Laboratory work No. 6

Purpose of the work:

1) Establish the dependence of the flight range of a body thrown horizontally on the height of the throw.

2) Experimentally confirm the validity of the law of conservation of momentum for two balls during their central collision.

Job description:

The ball rolls down a curved chute, bottom part which is horizontal. After separation from the chute, the ball moves along a parabola, the apex of which is at the point of separation of the ball from the chute. Let's choose a coordinate system as shown in Figure 1.

Initial height of the ball h and flight range / are related by the ratio . According to this formula, when the initial altitude decreases by 4 times, the flight range decreases by 2 times. Having measured h and /, you can find the speed of the ball at the moment of separation from the chute using the formula

Equipment: tripod with coupling and clamp, curved groove, metal ball, sheet of paper, sheet of carbon paper, plumb line, measuring tape.

Work progress:

1. Assemble the installation shown in the figure. Lower section
gutters should be horizontal and distance h from the bottom
the edge of the gutter to the table should be 40 cm. Clamping feet
should be located near the top end of the gutter.

2. Place a sheet of paper under the chute, weighing it down with a book so that
it did not move during the experiments. Mark on this sheet with
using a plumb point A, located on the same vertical with
the lower end of the gutter.

3. Place the ball in the groove so that it touches the clamp, and release the ball without pushing. Notice (roughly) the place on the table where the ball lands as it rolls off the chute and flies through the air. Place a sheet of paper on the marked place, and on it - a sheet of copy paper with the “working” side down. Press down these sheets with a book so that they do not move during experiments.

4. Place the ball back into the groove so that it touches the clamp and release without pushing. Repeat this experiment 5 times, making sure
so that the sheet of copy paper and the sheet underneath it
didn't move. Carefully remove the sheet of carbon paper without
moving the sheet underneath, and mark any point lying between the prints. Please note that visible
there may be less than 5 prints because some
fingerprints may merge.

5. Measure the distance l from the marked point to point A.

6. Repeat steps 1-5, lowering the gutter so that the distance from
the lower edge of the gutter to the table was 10 cm (initial height). Measure the corresponding value of the flight range and calculate the ratios and .

Write down the results of measurements and calculations in the table:

Task 1. Study of the motion of a body thrown horizontally

As the body under study, we use a steel ball, which is launched from the upper end of the chute. Then we release the ball. We repeat the launch of the ball 5-7 times and find S avg. Then we increase the height from the floor to the end of the gutter and repeat the ball launch.

We enter the measurement data into the table:

For height H = 81 cm.

Experience no.	S, mm	S avg., mm	N, mm	, mm	S avg / , mm
		40,6		28,5	1,42

For height H = 106 cm.

Experience no.	S, mm	S avg., mm	N, mm	, mm	S avg / , mm
				32,6	1,41
	47,5
	48,5

Task 2. Study of the law of conservation of momentum

We measure the mass of the steel ball m 1 and m 2 on the scales. We attach a device to the surface of the work table to study the motion of a body thrown horizontally. Place it where the ball fell blank slate white paper, glue it with tape and cover it with carbon paper. Use a plumb line to determine the point on the floor above which the edges are located. horizontal section gutters. The ball is launched and its flight range in the horizontal direction l 1 is measured. According to the formula

We calculate the speed of the ball and its momentum P 1 .

Next, we install another ball opposite the lower end of the gutter, using a knot with a support. The steel ball is launched again, the flight range l 1 ’ and the second ball l 2 ’ are measured. Then the velocities of the balls after the collision V 1 ’ and V 2 ’, as well as their impulses p 1 ’ and p 2 ’, are calculated.

Let's find the average value and absolute measurement error using the formulas

, .

Let's calculate the relative measurement error

.

We will enter the data into a table.

Experience no.	m 1, kg	m 2, kg	l 1, m	V 1, m/s	P 1, kg m/s	l 1 ', m	l 2 ', m	V 1 ', m/s	V 2 ', m/s	H, m	P 1 ', kg m/s	P 2 ', kg m/s
1.	0,0076	0,0076	0,47	1,15	0,0076	0,235	0,3	0,5	0,74	0,81	0,004	0,005

1.15 m/s

0.5 m/s

0.74 m/s

P 1 = m 1 V 1 = 0.0076 1.15 = 0.009 m/s

P 1 ’ = m 1 V 1 ’ = 0.0076 0.5 = 0.004 m/s

P 2 ' = m 2 V 2 ' = 0.0076 0.74 = 0.005 m/s

Here – initial body speed, is the speed of the body at the moment of time t, s– horizontal flight range, h– the height above the surface of the earth from which a body is thrown horizontally with speed .

1.1.33. Kinematic equations for velocity projection:

1.1.34. Kinematic coordinate equations:

1.1.35. Body speed at a point in time t:

At the moment falling to the ground y = h, x = s(Fig. 1.9).

1.1.36. Maximum horizontal flight range:

1.1.37. Height above ground level, from which the body is thrown

horizontally:

Motion of a body thrown at an angle α to the horizontal
with initial speed

1.1.38. The trajectory is a parabola(Fig. 1.10). Curvilinear motion along a parabola is caused by the addition of two rectilinear motions: uniform motion along the horizontal axis and uniform motion along the vertical axis.

Rice. 1.10

( – initial speed of the body, – projections of velocity on the coordinate axes at the moment of time t, – body flight time, hmax– maximum body lifting height, s max– maximum horizontal flight range of the body).

1.1.39. Kinematic projection equations:

;

1.1.40. Kinematic coordinate equations:

;

1.1.41. Height of lifting the body to the top point of the trajectory:

At time , (Figure 1.11).

1.1.42. Maximum lifting height:

1.1.43. Body flight time:

At a moment in time , (Fig. 1.11).

1.1.44. Maximum horizontal body flight range:

1.2. Basic equations of classical dynamics

Dynamics(from Greek dynamis– force) is a branch of mechanics devoted to the study of the movement of material bodies under the influence of forces applied to them. Classical dynamics are based on Newton's laws . From these we obtain all the equations and theorems necessary for solving dynamics problems.

1.2.1. Inertial reporting system – This is a frame of reference in which the body is at rest or moves uniformly and rectilinearly.

1.2.2. Strength- is the result of the interaction of the body with environment. One of the simplest definitions of force: the influence of a single body (or field) that causes acceleration. Currently, four types of forces or interactions are distinguished:

· gravitational(manifest in the form of universal gravitational forces);

· electromagnetic(existence of atoms, molecules and macrobodies);

· strong(responsible for the connection of particles in nuclei);

· weak(responsible for particle decay).

1.2.3. Principle of superposition of forces: if several forces act on a material point, then the resulting force can be found using the vector addition rule:

.

Body mass is a measure of body inertia. Any body exhibits resistance when trying to set it in motion or change the module or direction of its speed. This property is called inertia.

1.2.5. Pulse(momentum) is the product of mass T body by its speed v:

1.2.6. Newton's first law: Any material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it (it) to change this state.

1.2.7. Newton's second law(basic equation of the dynamics of a material point): the rate of change of the momentum of the body is equal to the force acting on it (Fig. 1.11):

Rice. 1.11	Rice. 1.12

The same equation in projections onto the tangent and normal to the trajectory of a point:

And .

1.2.8. Newton's third law: the forces with which two bodies act on each other are equal in magnitude and opposite in direction (Fig. 1.12):

1.2.9. Law of conservation of momentum For closed system: the impulse of a closed-loop system does not change over time (Fig. 1.13):

,

Where n– the number of material points (or bodies) included in the system.

Rice. 1.13

The law of conservation of momentum is not a consequence of Newton's laws, but is fundamental law of nature, which knows no exceptions, and is a consequence of the homogeneity of space.

1.2.10. The basic equation for the dynamics of translational motion of a system of bodies:

where is the acceleration of the center of inertia of the system; – total mass of the system from n material points.

1.2.11. Center of mass of the system material points (Fig. 1.14, 1.15):

.

Law of motion of the center of mass: the center of mass of a system moves like a material point, the mass of which is equal to the mass of the entire system and which is acted upon by a force equal to the vector sum of all forces acting on the system.

1.2.12. Impulse of a system of bodies:

where is the speed of the center of inertia of the system.

Rice. 1.14	Rice. 1.15

1.2.13. Theorem on the motion of the center of mass: if the system is in an external stationary uniform field of forces, then no actions within the system can change the movement of the center of mass of the system:

.

1.3. Forces in mechanics

1.3.1. Body weight connection with gravity and ground reaction:

Acceleration of free fall (Fig. 1.16).

Rice. 1.16

Weightlessness is a state in which body weight is zero. IN gravitational field weightlessness occurs when a body moves only under the influence of gravity. If a = g, That P = 0.

1.3.2. Relationship between weight, gravity and acceleration:

1.3.3. Sliding friction force(Fig. 1.17):

where is the sliding friction coefficient; N– normal pressure force.

1.3.5. Basic relations for a body on an inclined plane(Fig. 1.19). :

· friction force: ;

· resultant force: ;

· rolling force: ;

· acceleration:

Rice. 1.19

1.3.6. Hooke's law for a spring: spring extension X proportional to the elastic force or external force:

Where k– spring stiffness.

1.3.7. Potential energy of an elastic spring:

1.3.8. Work done by a spring:

1.3.9. Voltage– a measure of internal forces arising in a deformable body under the influence external influences(Fig. 1.20):

where is the cross-sectional area of ​​the rod, d– its diameter, – the initial length of the rod, – the increment in the length of the rod.

Rice. 1.20	Rice. 1.21

1.3.10. Strain diagram – dependency graph normal voltage σ = F/S from relative elongation ε = Δ l/l when the body is stretched (Fig. 1.21).

1.3.11. Young's modulus– quantity characterizing the elastic properties of the rod material:

1.3.12. Bar length increment proportional to voltage:

1.3.13. Relative longitudinal tension (compression):

1.3.14. Relative transverse tension (compression):

where is the initial transverse dimension of the rod.

1.3.15. Poisson's ratio– the ratio of the relative transverse tension of the rod to the relative longitudinal stretching :

1.3.16. Hooke's law for a rod: the relative increment in the length of the rod is directly proportional to the stress and inversely proportional to Young’s modulus:

1.3.17. Volumetric potential energy density:

1.3.18. Relative shift ( fig1.22, 1.23 ):

where is the absolute shift.

Rice. 1.22	Fig.1.23

1.3.19. Shear modulusG- a quantity that depends on the properties of the material and is equal to the tangential stress at which (if such huge elastic forces were possible).

1.3.20. Tangential elastic stress:

1.3.21. Hooke's law for shear:

1.3.22. Specific potential energy bodies in shear:

1.4. Non-inertial frames of reference

Non-inertial reference frame– an arbitrary reference system that is not inertial. Examples of non-inertial systems: a system moving in a straight line with constant acceleration, as well as a rotating system.

Inertial forces are caused not by the interaction of bodies, but by the properties of the non-inertial reference systems themselves. Newton's laws do not apply to inertial forces. Inertial forces are non-invariant with respect to the transition from one frame of reference to another.

In a non-inertial system, you can also use Newton's laws if you introduce inertial forces. They are fictitious. They are introduced specifically to take advantage of Newton's equations.

1.4.1. Newton's equation for a non-inertial reference frame

where is the acceleration of the body of mass T relative to a non-inertial system; – inertial force is a fictitious force due to the properties of the reference system.

1.4.2. Centripetal force– inertial force of the second kind, applied to a rotating body and directed radially to the center of rotation (Fig. 1.24):

,

where is the centripetal acceleration.

1.4.3. Centrifugal force– inertia force of the first kind, applied to the connection and directed radially from the center of rotation (Fig. 1.24, 1.25):

,

where is the centrifugal acceleration.

Rice. 1.24	Rice. 1.25

1.4.4. Gravity acceleration dependence g depending on the latitude of the area is shown in Fig. 1.25.

Gravity is the result of the addition of two forces: and ; Thus, g(and therefore mg) depends on the latitude of the area:

,

where ω is the angular velocity of the Earth's rotation.

1.4.5. Coriolis force– one of the forces of inertia that exists in a non-inertial reference system due to rotation and the laws of inertia, which manifests itself when moving in a direction at an angle to the axis of rotation (Fig. 1.26, 1.27).

where is the angular velocity of rotation.

Rice. 1.26	Rice. 1.27

1.4.6. Newton's equation for non-inertial reference systems taking into account all forces will take the form

where is the inertial force due to the translational motion of the non-inertial reference frame; And – two inertia forces caused by the rotational motion of the reference system; – acceleration of the body relative to a non-inertial reference frame.

1.5. Energy. Job. Power.
Conservation laws

1.5.1. Energy– universal measure various forms movement and interaction of all types of matter.

1.5.2. Kinetic energy– function of the state of the system, determined only by the speed of its movement:

Kinetic energy of a body is scalar physical quantity, equal to half the product of mass m body per square of its speed.

1.5.3. Theorem on the change in kinetic energy. The work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body, or, in other words, the change in the kinetic energy of the body is equal to the work A of all forces acting on the body.

1.5.4. Relationship between kinetic energy and momentum:

1.5.5. Work of force– quantitative characteristic of the process of energy exchange between interacting bodies. Mechanical work .

1.5.6. Constant force work:

If a body moves in a straight line and is acted upon by a constant force F, which makes a certain angle α with the direction of movement (Fig. 1.28), then the work of this force is determined by the formula:

,

Where F– force module, ∆r– module of displacement of the point of application of force, – angle between the direction of force and displacement.

If< /2, то работа силы положительна. Если >/2, then the work done by the force is negative. When = /2 (the force is directed perpendicular to the displacement), then the work done by the force is zero.

Rice. 1.28	Rice. 1.29

Constant force work F when moving along the axis x to a distance (Fig. 1.29) is equal to the projection of force on this axis multiplied by the displacement:

.

In Fig. Figure 1.27 shows the case when A < 0, т.к. >/2 – obtuse angle.

1.5.7. Elementary work d A strength F on elementary displacement d r is a scalar physical quantity equal to the scalar product of force and displacement:

1.5.8. Variable force work on trajectory section 1 – 2 (Fig. 1.30):

Rice. 1.30

1.5.9. Instantaneous power equal to the work done per unit time:

.

1.5.10. Average power for a period of time:

1.5.11. Potential energy body at a given point is a scalar physical quantity, equal to the work done by a potential force when moving a body from this point to another, taken as the zero potential energy reference.

Potential energy is determined up to some arbitrary constant. This is not reflected in the physical laws, since they include either the difference in potential energies in two positions of the body or the derivative of potential energy with respect to coordinates.

Therefore, the potential energy at a certain position is considered equal to zero, and the energy of the body is measured relative to this position (zero reference level).

1.5.12. Principle of minimum potential energy. Any closed system tends to transition to a state in which its potential energy is minimal.

1.5.13. The work of conservative forces equal to the change in potential energy

.

1.5.14. Vector circulation theorem: if the circulation of any force vector is zero, then this force is conservative.

The work of conservative forces along a closed contour L is zero(Fig. 1.31):

Rice. 1.31

1.5.15. Potential energy of gravitational interaction between the masses m And M(Fig. 1.32):

1.5.16. Potential energy of a compressed spring(Fig. 1.33):

Rice. 1.32	Rice. 1.33

1.5.17. Total mechanical energy of the system equal to the sum of kinetic and potential energies:

E = E k + E p.

1.5.18. Body potential energy on top h above ground

E n = mgh.

1.5.19. Relationship between potential energy and force:

Or or

1.5.20. Law of conservation of mechanical energy(for a closed system): the total mechanical energy of a conservative system of material points remains constant:

1.5.21. Law of conservation of momentum for a closed system of bodies:

1.5.22. Law of conservation of mechanical energy and momentum with an absolutely elastic central impact (Fig. 1.34):

Where m 1 and m 2 – body masses; and – the speed of the bodies before the impact.

Rice. 1.34	Rice. 1.35

1.5.23. Speeds of bodies after an absolutely elastic impact (Fig. 1.35):

.

1.5.24. Speed ​​of bodies after a completely inelastic central impact (Fig. 1.36):

1.5.25. Law of conservation of momentum when the rocket is moving (Fig. 1.37):

where and are the mass and speed of the rocket; and the mass and speed of the emitted gases.

Rice. 1.36	Rice. 1.37

1.5.26. Meshchersky equation for a rocket.

Theory

If a body is thrown at an angle to the horizon, then in flight it is acted upon by the force of gravity and the force of air resistance. If the resistance force is neglected, then the only force left is gravity. Therefore, due to Newton's 2nd law, the body moves with acceleration equal to the acceleration of gravity; acceleration projections on the coordinate axes are equal a x = 0, and y= -g.

Any complex movement of a material point can be represented as a superposition of independent movements along the coordinate axes, and in the direction of different axes the type of movement may differ. In our case, the motion of a flying body can be represented as the superposition of two independent motions: uniform motion along the horizontal axis (X-axis) and uniformly accelerated motion along the vertical axis (Y-axis) (Fig. 1).

The body's velocity projections therefore change with time as follows:

,

where is the initial speed, α is the throwing angle.

The body coordinates therefore change like this:

With our choice of the origin of coordinates, the initial coordinates (Fig. 1) Then

The second time value at which the height is zero is zero, which corresponds to the moment of throwing, i.e. this value also has a physical meaning.

We obtain the flight range from the first formula (1). Flight range is the coordinate value X at the end of the flight, i.e. at a time equal to t 0. Substituting value (2) into the first formula (1), we get:

.

(3)

From this formula it can be seen that the greatest flight range is achieved at a throwing angle of 45 degrees.

Greatest height lifting of the thrown body can be obtained from the second formula (1). To do this, you need to substitute a time value equal to half the flight time (2) into this formula, because It is at the midpoint of the trajectory that the flight altitude is maximum. Carrying out calculations, we get

Laboratory work No. 5 in physics, grade 9 (answers) - Study of the motion of a body thrown horizontally

5. Measure the height of the fall and the flight distance of the ball in all five experiments. Enter the data into the table.

Experience	h	l	v
1	0.33 m	0.195 m
2	0.32 m	0.198 m
3	0.325 m	0.205 m
4	0.33 m	0.21 m
5	0.32 m	0.22 m
Wed.	0.325 m	0.206 m	0,8

7. Calculate the absolute and relative errors of direct measurement of the ball’s flight range. Write down the measurement results in interval form.

Answer security questions

1. Why is the trajectory of a body thrown horizontally half a parabola? Provide evidence.

The speed of a body thrown horizontally along the x axis does not change, but along the y axis it increases due to the action of force g on the body (gravitational acceleration).

2. What is the direction of the velocity vector at various points in the trajectory of a body thrown horizontally?

The vector of a body thrown horizontally is directed tangentially.

3. Is the motion of a body thrown horizontally uniformly accelerated? Why?

Is. The path of a ball thrown horizontally is curvilinear and uniformly accelerated, since this path is characterized by two independent directions: horizontal and the direction of free fall g, which exerts permanent action on the body.

Conclusions: learned to calculate the modulus of the initial velocity of a body thrown in a horizontal direction and under the influence of gravity.

Super task

Using the results of the work, determine the final speed of the ball (before its resistance with a sheet of paper). What angle does this speed make with the surface of the sheet?

10th grade

Laboratory work No. 1

Determination of free fall acceleration.

Equipment: ball on a string, tripod with coupling and ring, measuring tape, clock.

Work order

The model of a mathematical pendulum is a metal ball of small radius suspended on a long thread.

Pendulum length determined by the distance from the suspension point to the center of the ball (by formula 1)

Where - the length of the thread from the point of suspension to the place where the ball is attached to the thread; - ball diameter. Thread length measured with a ruler, ball diameter - caliper.

Leaving the thread taut, the ball is moved from the equilibrium position to a distance very small compared to the length of the thread. Then the ball is released without giving it a push, and at the same time the stopwatch is started. Determine the period of timet , during which the pendulum makesn = 50 complete oscillations. The experiment is repeated with two other pendulums. Obtained experimental results ( ) are entered into the table.

Measurement number

t , With

T, s

g, m/s

According to formula (2)

calculate the period of oscillation of the pendulum, and from the formula

(3) calculate the acceleration of a freely falling bodyg .

(3)

The measurement results are entered into the table.

Calculate the arithmetic mean from the measurement results and mean absolute error .The final result of measurements and calculations is expressed as .

10th grade

Laboratory work № 2

Study of the motion of a body thrown horizontally

Purpose of the work: measure the initial speed of a body thrown horizontally, to study the dependence of the flight range of a body thrown horizontally on the height from which it began to move.

Equipment: tripod with coupling and clamp, curved channel, metal ball, a sheet of paper, a sheet of carbon paper, a plumb line, a measuring tape.

Work order

The ball rolls down a curved chute, the lower part of which is horizontal. Distanceh from the bottom edge of the gutter to the table should be 40 cm. The clamping legs should be located near the upper end of the gutter. Place a sheet of paper under the gutter, weighing it down with a book so that it does not move during experiments. Mark a point on this sheet using a plumb lineA located on the same vertical with the lower end of the gutter. Release the ball without pushing. Note (roughly) the place on the table where the ball will land as it rolls off the chute and flies through the air. Place a sheet of paper on the marked place, and on it - a sheet of copy paper with the “working” side down. Press down these sheets with a book so that they do not move during experiments. Measure the distance from marked point to pointA . Lower the gutter so that the distance from the bottom edge of the gutter to the table is 10 cm, repeat the experiment.

After separation from the chute, the ball moves along a parabola, the apex of which is at the point of separation of the ball from the chute. Let's choose a coordinate system as shown in the figure. Initial height of the ball and flight range related by the relation According to this formula, when the initial altitude decreases by 4 times, the flight range decreases by 2 times. Having measured And you can find the speed of the ball at the moment of separation from the chute according to the formula