Exam in physics Newton's laws. Mechanical vibrations and waves

In the second task of the Unified State Exam in physics, it is necessary to solve a problem on Newton’s laws or related to the action of forces. Below we present the theory with formulas that are necessary to successfully solve problems on this topic.

Theory for task No. 2 of the Unified State Exam in Physics

Newton's second law

Newton's second law formula F =ma . Here F And a – vector quantities. Magnitude a – This is the acceleration of a body's motion under the influence of a specified force. It is directly proportional to the force acting on a given body and is directed in the direction of the force.

Resultant

Resultant force is a force whose action replaces the action of all forces applied to the body. Or, in other words, the resultant of all forces applied to the body is equal to the vector sum of these forces.

Friction force

F tr =μN , Where μ μ, which is a constant value for a given case. Knowing the friction force and the normal pressure force (this force is also called the support reaction force), you can calculate the friction coefficient.

Gravity

The vertical component of movement depends on the forces acting on the body. Knowledge of the gravity formula is required F=mg, since, as a rule, only it acts on a body thrown at an angle to the horizontal.

Elastic force

Elastic force is a force that arises in a body as a result of its deformation and tends to return it to its original (initial) state. For elastic force, Hooke's law is used: F = kδl, Where k— elasticity coefficient (body stiffness), δl— magnitude of deformation.

Law of Gravity

The force F of gravitational attraction between two material points of mass m1 and m2, separated by a distance r, is proportional to both masses and inversely proportional to the square of the distance between them:

Analysis of typical options for tasks No. 2 of the Unified State Exam in Physics

Demo version 2018

The graph shows the dependence of the sliding friction force modulus on the normal pressure force modulus. What is the coefficient of friction?

Solution algorithm:

1. Let us write down a formula connecting these forces. Express the coefficient of friction.
2. We examine the graph and set a pair of corresponding values ​​of the forces of normal pressure N and friction.
3. We calculate the coefficient based on the force values ​​taken from the graph.
4. We write down the answer.

Solution:

1. The friction force is related to the normal pressure force by the formula F tr=μ N, Where μ – friction coefficient. From here, knowing the magnitude of the friction force and pressure normal to the surface, we can determine μ, which is a constant value for a given case. Knowing the friction force and the normal pressure force (this force is also called the support reaction force), you can calculate the friction coefficient. From the above formula it follows that: μ = F tr: N
2. Let's look at the dependence graph. Let's take any point on the graph, for example, when N = 12 (N), and F tr = 1.5 (N).
3. Let's take the selected force values ​​and calculate the coefficient value μ : μ= 1,5/12 = 0,125

Answer: 0.125

First version of the task (Demidova, No. 3)

Force F imparts an acceleration a to a body of mass m in the inertial frame of reference. Determine the acceleration of a body of mass 2m under the influence of a force of 0.5F in this frame of reference.

1) ; 2) ; 3) ; 4)

Solution algorithm:

1. Let's write down Newton's second law. We express the acceleration from the formula.
2. We substitute the changed values ​​of mass and force into the resulting expression and find the new value of acceleration, expressed through its original value.
3. Choose the correct answer.

Solution:

1. According to Newton's second law F=m a, force F, which acts on a body of mass m, imparts acceleration to the body A. We have:

2. By condition m 2 = 2m, F 2 =0,5F.

Then the changed acceleration will be equal to:

In vector form the notation is similar.

Second version of the task (Demidova, No. 9)

A stone weighing 200 g is thrown at an angle of 60° to the horizontal with an initial speed v = 20 m/s. Determine the modulus of gravity acting on the stone at the top point of the trajectory.

If a body is thrown at an angle to the horizontal and the drag force can be neglected, the resultant of all forces is constant. The vertical component of movement depends on the forces acting on the body. It is necessary to know the formula of gravity F=mg, since, as a rule, only it acts on a body thrown at an angle to the horizontal.

Solution algorithm:

1. Convert the mass value to SI.
2. We determine what forces act on the stone.
3. We write down the formula for gravity. We calculate the magnitude of the force.
4. We write down the answer.

Solution:

1. Stone mass m=200 g=0.2 kg.
2. A thrown stone is affected by gravity F T = mg. Since the condition does not stipulate otherwise, air resistance can be neglected.
3. The force of gravity is the same at any point in the trajectory of the stone. This means the data in the condition (initial speed v and the angle to the horizon at which the body is thrown) are redundant. From here we get: F T = 0.2∙10 =2 N.

Answer : 2

Third version of the task (Demidova, No. 27)

A constant horizontal force of F = 9 N is applied to a system of a cube weighing 1 kg and two springs (see figure). The system is at rest. There is no friction between the cube and the support. The left edge of the first spring is attached to the wall. The stiffness of the first spring k1 = 300 N/m. The stiffness of the second spring is k2 = 600 N/m. What is the elongation of the second spring?

Solution algorithm:

1. We write down Hooke's law for the 2nd spring. We find its connection with the force F given in the condition.
2. From the resulting equation we express the elongation and calculate it.
3. We write down the answer.

Solution:

1. According to Hooke's law, the elongation of a spring is related to the spring stiffness k and the force applied to it F expression F= k∆ l. The second spring is subject to a tensile force F 2 = k2∆ l. 1st spring is stretched by force F. By condition F=9 H. Since the springs form a single system, the force F also stretches the 2nd spring, i.e. F 2 =F.
2. Elongation Δ l is defined like this:

According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of a curve.

Path is the length of the trajectory that a material point describes over a given period of time. This is a scalar quantity.
Moving is a vector connecting the initial position of a material point with its final position (see figure).

It is very important to understand how a path differs from a movement. The most important difference is that movement is a vector with a beginning at the point of departure and an end at the destination (it does not matter at all what route this movement took). And the path is, on the contrary, a scalar quantity that reflects the length of the trajectory traveled.

Uniform linear movement called a movement in which a material point makes the same movements over any equal periods of time
Speed ​​of uniform linear motion is called the ratio of movement to the time during which this movement occurred:

For uneven motion they use the concept average speed. Average speed is often introduced as a scalar quantity. This is the speed of such uniform motion in which the body travels the same path in the same time as during uneven motion:

Instant speed call the speed of a body at a given point in the trajectory or at a given moment in time.
Uniformly accelerated linear motion- this is a rectilinear movement in which the instantaneous speed for any equal periods of time changes by the same amount

The dependence of the body coordinates on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
Free fall called uniformly accelerated motion with constant acceleration g = 9.8 m/s 2, independent of the mass of the falling body. It occurs only under the influence of gravity.

Free fall speed is calculated using the formula:

Vertical movement is calculated using the formula:

One type of motion of a material point is motion in a circle. With such movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). You can describe the position of a body on a circle using a radius drawn from the center of the circle to the body. The displacement of a body when moving in a circle is described by turning the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the period of time during which this rotation occurred characterizes the speed of movement of the body in a circle and is called angular velocity ω:

Angular velocity is related to linear velocity by the relation

where r is the radius of the circle.
The time it takes a body to complete a complete revolution is called circulation period. The reciprocal of the period is the circulation frequency - ν

Since during uniform motion in a circle the velocity module does not change, but the direction of the velocity changes, with such motion there is acceleration. He is called centripetal acceleration, it is directed radially to the center of the circle:

Basic concepts and laws of dynamics

The part of mechanics that studies the reasons that caused the acceleration of bodies is called dynamics

Newton's first law:
There are reference systems relative to which a body maintains its speed constant or is at rest if other bodies do not act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform linear motion with balanced external forces acting on it is called inertia. The phenomenon of maintaining the speed of a body under balanced external forces is called inertia. Inertial reference systems are systems in which Newton's first law is satisfied.

Galileo's principle of relativity:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. subject to the same laws
Weight is a measure of body inertia
Force is a quantitative measure of the interaction of bodies.

Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$

The addition of forces consists of finding the resultant of several forces, which produces the same effect as several simultaneously acting forces.

Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $

Newton's III law emphasizes that the action of bodies on each other is in the nature of interaction. If body A acts on body B, then body B acts on body A (see figure).

Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves press into the ground must be greater than the frictional force of the sled on the ground. Otherwise, the hooves will slip and the horse will not move.
If a body is subjected to deformation, forces arise that prevent this deformation. Such forces are called elastic forces.

Hooke's law written in the form

where k is the spring stiffness, x is the deformation of the body. The “−” sign indicates that the force and deformation are directed in different directions.

When bodies move relative to each other, forces arise that impede the movement. These forces are called friction forces. A distinction is made between static friction and sliding friction. Sliding friction force calculated by the formula

where N is the support reaction force, µ is the friction coefficient.
This force does not depend on the area of ​​the rubbing bodies. The friction coefficient depends on the material from which the bodies are made and the quality of their surface treatment.

Static friction occurs if the bodies do not move relative to each other. The static friction force can vary from zero to a certain maximum value

By gravitational forces are the forces with which any two bodies are attracted to each other.

Here R is the distance between the bodies. The law of universal gravitation in this form is valid either for material points or for spherical bodies.

Body weight called the force with which the body presses on a horizontal support or stretches the suspension.

Gravity- this is the force with which all bodies are attracted to the Earth:

With a stationary support, the weight of the body is equal in magnitude to the force of gravity:

If a body moves vertically with acceleration, its weight will change.
When a body moves with upward acceleration, its weight

It can be seen that the weight of the body is greater than the weight of the body at rest.

When a body moves with downward acceleration, its weight

In this case, the weight of the body is less than the weight of the body at rest.

Weightlessness is the movement of a body in which its acceleration is equal to the acceleration of gravity, i.e. a = g. This is possible if only one force acts on the body - gravity.
Artificial Earth satellite- this is a body that has a speed V1 sufficient to move in a circle around the Earth
There is only one force acting on the Earth's satellite - the force of gravity directed towards the center of the Earth
First escape velocity- this is the speed that must be imparted to the body so that it revolves around the planet in a circular orbit.

where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is equal to

1.3. Basic concepts and laws of statics and hydrostatics

Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
Pressure is a physical quantity equal to the ratio of the force acting on a platform perpendicular to this force to the area of ​​the platform:

Valid for liquids and gases Pascal's law:
pressure spreads in all directions without changes.
If a liquid or gas is in a gravity field, then each layer above presses on the layers below, and as the liquid or gas is immersed inside, the pressure increases. For liquids

where ρ is the density of the liquid, h is the depth of penetration into the liquid.

A homogeneous liquid in communicating vessels is established at the same level. If liquid with different densities is poured into the elbows of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case

The heights of liquid columns are inversely proportional to densities:

Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. The pistons have different areas. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then

Then A1 = A2.
A body immersed in a liquid or gas is acted upon by an upward buoyant force from the side of this liquid or gas, which is called by the power of Archimedes
The magnitude of the buoyancy force is determined by Archimedes' law: a body immersed in a liquid or gas is acted upon by a buoyant force directed vertically upward and equal to the weight of the liquid or gas displaced by the body:

where ρ liquid is the density of the liquid in which the body is immersed; V submergence is the volume of the submerged part of the body.

Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.

1.4. Conservation laws

Momentum is a vector quantity. [p] = kg m/s. Along with body impulse, they often use impulse of power. This is the product of force and the duration of its action
The change in the momentum of a body is equal to the momentum of the force acting on this body. For an isolated system of bodies (a system whose bodies interact only with each other) law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before interaction is equal to the sum of the impulses of the same bodies after the interaction.
Mechanical work called a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:

Power is the work done per unit of time:

The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula

If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated using the formula

where h is the lift height

Compressed spring energy:

where k is the spring stiffness coefficient, x is the absolute deformation of the spring.

The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, law of conservation of mechanical energy: if there are no frictional forces between the bodies of an isolated system (or other forces leading to energy dissipation), then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during interaction part of the mechanical energy of the bodies turns into internal energy.

1.5. Mechanical vibrations and waves

The quantity A equal to the largest absolute value of the fluctuating physical quantity x is called amplitude of oscillations. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T is the time it takes for an oscillating body to complete one complete oscillation. Frequency of periodic oscillations The number of complete oscillations completed per unit of time is called:

Frequency is measured in s -1. This unit is called hertz (Hz).

Mathematical pendulum is a material point of mass m suspended on a weightless inextensible thread and oscillating in a vertical plane.
If one end of the spring is fixed motionless, and a body of mass m is attached to its other end, then when the body is removed from the equilibrium position, the spring will stretch and oscillations of the body on the spring will occur in the horizontal or vertical plane. Such a pendulum is called a spring pendulum.

Period of oscillation of a mathematical pendulum determined by the formula

where l is the length of the pendulum.

Period of oscillation of a load on a spring determined by the formula

where k is the spring stiffness, m is the mass of the load.

Propagation of vibrations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves are the process of propagation of vibrations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of propagation of the wave. The wave is called longitudinal, if the vibrations of the particles of the medium occur in the direction of wave propagation.
Wavelength is the distance between two closest points oscillating in the same phase:

where v is the speed of wave propagation.

Sound waves are called waves in which oscillations occur with frequencies from 20 to 20,000 Hz.
The speed of sound varies in different environments. The speed of sound in air is 340 m/s.
Ultrasonic waves are called waves whose oscillation frequency exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.

« Physics - 10th grade"

Let's get acquainted with problems for which you do not need to know how forces depend on the distances between interacting bodies (or parts of one body) and on their velocities. The only thing we need is an expression for the force of gravity near the Earth's surface: τ = m.

Task 1.

A force F = 1.5 N is applied to the center of a homogeneous ball with a mass m = 0.2 kg. Determine the magnitude and direction of the force 1 that must be applied to the center of the ball in addition to the force so that the ball moves with an acceleration a = 5 m/s 2 directed the same as force (Fig. 2.17).

Solution.

Two forces act on the ball: force and the desired force 1.
Since the magnitude and direction of the force are unknown, we can first depict only the force in the figure (see Fig. 2.17).
According to Newton's second law, m = + 1.
Hence 1 = m - .
Since the vectors m and at any moment of time must be located on the same straight line, then the force 1, being their difference, is located on the same straight line.

Thus, the desired force can be directed either in the same way as the force or opposite to it.
To determine the magnitude and direction of force 1, we find its projection onto the X axis, the direction of which coincides with the force.
Considering that F x = F and a x = a, the expression for force 1 in projections on the X axis can be written as F 1x = ma - F.

Let's analyze the last expression.
If ma > F, then F 1x > 0, i.e. force 1 is directed in the same way as the X axis.
If ma< F, то F 1x < 0, т. е. сила F 1 направлена противоположно направлению оси X. Для рассматриваемого случая

F 1x - 0.2 5N - 1.5 N = -0.5 N.

Task 2.

As a result of the received push, the block began to slide up the inclined plane from point O with an initial speed υ 0 = 4.4 m/s. Determine the position of the block relative to point O after a period of time t 1 - 2 s after the start of its movement, if the angle of inclination of the plane to the horizon is α = 30°. Ignore friction.

Solution.

Since we need to find the position of the block relative to point O, we take the origin of coordinates at this point. The X axis will be directed downwards along the inclined plane, and the Y axis will be directed upwards perpendicular to this plane (Fig. 2.19). When the block moves, two forces act on it: the force of gravity m and the reaction force of the support of the inclined plane, perpendicular to the latter. This force is sometimes called the normal reaction force. It is always perpendicular to the surface on which the body is located.

According to Newton's second law, m = m +. Since constant forces act on the block, it will move along the X axis with constant acceleration. Therefore, to determine the position of the block relative to point O, you can use the kinematic equation

With the choice of the direction of the X axis and the origin of coordinates, we have x 0 = 0 and υ 0x = -υ 0. We find the projection of acceleration a x on the X axis using Newton’s second law. For the case under consideration, ma x = mg x + N x. Considering that g x = g sinα and Nx = 0, we obtain a x = g sinα. Thus,

Task 3.

Two bodies with masses m 1 = 10 g and m 2 = 15 g are connected by an inextensible and weightless thread thrown over a weightless block installed on an inclined plane (Fig. 2.20). The plane forms an angle α = 30° with the horizon. Determine the acceleration with which these bodies will move. Ignore friction.

Solution.

Let us assume that a body of mass m 1 is pulling.
Let us choose the coordinate axes as shown in Figure 2.21.
In projections on the X1 and X axes, we write the equations of motion of bodies in the form:

m 1 a x1 = m 1 g - T 1,

m 2 a x = T 2 - m 2 g sinα,

|a x | =|a x1 |, since the thread is inextensible.

The tension forces of the thread are equal, since the thread and the block are weightless.
Adding the left and right sides of the equation, we get
Since a x > 0, the movement of bodies occurs in the chosen direction.

Task 4.

A car weighing m = 1000 kg moves at a speed v = 36 km/h on a convex bridge having a radius of curvature R = 50 m. With what force F does the car press on the bridge in the middle? At what minimum speed umin must the car move so that at the top point it stops exerting pressure on the bridge?

The forces acting on the car along the radius of the bridge are shown in Figure 2.22:
m - gravity;
- normal reaction force of the bridge.
According to Newton's third law, the required pressure force is equal in magnitude to the reaction force of the bridge.
When a body moves in a circle, we always direct one of the coordinate axes from the body to the center of the circle.
According to Newton's second law, the centripetal acceleration of a car is determined by the sum of the forces acting on it along the radius of the circle along which it is moving:

mυ 2 /R = mg - N.

F = N = m(g - υ 2 /R) = 7.8 kN.

The pressure force on the bridge will become zero at mυ 2 min /R = mg, so that υ min = 80 km/h.
At a speed exceeding υ min, the car will break away from the bridge surface.

Topics of the Unified State Exam codifier: laws of dynamics, force, principle of superposition of forces, Newton’s second law, Newton’s third law.

The interaction of bodies can be described using the concept of force. Force is a vector quantity that is a measure of the influence of one body on another.

Being a vector, force is characterized by its modulus (absolute value) and direction in space. In addition, the point of application of the force is important: the same force in magnitude and direction, applied at different points of the body, can have different effects. So, if you grab the rim of a bicycle wheel and pull tangentially to the rim, the wheel will begin to rotate. If you pull along the radius, there will be no rotation.

Superposition principle.

Experience shows that if several other bodies act on a given body, then the corresponding forces add up as vectors. More precisely, the principle of superposition is valid.
The principle of superposition of forces .Let the forces act on the body. If you replace them with one force then the result of the impact will not change.

The force is called resultant strength

Newton's second law.

If the resultant of the forces applied to the body is equal to zero (that is, the effects of other bodies compensate each other), then, by virtue of Newton’s first law, there will be such reference systems (called inertial) in which the movement of the body will be uniform and rectilinear. But if the resultant does not vanish, then the body will experience acceleration in the inertial frame of reference.
Newton's second law provides a quantitative relationship between acceleration and force.

Newton's second law. The product of the body mass and the acceleration vector is the resultant of all forces applied to the body:.

We emphasize that Newton’s second law relates vectors acceleration and force. This means that the following statements are true.

1. , where is the module of acceleration, is the module of the resultant force.

2. The acceleration vector is codirectional with the resultant force vector, since the mass of the body is positive.

For example, if a body moves uniformly in a circle, then its acceleration is directed towards the center of the circle. Therefore, the resultant of all forces applied to the body is also directed towards the center of the circle. Newton's second law is not valid in any frame of reference. Let us remember the staggering observer ( Newton's first law): relative to it, the house moves with acceleration, although the resultant of all forces applied to the house is equal to zero. Newton's second law is satisfied only in inertial frames of reference, the fact of whose existence is established by Newton's first law.

Newton's third law.

Experience shows that if body A acts on body B, then body B acts on body A. The quantitative relationship between the actions of bodies on each other is given by Newton’s third law (“action is equal to reaction”).

Newton's third law. Two bodies act on each other with forces equal in magnitude and opposite in direction. These forces have the same physical nature and are directed along a straight line connecting their points of application.

For example, if a pencil acts on the table with a force directed downwards, then the table acts on the pencil with a force directed upwards (Fig. 1). These forces are equal in absolute magnitude.

Rice. 1.

The forces and , as we see, are applied to different bodies and therefore cannot balance each other (there is no point in talking about their resultant).
Newton's third law, like the second, is valid only in inertial frames of reference.
Mechanics based on Newton's laws is called classical mechanics. Classical mechanics, however, has a limited range of applicability. Within the framework of classical mechanics, motion is well described not very small bodies with not very high speeds. When describing atoms and elementary particles, classical mechanics is replaced by quantum mechanics. The movement of objects at speeds close to the speed of light occurs according to the laws theory of relativity.