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INTRODUCTION
Basic goal of the laboratory work manual on Physics is to enable students to learn important physical phenomena by experience. Laboratory work description does not try to give students a complete picture of the studied phenomena. Such presentation can only be achieved as a result of study of lectures and textbooks.
Large attention in the laboratory work manual on Physics for the students of technical professions is devoted to the handling of the measured results. Prior independent preparation, above all theoretical, is needed for successful completion of the work.
Every laboratory work is supposed to take two academic hours. Before the class a student must prepare a protocol of laboratory work and learn appropriate theoretical material.
During the class students do the necessary measurements, execute calculations and take the report to the conclusion. Measured results are discussed with a teacher and confirmed.
Fully designed report on laboratory work should be given to the teacher before the end of the class. It must include: title sheet, laboratory work number and name, list of devices and installations, purpose of work, drawing of the setting, calculation formulae, table of the measurement results and calculations, conclusions, as the result of the work. Graphs must be done on a millimeter`s paper.
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If a student does not have time to support the laboratory work before the end of the class, he/she is allowed to design a report (graph) with the use of the computer programs (Excel, Origin) for the next class.
Laboratory work is considered done after the successful speech in support in front of a teacher (report explanation + mark for theoretical material).
Support of report: purpose of work + experimental method + conclusions.
Theoretical material: knowledge of the physical phenomena, which was studied in this laboratory work (laws, formulas).
41
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(.1.1) 1, 2 4 3. 5, 6.
(.1.2) . xo x F, F = -kx, k .
, 
. (1.1)
г (1.1)
. (1.2)
,
. (1.3)
г (1.3) .
,
, (1.4)
- t.
.
,
(1.5)
2π .
(1.6)
, , 2π.
(1.7)
1 .
(1.5) (1.6)
. (1.8)
, , . F = mg = kxo, 
. (1.9)
(1.8)
(1.10), 
. (1.11)
, , , 2 = f(xo) (. 1.11) . .
(1.8), 
.
1. ( ).
2. ϳ , , m ( ) ( ). 11 , .
|
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3. m , , (2÷3 ), 20 . 1.1. ʳ 9÷10.
1.1
| m | Xo | t | T | 2 2 | 
0,5
| 
/
| k / | |
| - | ||||||||
| - |
4. k (1.9) .
5. : 1) 2 = f(xo); 2) 
(.1.3) 1-2 3-4, , , (1.12) (1.13) g k.
 |
(1.12) 
(1.13)
6. , g , k, (1.13), , (1.9). 볭 , ?
1. ?
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4. .
5. .
6. , .
7. .
˳
1. .. Գ.- .: , 2003.- .77-80.
2. .. . - .1, .: ,1982.- .195-196.
3. .. .- : , 1990.- .222-223.
..
Laboratory work 41
SPRING PENDULUM
Purpose of the work is to study dependence of period of oscillations on mass for a spring pendulum.
Task: to define acceleration offree falling; experimentally to check up the theoretical formula of period of the spring pendulum.
Theory
Oscillations of a spring pendulum are acted upon by the elastic force
, (2.1)
where x is a deviation from state of stable equilibrium, k is a rigidity of the spring. The period of oscillations is
, (2.2)
where m is mass of the body fixed on the spring.
Spring pendulum (fig.2.1) it a body is suspended on a spring. At the leadingout of it from position of equilibrium of xo on distance of x there is force of elasticity F, which by Hook`s law is evened F = -kx, where k is rigidity of spring. This force gives the acceleration
, or . (2.3)
Equation (2.1) can be write down in such a way
. (2.4)
Designating , get . (2.5)
Equation (2.3) is named differential equation of undamped free harmonic oscillations. The decision of this equation is a harmonic function
, or (2.6)
what sets the coordinate of x load in any moment of time t.
Will consider descriptions of harmonic oscillations. Amplitude of Ao is most deviation of point from position of equilibrium
Cyclic frequency of oscillations (2.7)
it is an amount of oscillations for 2π seconds.
Period of oscillations (2.8)
it is time of one full-oscillate, or time for which the phase of oscillation changes on 2π.
Frequency of oscillations (2.9)
it is an amount of oscillations for 1 second.
From (2.5) and (2.6) get the period of oscillation of the spring pendulum
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. (2.10)
When load hangs on a spring, which does not carry out oscillations, it deforms a spring on the size of . By Hook`s law of F = mg = kxo, it is possible to find rigidity of spring
. (2.11)
Then a formula (2.8) for the period of oscillations assumes an air
(2.12), or . (2.13)
Thus, experimentally measuring the period of oscillations of T and lengthening of spring of , from its undeformed state to position of equilibrium at the different loadings, it is possible on the angle of slope of graph of T2 = f(xo) (see a formula 2.11) to find the acceleration of the free falling. The first task is so executed.
The second task of work is executed on the basis of formula (2.8), that the linearness of dependence is checked up .
Order of execution
Figure 2.1
1. Varying the weights on the spring measure time of 50 oscillations by the stop-watch (once for each weight). Then calculate a period of oscillations for each measurement.
2. Represent graphically the dependence of period 
on 
.
3. Using the graph determine a rigidity of the spring.
To take off from a spring fastening of loads and to mark position of end of spring on a line (points of hang up of fastening).
1. To hang up fastening and, combining the set of loads, to change general mass of m of a spring pendulum from the least (one fastening to most, all loads is suspended). Mass of fastening is 11 g, the masses of loads see on them.
2. With the chosen mass of m to measure deformation of spring to position of equilibrium, and then, giving small amplitude (2 ÷ 3 sm), to measure a stop-watch time 20 oscillations. To add the results of measurings to the table 2.1. An amount of combinations of the masses must be not less 9 ÷ 10.
Table 2.1
| n | m, kg | xo, sm | t, s | T, s | T2, s2 | ,
kg0.5
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| ka |
| - | ||||||||
| - |
On results of every experience to expect rigidity of spring k on formula (2.9) and to find it mean value.
3. To build graphics: 1) 2 = f(xo); 2)
4. On lineal parts of graph (2.3) to choose for two points 1 ÷ 2 and 3 ÷ 4, to define their coordinates for to the axes, but not from a table, and after formulas (2.12) and (2.13) to expect g and k.
Figure 2.2
(2.14) (2.15)
5. To draw conclusion, comparing the value of g to tabular, and k, got after a formula (2.3), with a mean value, found after a formula (2.9). In a conclusion also to mark or are lines graphic got in experiments and if so, then what does it testify to?
Questions
1. What kind of oscillations is called harmonic? Write the dependence of deviation upon time for harmonic oscillations.
2. Give definitions for period, amplitude, frequency, cyclic frequency and phase.
3. Write the differential equation of harmonic oscillations.
4. Write the formula for period of a spring pendulum. What is a rigidity of a spring?
Translator: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.
Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences.
Approved by the chair of physics. Protocol 6 from 30.03.2009.
42
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(.3.1) 1, 3, 4, 5. 2 , 3 7. L 5, 6. 7.
m, , . . Ⳮ , mg, Lsinα (.3.2). ϳ 䳺 m Ⳮ .
. (3.1)
: 
- , 
. , mg α.
. (3.2)
α ( 5) , sin α = α. 
. (3.3)
, (3.4)
- , . L = L0 + ΔL,
. (3.5)
ϳ
. (3.6)
, 2 ΔL , g.
(3.4)
. (3.7)
1. ³ 5 50 .
2. , , 5.
3. , , 20 . 3.1. ΔL = 0.
3.1
| t, . | = t/20, . | 2, 2 | ΔL, |
4. 2 7. ³ 6 ΔL = 10 . . .
5. . 2 - 5 ( ). , .
6. = t/20 . 3.1.
7. 
2 ΔL(.3.3). ˳ (3.5) ~ 
.
8. 1 2,
9. 
. :
. (3.8)
10. 9,8 /2. .
1. ?
2. .
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4. , .
5. .
˳
1. .. Գ.- .: , 2003.- .77-80.
2. .. . - .1, .: ,1982.- .195-196.
3. .. .- : , 1990.- .222-223.
. ..
Laboratory work 42
MATHEMATICAL PENDULUM
Purpose of work: to measure the free fall acceleration.
INSTRUMENTATION AND APPLIANCES: mathematical pendulum; the stop-watch, straightedge.
Short theory
The point mass suspended by means of an unelastic weightless thread is called the mathematical pendulum. The restoring force is the projection of the force of gravity P=mg on the direction of motion of the point mass. In this case
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where a = x/l, the angle between positions of l in free state and deviation state.
Figure 4.1
If the angles a are so small that sina a, then
.
Since this force is always directed to the equilibrium position and that is why it has a sign opposite to that of x:
In this case the oscillations are harmonic. The second Newtons law is
.
Finally we obtain the differential equation of harmonic oscillations:
.
If g/l=w2, then
.
Solution of this equation is
The cyclic frequency is
,
and period is equal
,
i. e., the frequency and the period of oscillations do not depend on the mass of the oscillating body; they are determined only by the length of the thread and the free fall acceleration. The measurement of the period of a mathematical pendulum is used for determine g:
.
Experimental part
1. Fix the end of the cord in the first position (l = 50 cm). Make the pendulum vibrate (the amplitude of vibration must not be more than the ball diameter). Determine time of 50 oscillations by the stop-watch three times and calculate the period T using the average value of the time.
2. Determine the free fall acceleration g.
3. Calculate an error.
4. Investigate relation between the period T and the length l. Put the end of the cord in the second position (l = 150 cm). Calculate period of oscillations 
in the same way and compare with T.
5. Increase the amplitude of oscillations twice. Test independence of the period on the amplitude of oscillations (small amplitude and big amplitude).
6. Make conclusions.
Control questions
1. What is a mathematical pendulum?
2. What kind of oscillations is called harmonic?
3. Write the differential equation of harmonic oscillations.
4. Write the dependence of deviation upon time for harmonic oscillations. Show this dependence in figure.
5. Give definitions for a period, an amplitude, a frequency, a cyclic frequency and a phase of vibration.
Translator: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.
Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences.
Approved by the chair of physics. Protocol 6 from 30.03.2009.
43.1
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ϳ , . ϳ ϔ .
Գ , , . ϳ 䳺 mg, Lsinα, (.5.2). L ( ) - .
, (5.1)
I - , - . , mg α. ,
. (5.2)
α ( 5) , sin α = α. 
(5.3)
, (5.4)
.
(5.5)
(5.4)
(5.6)
. 8 7 (.5.1). , L, .
( ) I = Z2 . , ອ I = 1/12 mb2 + m(b/2 a)2 (.5.3). , Z
I = Z2 + 1/12 mb2 + m(b/2 a)2 (5.7)
, L Z. (.5.3)
mgZ1 = MgZ2 (5.8)
.5.3 , L = Z Z2, Z1 + b/2 =L + a. (5.9)
(5.8) (5.9)
(5.10)
ϳ (5.7) (5.10) (5.5) (5.5)
, (5.11)
, 
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. (5.12)
, Y = f(X) . Z . Y = f(X). , (5.5) (5.7), g. 9,8 /2 .
1. 220 ܔ.
2. 6, 9 8 10 . ³ ( ) 1 , (2 ). .
3. ϳ 6. , .
4. ³ 4 , 7 5, .
5. , , 5.
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7. 9, ϔ. ϳ 10- .
8. , , 10. ³ Z 5.1.
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| - | |||
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9. . 2 ÷ 8, 2 ( ) .
10. (5.12) 5.1.
11. 
: m/M = 0,3; = 5 ; b = 59 . .
Y=f(X).
12. 1 2, (.5.4) , , Ⳮ
, /2. (5.13)
13. 980 /2. .
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˳
1. .. Գ.- .: , 2003.- .77-80.
2. .. . - .1, .: ,1982.- .196-199.
3. .. .- : , 1990.- .222-223.
. ..
6 LABORATORY Work 43.1
PHYSICAL PENDULUM
Purpose of work: check the dependence of physical pendulum free oscillations period from its moment of inertia; determine the value of the free fall acceleration.
DEVICES AND EQUIPMENT: physical pendulum, stopwatch, straightedge.
The experimental setting(fig.6.1 ) consists of the base 1, alignment of which is carried out by legs 2. The rack is fastened in the base 3, on which the bottom supporting arm 4 is fixed with photoelectric sensor 5. On the top supporting arm 6 is suspended the physical pendulum, which includes the rod 7, weight 8 and detent 9. On the rod 7 in 10 mm collar marks are made for the exact determination of the length and exact weight 8 fixation.
On the front gage panel there are: 10 oscillation indicator, 14 indicator of time, switches: -11 , 12 , 13 .
Figure 6.1
During the pendulum motion the flow of light from the lamp of the photoelectric sensor overlap and the electronic computation circuit of oscillations number and stopwatch actuates. After pressing the switch STOP the stopwatch stanching after the end of full current oscillations.
Theoretical part
Physical pendulum is the solid, which can rotate relatively to the arbitrary of the horizontal axis that doesnt pass through the center of mass. Moment of the gravity force mg, the arm of which is equal Lsin α. Value of L is the distance from pivot (suspension center) to point the center of the mass of the body. Under the action of this moment the body turns round the suspension center .
Figure 6.2
Write down the fundamental equation of the rotational motion dynamics
, (6.1)
where I is the moment of inertia of the body, 
is angular acceleration, minus accounts that the moment of force of mg reduces the angle α. Thus, we get the differential equation of physical pendulum free oscillations
. (6.2)
If angle α is small (less than 5) we can consider that sin α = α. We get
. (6.3) Comparing the received equation with the general equation of free harmonic oscillations
, (6.4) where 
- is cyclic frequency of oscillations, period. Lets get the equation of the period of oscillations
. (6.5) Equation solution (6.4) is the harmonic function which is the equation of free harmonic oscillations
. (6.6) For performing the first task point we need to change the moment of inertia of the pendulum. It is carried out by moving the weight 8 along the rod 7. But in this way the mass center position changes, it is the distance L.
Figure 6.3
The moment of inertia of the pendulum relatively to the point is equal to the amount of moment of inertia of the load and rod.
Taking into account Steiner theorem, we obtain
.
Thus, the moment of inertia of the pendulum as function of distance Z from point of suspension to weight center
. (6.7) Lets find the position of point the mass center of pendulum which is the distance L as function Z. By the law of moments relatively to the point C we have:
. (6.8)
From figure 3 we can see that
, 
. (6.9) From equations (6.8) (6.9) we get
. (6.10) Substitution of (6.7) and (6.10) into (6.5) after squaring (6.5) gives us
, (6.11)
where
,
,
. (6.12)
Thus, dependence Y = f(X) according to the theory, must be linear. Experimentally researched is the dependence of oscillations period of physical pendulum and distance Z from weight to suspension center. We plot the graph (6.12)
Y=f(X).
If you get a rectilinear graph, it confirms validity of theoretical formulas (6.5) and (6.7), and on its slope we can find the free fall acceleration g. Its coincidence with tabulated value 9,8 m/s2 confirms truth of the theoretical ratio.
Work procedure
1. Connect a device to the mains 220 V and push the button .
2. Remove the pendulum from the support arm 6, release the detent 9 and station the weight 8 center on the distance 10 m from the suspension point. Measure the distance with the number of known collar marks on the rod from supportive prisms (suspension center) plus 1 m to the weight which includes the thickness of the load (2 sm). Seek a clear fixing of a load in marks.
3. Hang up the pendulum on the supportive arm 6. Watch after its mounting reliability.
4. Regulate the supportive arm 4 position so that the bottom rod end of physical pendulum 7 wont catch on photoelectric sensor 5, but block its ray.
5. Activate physical pendulum in oscillations, having rejected it on a corner less than 5.
6. After the pendulum make 1 ÷ 2 oscillations, push the button . Counting of time and oscillations number will start which will be visible on corresponding indicators.
7. When oscillations indicator shows up number 9, press the button . The last 10-th oscillation will finish and the stopwatch will stop.
8. Define the period of oscillations, dividing time for oscillations number that is 9. Distance Z and period write down to the table.
9. Repeat points 2÷8, removing the weight down in 2 cm to the possibly lowest weight position.
10. Under formula (6.12) calculate and write it down to the table. Pendulum parameters are: mass ratio 
0,3; Length of the bottom rod end to suspension point = 5 cm; total rod length b = 59 cm. Calculation it is more convenient to do on the computer.
11. Plot the dependence Y = f(X) and choose two points 1 and 2 on its rectilinear part and define its coordinate by axes, not from the table, on the graph inclination find the free fall acceleration by the formula
, cm/s2. (6.13) Compare the received value with reference 980 cm/s2. Write the conclusion.
Table 6.1
| Z, cm | Y=Z2, cm2 | , s | , cm×s2 |
Figure 6.4
Control questions
6. What is the physical pendulum?
7. Deduce and write down the differential equation of physical pendulums free harmonic oscillations.
8. Write down the oscillations equation which is the solution of physical pendulum differential equation. Plot the graph of this equation.
9. What is the name of values which are a part of physical pendulum oscillations equation? What units they have?
10. Write down formulas for period and physical pendulums cyclic oscillation frequency.
Literature
1. .. Գ.- .: , 2003.- .77-80.
2. .. . - . 1, .: ,1982.- .196-199.
3. .. .- : , 1990.- .222-223.
Translator: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.
Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences.
Approved by the chair of physics. Protocol 6 from 30.03.2009.
43.2
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LABORATORY WORK 43.2
OSCILLATION OF PEG
Purpose of work is to probe dependence of period of oscillations of peg on distance between the axis of rotation and center of peg.
Devices: peg, straightedge, stop-watch.
Task of work: 1) to build the theoretical and experimental graph of dependence of period of oscillations T of peg from dimensionless length
