ТОП 10:

Simple measurement and processing

Job number 1


Simple measurement and processing

Obrabotka results of repeated direct measurements

AccessoriesPendulum, stopwatch.

The purpose of familiarization with the method of processing the results of repeated direct measurements such as measurement data period mathematical pendulum, to determine the most probable value of the measured value, the standard deviation, the confidence interval for a given reliability.

Brief details of the theory of measurement are divided into direct and indirect. Direct measurements are made using instruments that measure itself investigated value. So, you can find lots of bodies with the help of weights, length - measured with a ruler, and time - stopwatch. Measurement of body density by their weight and volume, speed of the train - based on the path traveled for some time, belong to the indirect measurements.

Every dimension of the imperfection of human senses and instruments sopryazhno with errors. In all dimensions allowed some bugs, so the measurement results do not give us the true, but only the approximate value of the measured value. Establishment of allowable measurement error range in which lies the true value of the measured quantity is a prerequisite of the experiment reliability. Measurement errors are divided into blunders, systematic and random.

Slips occur due to a malfunction of the device or inattentive observer, in violation of the experimental procedure and conditions for its implementation. In most cases, faults are clearly visible, as matching their samples are very different from other similar samples. The result of the measurement comprising the mistake should not be taken into account when processing the data - it should just be discarded.

Systematic errors are caused by faults or inaccuracies calibration of measuring instruments, when used for the calculation of inaccurate data, and also due to the imperfections of the measurement method. These errors affect the measurement results are always one-sided (only increasing or decreasing them). It is obvious that the influence of the systematic errors can not be reduced by increasing the number of dimensions. However, if the nature and character of systematic errors are known, their influence on the measurement result can be accounted for by introducing amendments and deleted.

Random errors are caused by fluctuations of the measured values, their appearance can not be prevented, so they can have some impact on the individual measurements, the result of changing in both directions, that is, and increase or decrease them. They are subject to statistical laws, so the impact of random errors in the measurement result can be accounted for, or greatly reduced. By applying the laws of probability theory, we determined the most probable values of the measured values and possible deviations from these values.

Order of work

As an example, consider the processing of the results of measurements of the oscillation period of a simple pendulum (mathematical pendulum is called the body suspended on a weightless and inextensible thread, which is much greater than its size).

1. Use the stopwatch to measure the time tі 30 oscillations of a pendulum. The measurements were repeated n times (n is given by the teacher).

2. According to the formula


calculate the period of oscillation.

3. Definitions sredngo arithmetic mean value of the oscillation period

4. According to the formula (1.1.9) to find the standard deviation of T.

5. From the table to find the value of the Student's coefficient corresponding to reliability, said the teacher (Student coefficient table is available in the laboratory).

6. When the values found and the formula (1.1.10) to calculate the absolute error.

7. Calculate the relative error by the formula (1.1.12).

8. The results of measurement and computation recorded in the table

Table (1.1.2)

N п ti с Тi с Тi с



Control questions

1. What kinds of measurements are divided and associated errors.

2. How is the absolute error.

3. What is the Student factor is introduced.

4. What do the confidence interval and confidence level.

5. Why is it necessary to calculate the relative error.

6. What is called a mathematical pendulum.

7. What is called the period of oscillation.



1. Kassandrova ON, VV Lebedev Processing of observations. M .: Nauka, 1970.

2. Agekyan TA Fundamentals of the theory of errors for astronomers and physicists. M .: Nauka, 1972.

Order of work


It is necessary to calculate the density of regular geometric body shapes (cuboid, cylinder, sphere).

1. To determine body mass (box, cylinder, sphere). Record it with the precision of measurements.

2.Find body dimensions using calipers (the same body). Determine error.

3.Opredelit body density measured using the following relations:

a) a rectangular parallelepiped


where m-mass, h-height, b-width, the length of the box

b) for the cylinder


where m-mass, R-radius of the base, h-height of the cylinder;

c) for the ball


where mass m-, R- radius of the sphere.

4. According to the formula (1.2.11) to calculate the relative error, ie. E .:

a) a rectangular parallelepiped



b) the length of the cylinder


c) for the ball


5. According to the formula (1.2.13) to calculate the absolute error. The results are rounded based on measurement errors.

6. The final result is written in the form (1.2.12)

7. The results of measurements and calculations recorded in table 1.2.1



Table 1.2.1


value rev. body m, кг Dm кг h м b м l м R м Dh=Db= Dl=DR м r ± Dr кг/м3 e

Control questions

1. As determined by the instrument error? What is the accuracy of the readings? Specify price vernier division.

2. How are the absolute errors of the physical constants with non-periodic infinite fractions?

3. How is the accuracy of the indirect measurement?

4. Get a formula assessing the relative errors for functions of the form:

; ;

where a, b, g - some constants.


1. A. Seidel Elementary estimates of measurement errors. L .: Science, 1974.

2.Kassandrova ON, VV Lebedev Processing of observations. M .: Nauka, 1970.

3.Agekyan TA Fundamentals of the theory of errors for astronomers and physicists. M .: Nauka, 1972.

Job number №2

Order of work

1. Move the flexible arm to the specified height h teacher. Set it so that loads falling freely pass through the middle of the working window of photoelectric sensors.

2. Set the movable weight on the rods on said teacher division and the distance R from the axis of rotation to the center of the load.

3. Measure the diameter of the disc and a caliper to determine the radii Rb RM small and larger drives.

4. Insert the power cord into the power meter network. Press "Start".

5. Press the "Network", checking whether all indicators show zero meter and all LEDs are lit if both photoelectric sensors.

6. Attach to the thread wound on the disk radius RM, the initial load mass m.

7. Move the load to its highest position by winding the thread on the selected disk, and set the bottom edge of the load precisely with a dash on the housing upper photoelectric sensor and squeeze the buttons "Start". Check whether the lock has occurred.

8. Press the "Reset". Press "Start".

9. Read off the measured value of goods falling time on the way h.

10. Press the "Reset".

11. Measuring repeated 3-5 times and the average value to determine the time of movement of goods by the formula:

, (2.22)

where n is the number of executed measurements, ti-time measured value at i-volume measurement, t-time average value of loads along the path h.

12. Calculate the linear acceleration, and in formula (2.15) and the angular acceleration ε formula (2.16).

13. Calculate the torque from the formula (2.20).

14. Repeat steps. 6-12 2-3 cargoes of other masses.

11. According to the calculated values ​​and plotted on the ordinate , whose value lay at the x-axis - torque M. In this case you need to make sure that the resulting linear relationship.



12. From the graph, determine the moment of inertia of the system and the moment of friction force Mtr to the axis of rotation, finding ways which are shown in Fig. 2.5

13. Repeat steps 6-16 for selecting Rb radius pulley scale for values ​​and can be plotted on the small and large pulleys in the same coordinate system and , make sure that they are collinear.

14. Determine the moment of inertia of the system calculated by the equation (2.21). Pendulum inertia torque without cylindrical weights indicated on the unit.

15. Check the convergence of the experimental and theoretical values ​​of the moments of inertia obtained by the ratio of the system:


15. All data and calculation results recorded in Table 2.1


Table 2.1



Control questions

1. What is the rotational motion of a solid body?

2. Identify and write the formula of angular velocity and angular acceleration. Specify their areas and units.

3. What is the relationship between the linear and angular velocity? As the linear velocity is directed?

4. What is called the moment of inertia of the material point and the body? What characterizes it?

5. Formulate the theorem of Steiner. Write a formula.

6. What is the torque? What force creates a torque? As directed vector torque?

7. Write and explain the fundamental law of dynamics of rotational motion?

8. Describe applied in this study unit. Which part of the system moves forward, what is rotated?

9. Is the line graph?



1. Savelyev IV Zhalpy physics courses. A .: Mektep, 1977, vol.1.

2. Frisch SE, AV Timoreva Zhalpy physics courses. Volume 1, A .: Mektep 1971.

3. DV sivukhin The general course of physics. Vol.1, M .: Nauka, 1979.


Job number 4

Fitting Description

Pendulum FPM Maxwell - 03.


General view of the FPM Maxwell pendulum - 03 is shown in Figure 4.1.

The base (1) is equipped with adjustable feet (2), which allows to align the instrument. At the base of fixed column (3), to which is attached a fixed upper bracket (4) and a movable lower bracket (5). The upper arm is an electromagnet (6), a photoelectric sensor №1 (7) and the knob (8) for fastening and adjusting the length of the bifilar suspension pendulum.

Bottom bracket with attached thereto a photoelectric sensor №2 (9) can be moved along the column and fixed in an arbitrary position of favorites.

A pendulum (10) of the device - a roller mounted on an axle and bifilar manner superimposed on the various rings (11), thereby making different, the moment of inertia.

Pendulums overlaid ring is held in an upper position by an electromagnet. The length of the pendulum is determined on the millimeter scale on the device column.

unok 4.1

Theoretical introduction

With the release of the pendulum, he starts to move steadily downward and rotating around its axis of symmetry.

Rotation, continuing to lower inertia of the movement (when the thread already unwound), again leads to the filament winding on the rod, and hence to the rise of the pendulum. The movement of the pendulum then slows down, the pendulum stops and re-starts its downward movement, etc.


The equation of motion of the pendulum without considering friction forces are of the form:

M=mg-2T (1)

Ie=2Tr (2)

a=er (3)

where: m - mass of the pendulum; I is the moment of inertia of the pendulum; g - acceleration due to gravity; r is the radius of the rod; T - the thread tension (a); a - acceleration of translational motion of the center of mass of the pendulum; ; e - - angular acceleration.

The acceleration may be obtained from the measured travel time t and the distance traveled by the pendulum, h of the equation assuming that the uniformly accelerated motion forward:


Equations (1) - (3) gives:

2Т=m(g-a) (5)


Substituting (6) (5), we obtain:


or taking into account (4) finally will have:


where: I - moment of inertia of the pendulum in kg.m2

D=D0 + Dn Dn = 0.5 mm = 0,00005m

D - outside diameter of the pendulum axis in position with a thread wound thereon suspension in m; D - is measured with calipers; t - time to fall in; g-gravitational acceleration in m / s2; h - the length of the pendulum, which is equal to the height to which it rises in m; m - mass of the pendulum in place with a ring in kg, is calculated as follows:

m=m 1+m 2+m 3 (9)

where: m 1 - mass of the pendulum axle in kg; m 2- roller mass in kg;

m 3 - weight imposed on the ring roller in kg.


Order of work

The device is ready for operation immediately after switching on the voltage and the light does not need heating.

You need:

1. Measure the height h of the pendulum.

2. Squeeze the button "Start" GRM-15 millisekundomera.

3. Wind up on the pendulum axle suspension thread, paying attention to the fact that it wound evenly, one coil to another.

4. Fix the pendulum by an electromagnet, paying attention to the fact that the thread in this position was not too flat.

5. tuck the pendulum in the position of its movement through an angle of 5 and release.

6. Read the measured value of the fall of the pendulum.

7. Press the "Reset".

8. Press the "Start".

9. Repeat steps 3-6.

10. Experiments repeat 5-7 times.

11. Determine the average time of fall of the pendulum

12. Using the expressions (8) and (9), the moment of inertia of the pendulum to count.

13. Calculate the theoretical value of the moment of inertia of the pendulum

I=+I0+Ip+Id (11)

where: ; and D-outer diameter of the pendulum axis;

where: Dp - the outer diameter of the roller,

where: Dd - the external diameter of the ring.

To measure Dp and Dd is necessary to remove the ring from the roller. Weight, mp md set.

14. Calculate the relative deviation of the experimentally determined value of the moment of inertia of the theoretical


where: I-moment of inertia obtained from the experiment using the formulas (8) and (9). IT- moment of inertia calculated from the formula (11)

15. Rate error in determining the moment of inertia based on the experimental data, appearing in the expression (8). All measurements must be made with great caution, as the pendulum can be easily damaged. It is necessary to protect the pendulum from the blows.


Control questions

1. In what part of Maxwell pendulum movements?

2. How the experimentally determined moment of inertia of the pendulum?

3. How to calculate theoretically the moment of inertia of the pendulum?

4. What are the moments of inertia units SI?

5. What is the angular acceleration?

6. In what units is measured by the angular acceleration?

7. In what units is measured by linear acceleration?

8. What is the relationship between the linear and angular accelerations?

9. As recorded 2nd Newton's law for rotary motion?

10. Record the forces acting on the pendulum Maxwell.



1. Savelyev IV The general course of physics. A .: Science, 1977, volume 1.

2. Frisch SE, AV Timoreva General Physics Course Volume 1, A .: Mektep 1971.

3. DV sivukhin The general course of physics. Vol.1, M .: Nauka, 1979.


Job number 5

Order of work


Exercise 1: Determination of the gravitational acceleration using a working pendulum

1. Get acquainted with the design of a universal pendulum MTF-04.

2. Strengthen the goods and in the extreme division of the scale (2 cm.) From the prism G (see. Fig. 3.2.) And hang the pendulum of this prism.

3. Turn on the power meter cord to the mains supply.

4. Press the switch "Network", checking whether all indicators show the meter number zero, and if the photoelectric sensor light is on.

5. Reject the pendulum from the equilibrium position at an angle of no more than 5 degrees (the upper end of the rod must be free to counter cross beam sensor) and press "Reset" after the 3-5 full swing.

6. Press "Stop" After counting meter periods of 19 complete oscillations (instrument reading stops at 20).

7. Determine the working period of the pendulum by the formula:


where t-time measured millisekundomerom 20 complete oscillations.

8. To remove the pendulum and set it on the prism О/.

9. On this prism to repeat the experience of claims. 5-7.

10. Move the goods A 1 cm (one division on the web), and repeat the experiment Nos. 5-9.

11. Repeat the 5-10 experience for the division of the rod (6-7 cm).

12. plotted oscillation period T by dividing the rod d (cm), corresponding to the position of cargo A. Find the intersection point of the curves, which corresponds to the position of the load A with the closest match at the swing periods with respect to both poles. The measurement results are recorded in Table 5.1.


Table 5.1

n d, cм Swing on the O Swing on the O /
t, c T, c t, c T, c

8. Secure the load And found via the chart position. Find and distance. To do this, put the pendulum horizontally on a special supporting prism and achieve its equilibrium. Mark the balance point and measure the desired distance and (see para. 5.3.).

9. suspending the pendulum in one and the other prism (without changing the load position), to determine the appropriate periods of vibration Т1 and Т2 . Each period to determine at least 3 times and 50 complete oscillations.

10. Calculate the formula (3.19) is the acceleration of gravity. In this formula, Т1 and Т2 have mean values ​​of three measurements, respectively.

11. Calculate the ratio (3.21.) Measurement error. To determine the need to use the standard error of a certain reliability , ie


where -the coefficient of the Student; the mean square error is equal to (3.24).



12. The results of measurements and calculations recorded in Table 3.2.


Table 3.2.


n d, cм Swing on the O Swing on the O / м/с2
t1,c T1,c DT1,c t2 ,c T2, c

Exercise 2: Defining the free fall with the help of a mathematical pendulum


1. Turn the upper bracket for 1800.

2. Install the lower bracket with a photoelectric sensor at a distance (the length of the distance between the prisms О and О/ defined in Exercise 1).

3. Turn the screw on the top bracket to establish the mathematical pendulum length, with hell on the ball must be at the level of the body features of the photoelectric sensor.

4. Enter a mathematical pendulum in motion, the ball deflecting no more than 50 from the equilibrium position.

5. Press the "Reset" button.

6. After calculating the 19 meter fluctuations press "Stop" (Reading the meter stops at 20).

7. According to the formula (3.22) to determine the period of oscillation of the mathematical pendulum.

8. According to the formula (3.16) to calculate the acceleration of gravity.

9. Ensure that the length of the periods of physical and mathematical pendulums coincide.

Control questions

1. What body is called a physical pendulum?

2. Write down the equation of motion of a physical pendulum.

3. What is the period of oscillation of a physical pendulum?

4. What is the value of the reduced length is called a physical pendulum? What is the physical meaning of the reduced length?

5. Formulate the theorem of Steiner.

6. Under what simplifying assumptions derived a formula of a physical pendulum period (3.9)?

7. What is the essence of the method of working of the pendulum?

8. Tell the design working pendulum used in this paper.


1. Savelyev IV The general course of physics v.1. A .: Science, 1977.

2. sivukhin IV The general course of physics. v.1. M .: Nauka, 1979.

3. Gunners SP The general course of physics. Mechanics. M .: 1975.


Job number 1


simple measurement and processing

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