Newton's second law formula for translational motion. Newton's second law for rotational motion

Dynamics of a material point and translational motion of a rigid body

Newton's first law. Weight. Force

Newton's first law: any material point (body) maintains a state of rest or uniform rectilinear motion until the impact from other bodies makes it change this state. The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is also called law of inertia.

Newton's first law is not valid in any frame of reference, and those systems in relation to which it is performed are called inertial reference systems.

Weight bodies - a physical quantity, which is one of the main characteristics of matter, which determines its inertial ( inertial mass) and gravitational ( gravitational mass) properties. At present, it can be considered proven that the inertial and gravitational masses are equal to each other (with an accuracy not less than 10–12 of their values).

So, force- this is a vector quantity, which is a measure of the mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size.

Newton's second law

Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the action of forces applied to it.

a~ F (t = const) . (6.1)

a~ 1 /t (F = const). (6.2)

a =kF/ m. (6.3)

In SI, the proportionality factor k= 1. Then

(6.4)

(6.5)

Vector quantity

(6.6)

numerically equal to the product of the mass of a material point and its speed and having the direction of speed, is called momentum (momentum) this material point.

Substituting (6.6) into (6.5), we obtain

(6.7)

Expression (6.7) is called the equation of motion of a material point.

Unit of force in SI - newton(N): 1 N is a force that imparts an acceleration of 1 m / s 2 to a mass of 1 kg in the direction of the force:

1 N = 1 kg m/s 2 .

Newton's second law is valid only in inertial frames of reference. Newton's first law can be derived from the second.

In mechanics, it is of great importance principle of independence of action of forces: if several forces act simultaneously on a material point, then each of these forces imparts an acceleration to the material point according to Newton's second law, as if there were no other forces.

Newton's third law

The interaction between material points (bodies) is determined by Newton's third law.

F 12 = – F 21 , (7.1)

Newton's third law allows the transition from dynamics separate material point to dynamics systems material points.

Friction forces

In mechanics, we will consider various forces: friction, elasticity, gravity.

Friction forces, which prevent the sliding of contacting bodies relative to each other.

external friction called the friction that occurs in the plane of contact between two bodies in contact with their relative movement.

Depending on the nature of their relative motion, one speaks of sliding friction, rolling or spinning.

internal friction called friction between parts of the same body, for example, between different layers of a liquid or gas. If the bodies slide relative to each other and are separated by a layer of viscous fluid (lubrication), then friction occurs in the lubricant layer. In this case, one speaks of hydrodynamic friction(the lubricant layer is thick enough) and boundary friction (the thickness of the lubricant layer is 0.1 µm or less).

sliding friction force F tr is proportional to strength N normal pressure with which one body acts on another:

F tr = f N ,

where f - coefficient of sliding friction, depending on the properties of the contacting surfaces.

In the limiting case (the beginning of the sliding of the body) F=F tr. or P sin  0 = f N = f P cos  0 , where

f = tg 0 .

For smooth surfaces, intermolecular attraction begins to play a certain role. For them apply sliding friction law

F tr = f ist (N + Sp 0 ) ,

where R 0 - additional pressure due to the forces of intermolecular attraction, which rapidly decrease with increasing distance between the particles; S - contact area between bodies; f ist - the true coefficient of sliding friction.

A radical way to reduce the friction force is to replace sliding friction with rolling friction (ball and roller bearings, etc.). The rolling friction force is determined according to the law established by Coulomb:

F tr = f to N / r , (8.1)

where r- radius of the rolling body; f k - coefficient of rolling friction, having the dimension dim f to =L. From (8.1) it follows that the rolling friction force is inversely proportional to the radius of the rolling body.

Law of conservation of momentum. Center of mass

The set of material points (bodies) considered as a whole is called mechanical system. The forces of interaction between the material points of a mechanical system are called - internal. The forces with which external bodies act on the material points of the system are called external. A mechanical system of bodies that is not acted upon by external forces is called closed(or isolated). If we have a mechanical system consisting of many bodies, then, according to Newton's third law, the forces acting between these bodies will be equal and oppositely directed, i.e., the geometric sum of internal forces is equal to zero.

We write down Newton's second law for each of n bodies of the mechanical system:

Adding these equations term by term, we obtain

But since the geometric sum of the internal forces of a mechanical system is equal to zero according to Newton's third law, then

(9.1)

where is the momentum of the system. Thus, the time derivative of the momentum of a mechanical system is equal to the geometric sum of the external forces acting on the system.

In the absence of external forces (we consider a closed system)

The last expression is law of conservation of momentum: the momentum of a closed system is conserved, i.e. does not change over time.

Experiments prove that it is also true for closed systems of microparticles (they obey the laws of quantum mechanics). This law is universal, i.e. the law of conservation of momentum - fundamental law of nature.

The law of conservation of momentum is a consequence of a certain property of the symmetry of space - its homogeneity. Homogeneity of space lies in the fact that during parallel transfer in space of a closed system of bodies as a whole, its physical properties and laws of motion do not change, in other words, do not depend on the choice of the position of the origin of the inertial reference frame.

center of gravity(or center of inertia) system of material points is called an imaginary point With, the position of which characterizes the mass distribution of this system. Its radius vector is

where m i and r i- respectively mass and radius vector i-th material point; n- number of material points in the system; is the mass of the system. Center of mass speed

Given that pi = m i v i, a there is momentum R systems, you can write

(9.2)

i.e., the momentum of the system is equal to the product of the mass of the system and the velocity of its center of mass.

Substituting expression (9.2) into equation (9.1), we obtain

(9.3)

i.e., the center of mass of the system moves as a material point at which the mass of the entire system is concentrated and on which a force acts equal to the geometric sum of all external forces applied to the system. Expression (9.3) is law of motion of the center of mass.

1. The time derivative of the amount of motion K of a material point or a system of material points relative to a fixed (inertial) frame of reference is equal to the main vector F of all external forces applied to the system:
dK/dt = F or mac = F

where ac is the acceleration of the center of inertia of the system, and m is its mass.
In the case of translational motion of a rigid body with an absolute velocity v, the velocity of the center of inertia is vc = v. Therefore, when considering the translational motion of a rigid body, this body can be mentally replaced by a material point coinciding with the center of inertia of the body, possessing its entire mass and moving under the action of the main driver of external forces applied to the body.
In projections on the axes of a fixed rectangular Cartesian coordinate system, the equations of the basic law of the dynamics of the translational motion of the system have the form:
Fx = dK/dt, Fy = dK/dt, Fz = dK/dt

or
macx=Fx, macy=Fy, macz=Fz

2. The simplest cases of translational motion of a rigid body.
a) Coasting (F = 0):
mv = const, a=0.

b) Motion under the action of a constant force:
d/dt (mv) = F = const, mv = Ft + mv0,

where mv0 is the amount of motion of the body at the initial time t = 0.
c) Movement under the action of a variable force. The change in the momentum of the body over a period of time from t1 to t2 is
mv2 - mv1 = Fcp(t2 - t1)

where Fcp is the average value of the force vector in the time interval from t1 to t2.

Other entries

06/10/2016. Newton's first law

1. Newton's first law: any material point retains a state of rest or uniform and rectilinear motion until the impact from other bodies takes it out of this state. This ...

06/10/2016. Force

1. Force - a vector quantity, which is a measure of mechanical action on a material point or body from other bodies or fields. A force is fully specified if its numerical value, direction are indicated ...

06/10/2016. Newton's third law

1. The actions of two material points on each other are numerically equal and directed in opposite directions: Fij = - Fji, where i is not equal to j. These forces are applied to different points and can be mutually balanced ...

Chapter 2. ELEMENTS OF DYNAMICS

Dynamics studies the movement of bodies, taking into account those causes (interactions between bodies) that determine one or another character of movement. Classical (Newtonian) mechanics is based on three laws of dynamics formulated by I. Newton in the 17th century. Newton's laws arose as a result of the generalization of a large number of experimental facts. Their correctness is confirmed by the coincidence with experience of the consequences that follow from them.

Newton's first law is formulated as follows: every body is in a state of rest or uniform and rectilinear motion, until the impact from other bodies makes it change this state. Both these states are united by the fact that the acceleration of the body is zero.

Considering that the nature of motion depends on the choice of reference frame, it should be concluded that Newton's first law is not valid in every frame of reference. The frame of reference in which Newton's first law is fulfilled is commonly called inertial. The law itself is called the law of inertia. The frame of reference in which Newton's first law is not fulfilled is commonly called non-inertial. Any frame of reference ͵ moving uniformly and rectilinearly relative to the inertial frame is also an inertial frame. For this reason, there are an infinite number of inertial systems.

The property of bodies to maintain a state of rest or uniform and rectilinear motion is commonly called inertia(inertia). The measure of body inertia is its mass m. It does not depend on the speed of the body. taken as a unit of mass kilogram(kg) - mass of the reference body.

If the state of motion of a body or its shape and dimensions change, then it is said that other bodies act on the body. Force is a measure of the interaction of bodies. Any force manifests itself as a result of the action of one body on another, which is reduced to the appearance of an acceleration in the body or its deformation.

Newton's second law: the resultant force acting on a body is equal to the product of the mass of this body and its acceleration:

Since the mass is a scalar, it follows from formula (6.1) that .

Based on this law, the unit of force is introduced - newton(H): .

Newton's second law is valid only in inertial frames of reference.

Let us replace the acceleration in equation (6.1) with the time derivative of the velocity:

Vector quantity

called body momentum.

From formula (6.3) it follows that the direction of the momentum vector coincides with the direction of the velocity. Unit of impulse - kilogram meter per second(kg×m/s).

Combining expressions (6.2) and (6.3), we obtain

The resulting expression allows us to propose a more general formulation of Newton's second law: the force acting on the body is equal to the derivative of the momentum with respect to time.

Any action of bodies on each other has the character of interaction (Fig. 6.1). If the body acts on the body with a certain force, then the body, in turn, acts on the body with a force.

Newton's third law is formulated as follows: interacting bodies act on each other with forces equal in magnitude and opposite in direction.

These forces, applied to different bodies, act in one straight line and are forces of the same nature. The mathematical expression of Newton's third law is

The "-" sign in formula (6.5) means that the force vectors are opposite in direction.

As Newton himself put it, the third law is: "An action always has an equal and opposite reaction, otherwise the actions of two bodies on each other are equal and directed in opposite directions."

The rotation of the body through a certain angle can be specified as a segment, the length of which is equal to j, and the direction coincides with the axis around which the rotation is performed. The direction of rotation and the segment depicting it is connected by the rule of the right screw.

In mathematics, it is shown that very small rotations can be considered as vectors, denoted by the symbols or . The direction of the rotation vector is associated with the direction of rotation of the body; - the vector of the elementary rotation of the body - is a pseudovector, since it does not have an application point.

During the rotational motion of a rigid body, each point moves along a circle, the center of which lies on a common axis of rotation (Fig. 6). In this case, the radius vector R, directed from the axis of rotation to a point, rotates in time Dt to some angle DJ. To characterize the rotational motion, the angular velocity and angular acceleration are introduced.

angular velocity is called a vector quantity equal to the first derivative of the angle of rotation of the body with respect to time:

An angle of 1 radian is a central angle whose arc length is equal to the radius of the circle; 360 o \u003d 2p rad.

The direction of the angular velocity is given right screw rule: the angular velocity vector is co-directed with the vector , that is, with the translational movement of the screw, the head of which rotates in the direction of movement of the point along the circle.

The linear velocity of a point is related to the angular velocity:

In vector form.

If during rotation the angular velocity changes, then angular acceleration occurs.

Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time. The vector of the angular velocity is co-directed with the vector of the elementary change in the angular velocity that occurred during the time dt:

With accelerated motion, the vector is parallel (Fig. 7), with slow motion, it is opposite (Fig. 8).

Angular acceleration occurs in the system only when there is a change in the angular velocity, that is, when the linear speed of movement changes in magnitude. The change in velocity characterizes the tangential acceleration in magnitude.

Let's find the relationship between the angular and tangential accelerations:

.

A change in the direction of speed during curvilinear motion is characterized by normal acceleration:

.

Thus, the relationship between linear and angular quantities is expressed by the following formulas:

Rotary motion types:

a) variable- a movement in which and change:

b) equally variable– rotational motion with constant angular acceleration:

in) uniform– rotational movement with constant angular velocity:

.

Uniform rotational motion can be characterized by period and frequency of rotation.

Period is the time it takes for the body to complete one revolution.

Rotation frequency is the number of revolutions per unit of time.

For one turn:

, .

Newton's laws. The basic equation of the dynamics of translational motion.

Dynamics studies the movement of bodies, taking into account the causes that cause this movement.

Dynamics is based on Newton's laws.

I law. There are inertial reference systems (ISR) in which a material point (body) maintains a state of rest or uniform rectilinear motion until the impact from other bodies takes it out of this state.

The property of a body to maintain a state of rest or uniform rectilinear motion in the absence of influence of other bodies on it is called inertia.

ISO is a frame of reference in which a body, free from external influences, is at rest or moves uniformly in a straight line.

An inertial reference frame is one that is at rest or moves uniformly in a straight line with respect to any IFR.

The frame of reference, moving with acceleration relative to the IFR, is non-inertial.

Newton's first law, also called the law of inertia, was first formulated by Galileo. Its content boils down to 2 statements:

1) all bodies have the property of inertia;

2) there are ISO.

Galileo's principle of relativity: all mechanical phenomena in all ISOs occur in the same way, i.e. it is impossible to establish by any mechanical experiments inside the IFR whether the given IFR is at rest or moves uniformly in a straight line.

In most practical problems, the frame of reference, rigidly connected with the Earth, can be considered as ISO.

From experience it is known that under the same influences, different bodies change their speed unequally, i.e. acquire various accelerations, the acceleration of bodies depends on their mass.

Weight- a measure of the inertial and gravitational properties of the body. With the help of precise experiments, it has been established that the inertial and gravitational masses are proportional to each other. Choosing units in such a way that the proportionality coefficient becomes equal to one, we get that, therefore, they simply talk about body weight.

[m]=1kg - mass of platinum-iridium cylinder, diameter and height of which are h=d=39mm.

To characterize the action of one body on another, the concept of force is introduced.

Force- a measure of the interaction of bodies, as a result of which the bodies change their speed or deform.

Force is characterized by a numerical value, direction, point of application. The line along which the force acts is called line of force.

The simultaneous action of several forces on a body is equivalent to the action of one force, called resultant or the resulting force and equal to their geometric sum:

Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the motion of a body changes under the action of forces applied to it.

Date: __________ Deputy Director for OIA: ___________

Subject; Newton's second law for rotational motion

Target:

Educational: determine and write down in mathematical form Newton's second law; explain the relationship between the quantities included in the formulas of this law;

Developing: develop logical thinking, the ability to explain the manifestations of Newton's second law in nature;

Educational : to form interest in the study of physics, to cultivate diligence, responsibility.

Type of lesson: learning new material.

Demonstrations: the dependence of the acceleration of a body on the force acting on it.

Equipment: trolley with light wheels, rotating disk, set of weights, spring, block, bar.

DURING THE CLASSES

Organizing time

Updating the basic knowledge of students

Formula chain (reproduce formulas):

II. Motivation of educational activity of students

Teacher. With the help of Newton's laws, one can not only explain the observed mechanical phenomena, but also predict their course. Recall that the direct main task of mechanics is to find the position and speed of a body at any moment of time, if its position and speed at the initial moment of time and the forces that act on it are known. This problem is solved with the help of Newton's second law, which we will study today.

III. Learning new material

1. The dependence of the acceleration of a body on the force acting on it

A more inert body has a large mass, a less inert body has a smaller one:

2. Newton's second law

Newton's second law of dynamics establishes a connection between kinematic and dynamic quantities. Most often, it is formulated as follows: the acceleration that a body receives is directly proportional to the mass of the body and has the same direction as the force:

where - acceleration, - resultant of forces acting on the body, N; m - body weight, kg.

If we determine the force from this expression, then we obtain the second law of dynamics in the following formulation: the force acting on the body is equal to the product of the body's mass and the acceleration provided by this force.

Newton formulated the second law of dynamics somewhat differently, using the concept of momentum (body momentum). Impulse - the product of body mass and its speed (the same as the amount of motion) - one of the measures of mechanical movement: Impulse (momentum) is a vector quantity. Since the acceleration

Newton formulated his law as follows: the change in the momentum of a body is proportional to the acting force and occurs in the direction of the straight line along which this force acts.

It is worth considering another of the formulations of the second law of dynamics. In physics, a vector quantity is widely used, which is called the impulse of a force - this is the product of the force and the time of its action: Using this, we get . The change in momentum of a body is equal to the momentum of the force acting on it.

Newton's second law of dynamics summarized an extremely important fact: the action of forces does not cause actual motion, but only changes it; force causes a change in speed, i.e. acceleration, not speed itself. The direction of the force coincides with the direction of the velocity only in the partial case of rectilinear evenly accelerated (Δ 0) motion. For example, during the movement of a body thrown horizontally, the force of gravity is directed downward, and the velocity forms a certain angle with the force, which changes during the flight of the body. And in the case of uniform motion of the body in a circle, the force is always directed perpendicular to the speed of the body.

The SI unit of force is determined based on Newton's second law. The unit of force is called [H] and is defined as follows: a force of 1 newton imparts an acceleration of 1 m/s2 to a body of mass 1 kg. Thus,

Application examples of Newton's second law

As an example of the application of Newton's second law, one can consider, in particular, the measurement of body mass by weighing. An example of the manifestation of Newton's second law in nature can be a force that acts on our planet from the Sun, etc.

Limits of application of Newton's second law:

1) the reference system must be inertial;

2) the speed of the body must be much less than the speed of light (for speeds close to the speed of light, Newton's second law is used in impulsive form: ).

IV. Fixing the material

Problem solving

1. A body with a mass of 500 g is simultaneously affected by two forces 12 N and 4 N, directed in the opposite direction along one straight line. Determine the modulus and direction of acceleration.

Given: m = 500 g = 0.5 kg, F1 = 12 N, F2 = 4 N.

Find: a - ?

According to Newton's second law: , where Let's draw the axis Ox, then the projection F = F1 - F2. Thus,

Answer: 16 m/s2, the acceleration is in the direction of the greater force.

2. The coordinate of the body changes according to the law x = 20 + 5t + 0.5t2 under the action of a force of 100 N. Find the mass of the body.

Given: x = 20 + 5t + 0.5t2, F = 100H

Find: m - ?

Under the action of a force, the body moves with equal acceleration. Therefore, its coordinate changes according to the law:

According to Newton's second law:

Answer: 100 kg.

3. A body with a mass of 1.2 kg acquired a speed of 12 m/s at a distance of 2.4 m under the action of a force of 16 N. Find the initial speed of the body.

Given: = 12 m/s, s = 2.4m, F = 16H, m = 1.2 kg

Find: 0 - ?

Under the action of a force, the body acquires acceleration according to Newton's second law:

For evenly accelerated movement:

From (2) we express the time t:

and substitute for t in (1):

Substitute the expression for acceleration:

Answer: 8.9 m/s.

V. Lesson summary

Frontal conversation for questions

1. How are such physical quantities as acceleration, force and mass of a body related?

2. Or can it be stated by the formula that the force acting on a body depends on its mass and acceleration?

3. What is the momentum of the body (momentum)?

4. What is the impulse of force?

5. What formulations of Newton's second law do you know?

6. What important conclusion can be drawn from Newton's second law?

VI. Homework

Work through the relevant section of the textbook.

Solve problems:

1. Find the acceleration module of a body with a mass of 5 kg under the action of four forces applied to it, if:

a) F1 = F3 = F4 = 20 H, F2 = 16 H;

b) F1 = F4 = 20 H, F2 = 16 H, F3 = 17 H.

2. A body with a mass of 2 kg, moving in a straight line, changed its speed from 1 m/s to 2 m/s in 4 s.

a) What is the acceleration of the body?

b) What force acted on the body in the direction of its motion?

c) How has the momentum of the body (momentum) changed over the time considered?

d) What is the impulse of the force acting on the body?

e) What is the distance traveled by the body during the considered time of motion?