; it can be balanced by any force that creates an equal torque acting counterclockwise.

To make it easier to take into account the signs of moments in cases where a lot of forces act on the body, the moment of force F relative to any point O of the body (Fig. 4, b) can be considered as a vector L, equal to the vector product r ґ F position vector r to strength F. Thus, L = rґ F. It is not difficult to show that if solid there is a system of forces applied at points O 1, O 2,..., O n (Fig. 5), then this system can be replaced by the resultant R strength F 1 , F 2 ,..., F n applied at any point Oў of the body, and a pair of forces L, the moment of which is equal to the sum [ r 1 ґ F 1 ] + [r 2 ґ F 2 ] +... + [r nґ F n]. To verify this, it is enough to mentally apply at point Oў a system of pairs of equal but oppositely directed forces F 1 and - F 1 ; F 2 and - F 2 ;...; F n and - F n, which obviously will not change the state of the solid.

Carried F 1 applied at point O 1, and force – F 1 applied at point Oў form a pair of forces, the moment of which relative to point Oў is equal to r 1 ґ F 1 . Likewise the strength F 2 and - F 2 applied at points O 2 and Oў, respectively, form a pair with a moment r 2 ґ F 2, etc. Total moment L of all such pairs relative to the point Oў is given by the vector equality L = [r 1 ґ F 1 ] + [r 2 ґ F 2 ] +... + [r nґ F n]. Other forces F 1 , F 2 ,..., F n applied at point Oў, in total they give the resultant R. But the system cannot be in equilibrium if the quantities R And L are different from zero. Consequently, the condition for the values ​​to be equal to zero at the same time R And L is a necessary condition balance. It can be shown that it is also sufficient if the body is initially at rest. So, the equilibrium problem is reduced to two analytical conditions: R= 0 and L= 0. These two equations represent a mathematical representation of the principle of equilibrium.

Theoretical principles of statics are widely used in the analysis of forces acting on structures and structures. In the case of a continuous distribution of forces, the sums that give the resulting moment L and resultant R, are replaced by integrals and in accordance with the usual methods of integral calculus.

Problem 3.2.1

Determine the resultant of two forces F 1 =50N and F 2 =30N, forming an angle of 30° between themselves (Fig. 3.2a).

Figure 3.2

Let's move the force vectors F 1 and F 2 to the point of intersection of the action lines and add them according to the parallelogram rule (Fig. 2.2b). The point of application and direction of the resultant are shown in the figure. The module of the resulting resultant is determined by the formula:

Answer: R=77.44N

Problem 3.2.2

Determine the resultant system of converging forces F 1 =10N, F 2 =15N, F 3 =20N, if the angles formed by the vectors of these forces with the Ox axis are known: α 1 =30 °, α 2 =45 ° and α 3 =60 ° ( Fig.3.3a)

Figure 3.3

We project forces on the Ox and Oy axes:

Resultant module

Based on the obtained projections, we determine the direction of the resultant (Fig. 3.3b)

Answer: R=44.04N

Problem 3.2.3

At the point of connection of two threads, a vertical force P = 100N is applied (Fig. 3.4a). Determine the forces in the threads if, in equilibrium, the angles formed by the threads with the OY axis are equal to α=30°, β=75°.

Figure 3.4

The tension forces of the threads will be directed along the threads from the connection point (Fig. 3.4b). The system of forces T 1, T 2, P is a system of converging forces, because the lines of action of forces intersect at the point where the threads join. The equilibrium condition for this system:

We compose analytical equilibrium equations for a system of converging forces and project the vector equation onto the axes.

We solve the system of obtained equations. From the first we express T 2.

Let's substitute the resulting expression into the second one and determine T 1 and T 2 .

N,

Let's check the solution from the condition that the modulus P of the sum of forces T 1 and T 2 must be equal to P (Fig. 3.4c).

Answer: T 1 =100N, T 2 =51.76N.

Problem 3.2.4

Determine the resultant of the system of converging forces if their modules are given: F 1 =12N, F 2 =10N, F 3 =15N and angle α = 60 ° (Fig. 3.5a).

Figure 3.5

We determine the projections of the resultant

Resultant module:

Based on the obtained projections, we determine the direction of the resultant (Fig. 3.5b)

Answer: R=27.17N

Problem 3.2.6

Three rods AC, BC, DC are hingedly connected at point C. Determine the forces in the rods if the force F=50N, angle α=60° and angle β=75° are given. The force F is in the Oyz plane. (Fig. 3.6)

Figure 3.6

Initially, we assume that all the rods are stretched, and accordingly we direct the reactions in the rods from node C. The resulting system N 1, N 2, N 3, F is a system of converging forces. Equilibrium condition for this system.

Often, not one, but several forces act on the body at the same time. Let's consider the case when the body is affected by two forces ( and ). For example, a body resting on a horizontal surface is affected by the force of gravity () and the reaction of the surface support () (Fig. 1).

These two forces can be replaced by one, which is called the resultant force (). Find it as a vector sum of forces and:

Determination of the resultant of two forces

DEFINITION

Resultant of two forces called a force that produces an effect on a body similar to the action of two separate forces.

Note that the action of each force does not depend on whether there are other forces or not.

Newton's second law for the resultant of two forces

If two forces act on a body, then we write Newton’s second law as:

The direction of the resultant always coincides in direction with the direction of acceleration of the body.

This means that if a body is affected by two forces () at the same moment in time, then the acceleration () of this body will be directly proportional to the vector sum of these forces (or proportional to the resultant forces):

M is the mass of the body in question. The essence of Newton's second law is that the forces acting on a body determine how the body's speed changes, and not just the magnitude of the body's speed. Note that Newton's second law is satisfied exclusively in inertial frames of reference.

The resultant of two forces can be equal to zero if the forces acting on the body are directed in different sides and are equal in modulus.

Finding the magnitude of the resultant of two forces

To find the resultant, 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.

Let us assume that the body is acted upon by two forces that are directed along the same straight line (Fig. 1). It can be seen from the figure that they are directed in different directions.

The resultant forces () applied to the body will be equal to:

To find the modulus of the resultant forces, we select an axis, denote it X, and direct it along the direction of action of the forces. Then, projecting expression (4) onto the X axis, we obtain that the magnitude (modulus) of the resultant (F) is equal to:

where are the modules of the corresponding forces.

Let's imagine that two forces and are acting on the body, directed at a certain angle to each other (Fig. 2). We find the resultant of these forces using the parallelogram rule. The magnitude of the resultant will be equal to the length of the diagonal of this parallelogram.

Examples of problem solving

EXAMPLE 1

Exercise	A body with a mass of 2 kg is moved vertically upward by a thread, while its acceleration is equal to 1. What is the magnitude and direction of the resultant force? What forces are applied to the body?
Solution	The force of gravity () and the reaction force of the thread () are applied to the body (Fig. 3). The resultant of the above forces can be found using Newton's second law: In projection onto the X axis, equation (1.1) takes the form: Let's calculate the magnitude of the resultant force:
Answer	N, the resultant force is directed in the same way as the acceleration of the body, that is, vertically upward. There are two forces acting on the body and .