As a rule, the motion of a point body with acceleration in IFR occurs under the action of several bodies. For example, let the cart move with acceleration along a real horizontal road. It is affected by the person who pushes the cart, and the road that slows down the movement of the cart. Studying the motion of a body under the action of several bodies on it, Newton came to two conclusions:
1. The actions that other bodies have on a point body do not depend on each other.
2. The forces characterizing these actions can be added.
Let us formulate the rules for the addition of forces acting on a point body along one straight line.
1. If two forces F 1 and F 2 act on a point body, directed in one direction (Fig. 73), then their action is equal to the action of one force F. In this case:
2. If two forces F 1 and F 2 act on a point body, directed in opposite directions (Fig. 74, a, b), then their action is equal to the action of force F, which:
If three forces (or more) act on a point body, then first you need to add two of them. Then add a third force to the resulting force, and so on.
From rule 2, a very important conclusion can be drawn: if only two equal in absolute value, but oppositely directed forces act on a point body, then the total action of these forces is zero (Fig. 75). In this case, the forces F 1 and F 2 are said to compensate (balance) each other. It is clear that then the acceleration of this body in the inertial frame of reference will be equal to zero and its speed will be constant. This means that the body will rest in a given ISO or move uniformly in a straight line.
The converse is also true:
if a body in an inertial reference frame moves uniformly in a straight line or is at rest, then either no other bodies act on the body, or the sum of the forces acting on the body is zero.
Note that in this case it is experimentally impossible to determine which of these two conditions is satisfied: whether the sum of all forces acting on a point body is equal to zero, or whether nothing acts on it at all.
In the same way, it is impossible to distinguish experimentally whether a single force F acts on a point body, or several forces act on this body, the sum of which is equal to F.
We use the rules of adding forces to develop a recipe for measuring force.
First of all, we introduce the standard of force. To do this, select a specific spring. Stretch it by a certain amount and attach it to the body. We will assume that in this case a force acts on the body from the side of the spring, the modulus of which is equal to unity (Fig. 76). As a result, the body will acquire acceleration in ISO.
To prevent this from happening, we attach a second spring to this body from the opposite side, as shown in Fig. 77. At the same time, we stretch the second spring in such a way that its action balances (compensates) the action of the first (reference) spring. Then the body, on which both springs act simultaneously, will remain at rest. Consequently, the modulus of force with which the second spring acts on the body will be exactly equal to the modulus of force of unit magnitude. Let us fix the extension of the second spring. stretched to such a length, it will also become a standard of strength. Thus, you can get as many standards of force as you like.
Let's create a force, the modulus of which is, for example, half the unit of force. To do this, we balance the action of the reference spring on the body with two identical springs stretched to the same length (Fig. 78). In this case, the modulus of force with which any of two identical springs acts on the body will be equal to the modulus of half a unit of force.
Similarly, you can create a force whose modulus is a given number of times (for example, 3, 10, etc.) less than the modulus of a unit of force.
This way we can create a set of springs that, under known tensions, act with different forces. Now it will not be difficult for us to measure the modulus of any unknown force. To do this, it will be enough to balance its action with the action of an appropriate set of springs. An example of such a measurement is shown in Fig. 79. The force measured in this way, firstly, is equal in absolute value to the sum of the modules of forces created by a set of springs, and, secondly, is directed in the direction opposite to the direction of their action.
Results
Rules for the addition of forces acting on a body along one straight line.
1. If two forces F 1 and F 2 act on a point body, directed in one direction, then their action is equal to the action of one force F. In this case:
– force F is directed in the same direction as the forces F 1 and F 2 ;
– force module F is equal to the sum of force modules F 1 and F 2 .
2. If two forces F 1 and F 2 act on a point body, directed in opposite directions, then their action is equal to the action of force F, which:
- directed towards the greater force in modulus;
- has a modulus equal to the difference between the modules of the greater and lesser forces.
If the sum of all forces acting on a point body is zero, then these forces are said to balance (compensate) each other. In this case, the body in the IFR moves uniformly in a straight line or is at rest, i.e., does not change its mechanical state.
To measure an unknown force, its action must be balanced (compensated) by the action of a set of reference springs.
Questions
- Formulate the rules for adding forces acting along one straight line.
- When are forces said to balance each other?
Exercises
1. Determine what is equal to and where the sum of two forces acting on a point body is directed if the first force is directed in the positive direction of the X axis, and the second in the opposite direction. Force modules measured in reference units are: |F 1 | = 3, |F 2 | = 5.
2. Determine what is equal to and where the sum of three forces acting on a point body is directed if the first force is directed in the positive direction of the X axis, and the second and third in the opposite direction. Force modules measured in reference units are: |F 1 | = 30, |F 2 | =5, |F 3 | = 15.
3. Find what is equal to and where is the force F acting on a point body, if the sum of the three forces F, F 1 and F 2 acting on this body is zero. In this case, F 1 is directed in the positive direction of the X axis, and F 2 in the opposite direction. Force modules measured in reference units are: |F 1 | = 30, |F 2 | = 5.
4. A stone lying on the road (Fig. 80) is motionless in the frame of reference associated with the Earth. Answer the questions:
a) What is the sum of the forces acting on the rock?
b) does the speed change with time (is the acceleration equal to zero) of the stone in the frame of reference related:
- with a straight line evenly driving along the road by a bus;
- with a car accelerating relative to the road;
- with a cone that freely falls from a tree with an acceleration g?
c) which of these frames of reference are inertial and which are non-inertial?
Strength. Addition of forces
Any changes in nature occur as a result of the interaction between bodies. The ball lies on the ground, it will not start moving if you do not push it with your foot, the spring will not stretch if you attach a weight to it, etc. When a body interacts with other bodies, its speed changes. In physics, they often do not indicate which body and how it acts on a given body, but they say that "a force acts on a body."
Force is a physical quantity that quantitatively characterizes the action of one body on another, as a result of which the body changes its speed. Force is a vector quantity. That is, in addition to the numerical value, the strength is the direction. Force is denoted by the letter F and is measured in Newtons in the System Internationale. 1 newton is the force that a body of mass 1 kg at rest provides a speed of 1 meter per second in 1 second in the absence of friction. You can measure the force using a special device - a dynamometer.
Depending on the nature of the interaction in mechanics, three types of forces are distinguished:
- gravity,
- elastic force,
- friction force.
As a rule, not one, but several forces act on the body. In this case, consider the resultant of forces. The resultant force is a force that acts in the same way as several forces acting simultaneously on a body. Using the results of the experiments, we can conclude: the resultant of forces directed along one straight line in one direction is directed in the same direction, and its value is equal to the sum of the values of these forces. The resultant of two forces directed along the same straight line in opposite directions is directed towards the greater force and is equal to the difference in the values of these forces.
With the simultaneous action of several forces on one body, the body moves with an acceleration, which is the vector sum of the accelerations that would arise under the action of each force separately. The forces acting on the body, applied to one point, are added according to the rule of addition of vectors.
The vector sum of all forces acting simultaneously on a body is called resultant force.
The straight line passing through the force vector is called the line of action of the force. If the forces are applied to different points of the body and act not parallel to each other, then the resultant is applied to the point of intersection of the lines of action of the forces. If the forces act parallel to each other, then there is no point of application of the resulting force, and the line of its action is determined by the formula: (see figure).
Moment of power. Lever equilibrium condition
The main sign of the interaction of bodies in dynamics is the occurrence of accelerations. However, it is often necessary to know under what conditions a body, which is acted upon by several different forces, is in a state of equilibrium.
There are two types of mechanical movement - translation and rotation.
If the trajectories of movement of all points of the body are the same, then the movement progressive. If the trajectories of all points of the body are arcs of concentric circles (circles with one center - the point of rotation), then the movement is rotational.
Equilibrium of non-rotating bodies: a non-rotating body is in equilibrium if the geometric sum of the forces applied to the body is zero.
Equilibrium of a body with a fixed axis of rotation
If the line of action of the force applied to the body passes through the axis of rotation of the body, then this force is balanced by the elastic force from the side of the axis of rotation.
If the line of action of the force does not cross the axis of rotation, then this force cannot be balanced by the elastic force from the side of the axis of rotation, and the body rotates around the axis.
The rotation of a body around an axis under the action of one force can be stopped by the action of a second force. Experience shows that if two forces separately cause the rotation of the body in opposite directions, then with their simultaneous action the body is in equilibrium if the condition is met:
, where d 1 and d 2 are the shortest distances from the lines of action of the forces F 1 and F 2. The distance d is called shoulder of strength, and the product of the modulus of force by the arm is moment of force:
.
If a positive sign is assigned to the moments of forces that cause the body to rotate around an axis clockwise, and a negative sign to the moments of forces that cause counterclockwise rotation, then the equilibrium condition for a body with an axis of rotation can be formulated as moment rules: a body with a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body about this axis is zero:
The SI unit of torque is a moment of force of 1 N, the line of action of which is at a distance of 1 m from the axis of rotation. This unit is called newton meter.
The general condition for the equilibrium of a body:a body is in equilibrium if the geometric sum of all forces applied to it and the algebraic sum of the moments of these forces about the axis of rotation are equal to zero.
Under this condition, the body is not necessarily at rest. It can move uniformly and rectilinearly or rotate.
Statics studies the equilibrium conditions of a material point and an absolute rigid body.
An absolutely rigid body is a body whose dimensions and shape can be considered unchanged.
Equilibrium conditions are understood as the conditions under which the body, in the presence of an external influence, can be at rest relative to the inertial frame of reference; move progressively, evenly and rectilinearly; rotate uniformly about an axis passing through the center of mass.
Strength. Addition of forces
The main physical quantities used in statics are force and moment of force. Force as a vector quantity is characterized by its modulus, direction in space, and point of application.
The result of the action of a force on a material point depends only on its modulus and direction. A solid body has a certain size. Therefore, forces of the same magnitude and direction cause different motions of a rigid body depending on the point of application.
The point of application of a force can only be moved along a straight line along which this force acts. This must always be remembered when carrying out various operations on forces.
The force \(~\vec R\), which produces the same effect on the body as several forces simultaneously acting on it, is called resultant. It is equal to the geometric sum of these forces\[~\vec R = \sum^n_(i=1) \vec F_i\].
Adding forces means finding their resultant.
If two forces are applied to the body at one point, then the resultant is found according to the parallelogram rule (Fig. 1). The modulus of the resultant of two forces can be determined by the law of cosines
\(~R = \sqrt(F^2_1 + F^2_2 + 2F_1F_2 \cos \alpha)\)
or when α = 90° - according to the Pythagorean theorem.
If non-parallel forces are applied at different points of the body, then to find their resultant, these forces \(~\vec F_1\) and \(~\vec F_2\) are transferred to the point O intersection of the lines along which they act (Fig. 2), and then perform their vector addition according to the parallelogram rule. The point of application of the resultant force can be any point on the straight line along which it acts.
The addition of forces is carried out using the vector addition rule. Or the so-called parallelogram rule. Since the force is depicted as a vector, that is, it is a segment, the length of which shows the numerical value of the force, and the direction indicates the direction of the force. That is, the forces, that is, the vectors, are added using the geometric summation of vectors.
On the other hand, the addition of forces is the finding of the resultant of several forces. That is, when several different forces act on the body. Different in both size and direction. It is necessary to find the resulting force that will act on the body as a whole. In this case, the forces can be added in pairs using the parallelogram rule. First, add the two forces. We add one more to their resultant. And so on until all the forces are combined.
Figure 1 - Parallelogram rule.
The parallelogram rule can be described as follows. For two forces coming out of the same point, and having an angle between them other than zero or 180 degrees. You can build a parallelogram. By moving the beginning of one vector to the end of another. The diagonal of this parallelogram will be the resultant of these forces.
But you can also use the force polygon rule. In this case, the starting point is selected. From this point, the first vector of the force acting on the body comes out, then the next vector is added to its end, using the parallel transfer method. And so on until a polygon of forces is obtained. In the end, the resultant of all forces in such a system will be a vector drawn from the starting point to the end of the last vector.
Figure 2 - Polygon of forces.
If the body moves under the action of several forces applied to different points of the body. We can assume that it moves under the action of the resultant force applied to the center of mass of the given body.
Along with the addition of forces, to simplify the calculations of motion, the method of decomposition of forces is also used. As the name implies, the essence of the method lies in the fact that one force acting on the body is decomposed into component forces. In this case, the components of the force have the same effect on the body as the original force.
The expansion of forces is also carried out according to the parallelogram rule. They must come from the same point. From the same point from which the decomposing force emerges. As a rule, the decomposed force is presented in the form of projections onto perpendicular axes. For example, as the force of gravity and the force of friction acting on a bar lying on an inclined plane.
Figure 3 - Bar on an inclined plane.
