What is the movement of a material point. Mechanical movement

Section 1 MECHANICS

Chapter 1: BASIC KINEMATICS

Mechanical movement. Trajectory. Path and movement. Speed ​​addition

Mechanical body movement is called the change in its position in space relative to other bodies over time.

Mechanical movement of bodies studies Mechanics. The section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and acting forces is called kinematics .

Mechanical motion is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates of a material point, you must first select a reference body and associate a coordinate system with it.

Body of referencecalled a body relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: Land, building, car, ship, etc.

The coordinate system, the reference body with which it is associated, and the indication of the time reference form frame of reference , relative to which the movement of the body is considered (Fig. 1.1).

A body whose dimensions, shape and structure can be neglected when studying a given mechanical movement is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.

Trajectoryit is the line along which the body moves.

Depending on the type of trajectory, movements are divided into rectilinear and curvilinear

Pathis the length of the trajectory ℓ(m) ( fig.1.2)

The vector drawn from the initial position of the particle to its final position is called moving of this particle for a given time.

Unlike a path, displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in what direction the body has moved during a given time.

Motion vector module(that is, the length of the segment that connects the starting and ending points of the movement) can be equal to the distance traveled or less than the distance traveled. But the displacement module can never be greater than the distance traveled. For example, if a car moves from point A to point B along a curved path, then the magnitude of the displacement vector is less than the distance traveled ℓ. The path and the modulus of displacement are equal only in one single case, when the body moves in a straight line.

Speedis a vector quantitative characteristic of body movement

average speed– this is a physical quantity equal to the ratio of the vector of movement of a point to the period of time

The direction of the average speed vector coincides with the direction of the displacement vector.

Instant speed, that is, the speed at a given moment in time is a vector physical quantity equal to the limit to which the average speed tends as the time interval Δt decreases infinitely.

Mechanical movement is considered for material point and For solid body.

Motion of a material point

Forward movement an absolutely rigid body is a mechanical movement during which any straight line segment associated with this body is always parallel to itself at any moment in time.

If you mentally connect any two points of a rigid body with a straight line, then the resulting segment will always be parallel to itself in the process of translational motion.

During translational motion, all points of the body move equally. That is, they travel the same distance in the same amount of time and move in the same direction.

Examples of translational motion: the movement of an elevator car, mechanical scales, a sled rushing down a mountain, bicycle pedals, a train platform, engine pistons relative to the cylinders.

Rotational movement

During rotational motion, all points of the physical body move in circles. All these circles lie in planes parallel to each other. And the centers of rotation of all points are located on one fixed straight line, which is called axis of rotation. Circles that are described by points lie in parallel planes. And these planes are perpendicular to the axis of rotation.

Rotational movement is very common. Thus, the movement of points on the rim of a wheel is an example of rotational movement. Rotational motion is described by a fan propeller, etc.

Rotational motion is characterized by the following physical quantities: angular velocity of rotation, period of rotation, frequency of rotation, linear speed of a point.

Angular velocity A body rotating uniformly is called a value equal to the ratio of the angle of rotation to the period of time during which this rotation occurred.

The time it takes a body to complete one full revolution is called rotation period (T).

The number of revolutions a body makes per unit time is called speed (f).

Rotation frequency and period are related to each other by the relation T = 1/f.

If a point is located at a distance R from the center of rotation, then its linear speed is determined by the formula:

Trajectory description

It is customary to describe the trajectory of a material point using a radius vector, the direction, length and starting point of which depend on time. In this case, the curve described by the end of the radius vector in space can be represented in the form of conjugate arcs of varying curvature, located in the general case in intersecting planes. In this case, the curvature of each arc is determined by its radius of curvature, directed towards the arc from the instantaneous center of rotation, located in the same plane as the arc itself. Moreover, a straight line is considered as a limiting case of a curve, the radius of curvature of which can be considered equal to infinity. And therefore, in the general case, a trajectory can be represented as a set of conjugate arcs.

It is important that the shape of the trajectory depends on the reference system chosen to describe the movement of the material point. Thus, rectilinear motion in an inertial frame will generally be parabolic in a uniformly accelerating reference frame.

Relationship with speed and normal acceleration

The velocity of a material point is always directed tangent to the arc used to describe the point's trajectory. In this case, there is a connection between the speed v, normal acceleration a n and the radius of curvature of the trajectory ρ at a given point:

Connection with equations of dynamics

Representation of a trajectory as a trace left by movement material point, connects the purely kinematic concept of trajectory, as a geometric problem, with the dynamics of the movement of a material point, that is, the problem of determining the causes of its movement. In fact, solving Newton's equations (in the presence of a complete set of initial data) gives the trajectory of a material point. And vice versa, knowing the trajectory of the material point in an inertial reference frame and its speed at each moment of time, you can determine the forces acting on it.

Trajectory of a free material point

In accordance with Newton's First Law, sometimes called the law of inertia, there must be a system in which a free body maintains (as a vector) its speed. Such a reference system is called inertial. The trajectory of such movement is a straight line, and the movement itself is called uniform and rectilinear.

Motion under the influence of external forces in an inertial reference frame

If in a known inertial system the speed of movement of an object with mass m changes in direction, even remaining the same in size, that is, the body turns and moves in an arc with a radius of curvature R, then the object experiences normal acceleration a n. The cause that causes this acceleration is a force directly proportional to this acceleration. This is the essence of Newton's Second Law:

Where is the vector sum of the forces acting on the body, its acceleration, and m- inertial mass.

In the general case, a body is not free in its movement, and its position, and in some cases its speed, are subject to restrictions - connections. If connections impose restrictions only on the coordinates of the body, then such connections are called geometric. If they also propagate at speed, then they are called kinematic. If the equation of a constraint can be integrated over time, then such a constraint is called holonomic.

The action of bonds on a system of moving bodies is described by forces called bond reactions. In this case, the force included in the left side of equation (1) is the vector sum of active (external) forces and the reaction of the connections.

It is important that in the case of holonomic connections it becomes possible to describe the motion of mechanical systems in generalized coordinates included in the Lagrange equations. The number of these equations depends only on the number of degrees of freedom of the system and does not depend on the number of bodies included in the system, the position of which must be determined to fully describe the motion.

If the bonds operating in the system are ideal, that is, there is no transition of motion energy into other types of energy in them, then when solving the Lagrange equations, all unknown bond reactions are automatically eliminated.

Finally, if the acting forces belong to the class of potential forces, then with an appropriate generalization of concepts it becomes possible to use Lagrange’s equations not only in mechanics, but also in other areas of physics.

In this understanding, the forces acting on a material point unambiguously determine the shape of the trajectory of its movement (under known initial conditions). The converse statement is not true in the general case, since the same trajectory can take place with different combinations of active forces and coupling reactions.

Motion under the influence of external forces in a non-inertial reference frame

If the reference system is non-inertial (that is, it moves with a certain acceleration relative to the inertial reference system), then it is also possible to use expression (1), however, on the left side it is necessary to take into account the so-called inertial forces (including centrifugal force and Coriolis force associated with rotation of a non-inertial reference system).

Illustration

Trajectories of the same movement in different reference systems. At the top of the inertial frame, a leaky bucket of paint is carried in a straight line above a rotating stage. Below in non-inertial (paint trace for an observer standing on the stage)

As an example, consider a theater worker moving in the grate space above the stage in relation to the theater building evenly And straight forward and carrying over rotating stage with a leaky bucket of paint. It will leave a mark on it from falling paint in the form unwinding spiral(if moving from stage rotation center) and twisting- in the opposite case. At this time, his colleague, who is responsible for the cleanliness of the rotating stage and who is located on it, will therefore be forced to carry a leak-free bucket under the first one, constantly being under the first one. And its movement in relation to the building will also be uniform And straightforward, although in relation to the scene, which is non-inertial system, its movement will be twisted And uneven. Moreover, in order to counteract the drift in the direction of rotation, he must overcome the action of the Coriolis force by muscular effort, which his upper colleague above the stage does not experience, although the trajectories of both are in inertial system the theater buildings will represent straight lines.

But one can imagine that the task of the colleagues considered here is precisely to apply straight lines on rotating stage. In this case, the lower one must require the upper one to move along a curve that is a mirror image of the trace of the previously spilled paint. Hence, rectilinear movement V non-inertial system countdown will not be such for the observer in an inertial frame.

Moreover, uniform body movement in one system, maybe uneven to another. So, two drops of paint that fell into different moments time from a leaky bucket, both in their own frame of reference and in the frame of the lower colleague stationary in relation to the building (on the stage that has already stopped rotating), will move in a straight line (towards the center of the Earth). The difference will be that for the lower observer this movement will be accelerated, and for his top colleague, if he stumbles, will fall, moving along with any of the drops, the distance between the drops will increase proportionally first degree time, that is, the mutual movement of drops and their observer in his accelerated the coordinate system will be uniform with speed v, determined by the delay Δ t between the moments of falling drops:

Where g- acceleration of gravity .

Therefore, the shape of the trajectory and the speed of movement of the body along it, considered in a certain frame of reference, about which nothing is known in advance, does not give an unambiguous idea of ​​the forces acting on the body. The question of whether this system is sufficiently inertial can be resolved only on the basis of an analysis of the causes of the appearance of the acting forces.

Thus, in a non-inertial frame:

• The curvature of the trajectory and/or the variability of the speed are insufficient arguments in favor of the statement that a body moving along it is acted upon by external forces, which in the final case can be explained by gravitational or electromagnetic fields.
• The straightness of the trajectory is an insufficient argument in favor of the statement that no forces act on a body moving along it.

Notes

Literature

• Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova. M.: Nauka, 1989
• Frisch S. A. and Timoreva A. V. Course of general physics, Textbook for physics-mathematics and physics-technical faculties of state universities, Volume I. M.: GITTL, 1957

Links

• http://av-physics.narod.ru/mechanics/trajectory.htm [ unreputable source?] Trajectory and displacement vector, section of a physics textbook

A basic level of
Option 1

A1. The trajectory of a moving material point in a finite time is

1. 
   line segment
2. 
   part of the plane
3. 
   finite set of points
4. 
   among answers 1,2,3 there is no correct one

A3. A swimmer swims against the current of the river. The speed of the river is 0.5 m/s, the speed of the swimmer relative to the water is 1.5 m/s. The speed modulus of the swimmer relative to the shore is equal to

1) 2 m/s 2) 1.5 m/s 3) 1 m/s 4) 0.5 m/s

A4. Moving in a straight line, one body covers a distance of 5 m every second. Another body, moving in a straight line in one direction, covers a distance of 10 m every second. The movements of these bodies

A5. The graph shows the dependence of the X coordinate of a body moving along the OX axis on time. What is the initial coordinate of the body?

3) -1 m 4) - 2 m

A6. What function v(t) describes the dependence of the velocity modulus on time for uniform rectilinear motion? (length is measured in meters, time in seconds)

1) v = 5t 2) v = 5/t 3) v = 5 4) v = -5

A7. The modulus of the body's velocity has doubled over some time. Which statement would be correct?

1. 
   body acceleration doubled
2. 
   acceleration decreased by 2 times
3. 
   acceleration hasn't changed
4. 
   body moves with acceleration

1) 1m/s 2 2) 1.2m/s 2 3) 2.0m/s 2 4) 2.4m/s 2

A9. When a body is in free fall, its speed (take g=10m/s 2)

1. 
   in the first second it increases by 5 m/s, in the second – by 10 m/s;
2. 
   in the first second it increases by 10 m/s, in the second – by 20 m/s;
3. 
   in the first second it increases by 10 m/s, in the second – by 10 m/s;
4. 
   in the first second it increases by 10m/s, and in the second – by 0m/s.

1) increased by 2 times 2) increased by 4 times

3) decreased by 2 times 4) decreased by 4 times
Option 2

A1. Two problems are solved:

A. the docking maneuver of two spacecraft is calculated;

b. the orbital period of spacecraft is calculated
around the Earth.

In what case can spaceships be considered as material points?

1. 
   only in the first case
2. 
   only in the second case
3. 
   in both cases
4. 
   neither in the first nor in the second case

1) 0 km 2) 109 km 3) 218 ​​km 4) 436 km

A3. When they say that the change of day and night on Earth is explained by the rising and setting of the Sun, they mean a reference system associated

1) with the Sun 2) with the Earth

3) with the center of the galaxy 4) with any body

A4. When measuring the characteristics of the rectilinear movements of two material points, the values ​​of the coordinates of the first point and the speed of the second point were recorded at the moments of time indicated in Tables 1 and 2, respectively:

What can be said about the nature of these movements, assuming that he hasn't changed in the time intervals between the moments of measurements?

1) both are uniform

2) the first is uneven, the second is uniform

3) the first is uniform, the second is uneven

4)both are uneven

A5. Using the graph of the distance traveled versus time, determine the speed
cyclist at time t = 2 s.
1) 2 m/s 2) 3 m/s

3) 6 m/s 4) 18 m/s

A6. The figure shows graphs of the distance traveled in one direction versus time for three bodies. Which body was moving with greater speed?
1) 1 2) 2 3) 3 4) the speeds of all bodies are the same
A7. The speed of a body moving rectilinearly and uniformly accelerated changed when moving from point 1 to point 2 as shown in the figure. What direction does the acceleration vector have in this section?

A8. Using the graph of the velocity modulus versus time shown in the figure, determine the acceleration of a rectilinearly moving body at the time t=2s.

1) 2 m/s 2 2) 3 m/s 2 3) 9 m/s 2 4) 27 m/s 2
A9. In a tube from which the air has been evacuated, a pellet, a cork and a bird feather are simultaneously dropped from the same height. Which body will reach the bottom of the tube faster?

1) pellet 2) cork 3) bird feather 4) all three bodies at the same time.

A10. A car on a turn moves along a circular path of radius 50 m with a constant absolute speed of 10 m/s. What is the acceleration of the car?

1) 1 m/s 2 2) 2 m/s 2 3) 5 m/s 2 4) 0 m/s 2
Answers.

Profile level
Option 1

A1. A body thrown vertically upward reached a maximum height of 10 m and fell to the ground. The displacement module is equal to

1) 20m 2) 10m 3) 5m 4) 0m

A2. A body thrown vertically upward reached a maximum height of 5 m and fell to the ground. The distance traveled by the body is

1) 2.5m 2) 10m 3) 5m 4) 0m

A3. Two cars are moving along a straight highway: the first at a speed V, the second at a speed 4 V. What is the speed of the first car relative to the second?

1) 5V 2) 3V 3) -3V 4) -5V

A4. A small object comes off at point A from an airplane flying horizontally at speed V. What line is the trajectory of this object in the reference frame associated with the airplane, if air resistance is neglected?

A5. Two material points move along the OX axis according to the laws:

x 1 = 5 + 5t, x 2 = 5 - 5t (x - in meters, t - in seconds). What is the distance between them after 2 s?

1) 5m 2) 10m 3) 15m 4) 20m

A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5 + 15t 2 (X is measured in meters, time in seconds). The initial velocity module is equal to

A7. Two material points move in circles of radii R, = R and R 2 = 2R with the same speeds. Compare their centripetal accelerations.

1) a 1 = a 2 2)a 1 =2a 2 3)a 1 =a 2 /2 4)a 1 =4a 2
Part 2.

IN 1. The graph shows the dependence of movement speed on time. What is the average speed during the first five seconds?

AT 2. A small stone thrown from a flat horizontal surface of the earth at an angle to the horizon reached a maximum height of 4.05 m. How much time passed from the throw to the moment when its speed became directed horizontally?
Part 3.

C1. The coordinates of a moving body change according to the law X=3t+2, Y=-3+7t 2. Find the speed of the body 0.5 s after the start of movement.
Option 2

A1. A ball thrown vertically down from a height of 3 m bounces off the floor vertically and rises to a height of 3 m. The path of the ball is

1) -6m 2) 0m 3) 3m 4) 6m

A2. A stone thrown from a second floor window from a height of 4 m falls to the ground at a distance of 3 m from the wall of the house. What is the modulus of movement of the stone?

1) 3m 2) 4m 3) 5m 4) 7m

A3. A raft floats uniformly down the river at a speed of 6 km/h. A person moves across a raft at a speed of 8 km/h. What is the speed of a person in the reference frame associated with the shore?

1) 2 km/h 2) 7 km/h 3) 10 km/h 4) 14 km/h

A4. The helicopter rises vertically upward evenly. What is the trajectory of a point at the end of a helicopter rotor blade in the reference frame associated with the helicopter body?

3) point 4) helix

A5. A material point moves in a plane uniformly and rectilinearly according to the law: X = 4 + 3t, ​​Y = 3 - 4t, where X,Y are the coordinates of the body, m; t - time, s. What is the speed of the body?
1) 1m/s 2) 3 m/s 3) 5 m/s 4) 7 m/s

A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5t+ 15t 2 (X is measured in meters, time in seconds).

The initial velocity module is equal to

1)0m/s 2) 5m/s 3) 7.5m/s 4) 15m/s

A7. The period of uniform motion of a material point along a circle is 2 s. After what minimum time does the direction of velocity change to the opposite?

1) 0.5 s 2) 1 s 3) 1.5 s 4) 2 s
Part 2.

IN 1. The graph shows the dependence of the speed V of the body on time t, describing the movement of the body along the OX axis. Determine the module of the average speed of movement in 2 seconds.
AT 2. A small stone was thrown from a flat horizontal surface of the earth at an angle to the horizon. What is the range of the stone if, 2 s after the throw, its speed was directed horizontally and equal to 5 m/s?
Part 3.

C1. A body emerging from a certain point moved with acceleration constant in magnitude and direction. Its speed at the end of the fourth second was 1.2 m/s, at the end of 7 seconds the body stopped. Find the path traveled by the body.
Answers.

Test on the topic “Newton’s Laws. Forces in mechanics."

A basic level of
Option 1

A1. Which equality correctly expresses Hooke's law for an elastic spring?

1) F=kx 2) F x =kx 3) F x =-kx 4) F x =k | x |

A2. Which of the following bodies are associated with reference systems that cannot be considered inertial?

A . A skydiver descending at a steady speed.

B. A stone thrown vertically upward.

B. A satellite moving in orbit with a constant absolute velocity.

1) A 2) B 3) C 4) B and C

A3. Weight has a dimension

1) mass 2) acceleration 3) force 4) speed

A4. A body near the Earth's surface is in a state of weightlessness if it moves with an acceleration equal to the acceleration of gravity and directed

1) vertically down 2) vertically up

3) horizontally 4) at an acute angle to the horizontal.

A5. How will the sliding friction force change when the block moves along a horizontal plane if the normal pressure force is doubled?

1) will not change 2) will increase by 2 times

3) will decrease by 2 times 4) will increase by 4 times.

A6. What is the correct relationship between static friction force, sliding friction force and rolling friction force?

1) F tr.p =F tr >F tr.k 2) F tr.p >F tr >F tr.k 3) F tr.p F tr.k 4) F tr.p >F tr =F tr. .To

A7. A paratrooper launches uniformly at a speed of 6 m/s. The force of gravity acting on it is 800N. What is the mass of the skydiver?

1) 0 2) 60 kg 3) 80 kg 4) 140 kg.

A8. What is the measure of interaction between bodies?

1) Acceleration 2) Mass 3) Impulse. 4) Strength.

A9. How are changes in speed and inertia of a body related?

A . If the body is more inert, then the change in speed is greater.

B. If the body is more inert, then the change in speed is less.

B. A body that changes its speed faster is less inert.

G . The more inert body is the one that changes its speed faster.

1) A and B 2) B and D 3) A and D 4) B and C.
Option 2

A1. Which of the following formulas expresses the law of universal gravitation?
1) F=ma 2) F=μN 3) F x =-kx 4) F=Gm 1 m 2 /R 2

A2. When two cars collided, the buffer springs with a stiffness of 10 5 N/m were compressed by 10 cm. What is the maximum elastic force with which the springs acted on the car?

1) 10 4 N 2) 2*10 4 N 3) 10 6 N4) 2*10 6 N

A3. A body of mass 100 g lies on a horizontal stationary surface. Body weight is approximately

1) 0H 2) 1H 3) 100N 4) 1000 N.

A4. What is inertia?

2) the phenomenon of conservation of the speed of a body in the absence of the action of other bodies on it

3) change in speed under the influence of other bodies

4) movement without stopping.

A5. What is the dimension of the friction coefficient?
1) N/kg 2) kg/N 3) no dimension 4) N/s

A7. The student jumped to a certain height and sank to the ground. On what part of the trajectory did he experience the state of weightlessness?

1) when moving up 2) when moving down

3) only at the moment of reaching the top point 4) during the entire flight.

A8. What characteristics determine strength?

A. Module.

B. Direction.

B. Application point.

1) A, B, D 2) B and D 3) B, C, D 4) A, B, C.

A9. Which of the quantities (speed, force, acceleration, displacement) during mechanical motion always coincide in direction?

1) force and acceleration 2) force and speed

3) force and displacement 4) acceleration and displacement.
Answers.

Profile level
Option 1

A1. What forces in mechanics retain their significance during the transition from one inertial system to another?

1) forces of gravity, friction, elasticity.

2) only gravity

3) only friction force

4) only elastic force.

A2. How will the maximum static friction force change if the force of normal pressure of the block on the surface is doubled?

1) Will not change. 2) Will decrease by 2 times.

3) Will increase by 2 times. 4) Will increase 4 times.

A3. A block of mass 200 g slides on ice. Determine the sliding friction force acting on the block if the coefficient of sliding friction of the block on ice is 0.1.

1) 0.2N. 2) 2H. 3) 4H. 4) 20N

A4. How and how many times do you need to change the distance between the bodies so that the gravitational force decreases by 4 times?

1) Increase by 2 times. 2) Reduce by 2 times.

3) Increase by 4 times. 4) Reduce by 4 times

A5. A load of mass m lies on the floor of an elevator starting to move downward with acceleration g.

What is the weight of this load?

1) mg. 2) m (g+a). 3) m (g-a). 4) 0

A6. After the rocket engines are turned off, the spacecraft moves vertically upward, reaches the top of the trajectory and then descends. At what part of the trajectory is the astronaut in a state of weightlessness? Neglect air resistance.

1) Only during upward movement. 2) Only during downward movement.

3) During the entire flight with the engine not running.

4) During the entire flight with the engine running.