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Description of installation and measuring method

 

The cross-shaped pendulum is Oberbeck's pendulum. It consist of two sheaves with radii are r1 and r2, and four crosswise shafts. Four loads with equal masses m are able to move along shafts. A thread is wound on one of the sheaves. A load P = mg is fasten to the other end of the thread. When the load P falls, the thread unwinds. A mechanical brake is possible to operate the motion of the load from the start to the end of its motion.

The displacement of the load can be measured by means of on indicator panel. It is not difficult to show that the acceleration of the load is

,

where h is the distance of the falling load; t is the time of the falling load.

The load is allowed to fall under the action of its own weight. It equals mg. The line acceleration of the point at the sheaf surface is equal to the load acceleration. Then angular acceleration can be represented as

, (15.1)

where r is radius of the sheaf. According to the second Newton's law

,

where N is pull of the thread. Then, pull of the thread is given by

.

Moment of force at the cross-shaped pendulum may be expressed as

.

In our experiment conditions usually a << g . Then

. (15.2)

The fundamental equation of motion for a rotating body is as follows

, (15.3)

where I is the moment of inertia of the pendulum; ε is the angular acceleration of the pendulum. From the formulas (15.2) and (15.3) it follows that

. (15.4)

Taking (15.1) into consideration of formula (15.4) one may receive

. (15.5)

If I = const, then

.

If M = const, then

.

Moment of inertia of the cross-shaped pendulum consists of the moment of inertia of the cross-shaped Il (as is shown on the device) and moment of inertia of four loads at the shafts

(15.6)

where m is the mean mass; R is the distance between the axis of revolution and the centre of gravity loads m.

Task 5.1

 

THE AIM is to determine the angular accelerations and moments of forces of the pendulum when values of falling loads are different and when moment of inertia is constant.

1. Put down the data of measurements in table 15.1.

 

 

Table 15.1

t, s m, kg r, m h, m ε, s-2 M, N˙m Note
            The measurement is done with the system of fife loads
   
   
   
   
        The load is decreased by one part.
        The load is decreased by two parts.
        The load is decreased by three parts.
        The load is decreased by four parts.

 

2. Calculate Δε/ε. Put down the results of calculations in table 15.2.

 

Table 15.2

N ti, s Δti,s Δti2, s2 Half-width of the trust interval Δε/ε
Δt, s Δh, m Δr, m Δε, s-2
               
     
     
     
     

 

Σti = ΣΔti2=

 

3. Put M versus ε on the graph.

4. Determine the moment of the friction forces and moment of inertia on the graph.

5. Make analysis of the experiment results.

Task 5.2

 

THE AIM is to determine the angular acceleration and moments of forces of the pendulum with different values of the sheaves radius and when moment of inertia is constant.

 

1. Put down the data of measurements in table 15.3.

 

Table 15.3

t1, s r, m m, kg h, m ε, s-2 M, N˙m Note
            The measurements are made when a thread is wound on big sheaf
   
   
   
   
        The measurements are made on other four sheaves
       
       
       

 

2. For the first five measurements use average time t, when you calculate ε.

3. Calculate for one measurement. Calculate the half-width of the confidence interval for r to use the main errors of measurement, m0 and g are table values.

4. Calculate the half-width of the confidence interval for t using the first five measurements.

5. Put down the data of measurements in the table 15.4.

 

Table 15.4

ti s Δti, s Δti2, s2 Half-width of the trust interval ΔM/M
Δt, s Δr, m Δm, kg Δg, m/s2 ΔM, N˙m
                   
                   
                   
                   
                   
  ΣΔti2=
                           

 

6. Put M versus ε on the graph.

7. Determine moment of inertia on the graph.

8. Make analysis of the experiment results.

 

Task 5.3

 

THE AIM is to determine moment of inertia of four loads at the shafts with dynamics and theoretical methods.

1. Put the data of measurements in the table 15.5.

 

Table 15.5

t, s , s m0, kg r, m h, m R, m , g I1, kg˙m2 I0, kg˙m2 , g˙m2
                   
                   
                   
                   
                   

 

2. Determine the moment of inertia of four loads at the shafts with the dynamics method.

,

where I0 is the moment of inertia of cross-shaped pendulum (I0 = 4,15×10-2 g×m2); k is a coefficient, it takes into consideration the mean forces of friction and errors of the time measurements (k = 1.1).

3. Calculate the moment of inertia of four loads at the shafts by the formula

,

where is the mean mass of four loads; R is distance between the axis of rotation and the centre of load's gravity.

4. Calculate ΔI/I as an error of indirect measurement. Calculate the half- width of the confidence interval for r and h as a direct measurement errors, m and g are the table values. Calculate the half-width of the confidence interval of t as five direct measurements error.

5. Put the results of calculation in table 15.6.

 

Table 15.6

N ti, s Δti, s Δti2, s2   ΔI/I
Δt, s Δl, m Δr, m Δm, kg Δg, m/s2 ΔI, kg˙m2
                   
       
       
       
       
  Σti = = ΣΔti2=

 

6. Analyse the experimental results and draw the conclusion.

m0 = 0,5 kg; m(I) = 0,592 kg; m(II) = 0,59 kg; m(III) = 0,575 kg;

m(IV) = 0,592 kg; I0 = 4,5˙10-2 kg˙m2

 

Task 5.4

THE AIM is to determine the angular accelerations and moments of inertia of the pendulum when the load masses at the shafts are changed and the moment of force is constant.

 

1. Put down the data of measurements in table 15.7.

 

 

Table 15.7

T,s s The middle mass of the load , kg r, m h, m R, m ε0, s-2 I, kgm2 1/I, kg-1m-2 Note
                Measurement with the first set of loads
   
   
   
   
                with the second set of loads.
                with the third set of loads.
                with the fourth set of loads.
                with the set of loads.

 

2. Calculate angular acceleration defined by

.

3. Calculate moments of inertia given by

,

where I0 is the moment of inertia of the pendulum without loads (I0 = 4,6×10-2 g/m2); m is the mean mass of four loads at the shafts; R is the distance between the axis of revolution and the centre if gravity loads.

4. Calculate and Δε for five measurements. Calculate the half-width of the confidence interval for r and h using the main errors of measurements. Calculate the half-width of the trust interval for t using five measurements.

5. Results of calculation carry in the table 15.8.

 

Table 15.8

ti, s Δt, s Δti2, s2 Half-width of the trust interval
Δt, s Δh, m Δr, m Δε, s-2
         
         
         
         
         
                 

∑ ti = ∑Δ ti2= =

 

6. Make analysis of the experiment results.

 

Task 5.5

THE AIM is to determine the angular accelerations and moments of inertia of the pendulum with changed distance of the loads at the shafts and a moment of force is constant.

 

1. Measure t and h for five different distances of the loads at the shafts.

2. Put down the data of measurements in table 15.9.

 

Table 15.9

r, m h, m , kg t, s R, m R2, m2 ε, s-2 I, kgm2 Note
                The measurementon the arrangement of the loads the first position.
 
 
 
 
          The measurement on the next position.
         
         
         

 

3. Calculate angular acceleration defined by

.

4. Calculate moment of inertia given by

,

where I0 is the moment of inertia of the pendulum without loads ( ) kg m2; m is the mean mass of four loads at the shafts; R is the distance between the axis of revolution and centre of gravity loads.

5. Calculate ΔI and for one measurement. Calculate the half-width of the confidence interval for R to use main errors of measurements, and m is the table values; ΔI0 =

6. Put down the results of calculations in table 15.10.

 

Table 15.10

ti, s Δti. s Δti2, s2 Half-width of the trust interval
Δt, s ΔR, m Δm, kg Δ I0, kgm2 Δ I,kgm2
                 
                 
                 
                 
                 

∑ ti = ∑Δ ti2= =

 

7. Put I versus R2 on the graph.

8. Put I versus ε on the graph.

9. Make analysis of the experimental results.

 

Task 5.6

 

THE AIM is to determine moment of inertia of the pendulum with dynamics and theoretical methods.

NSTRUMENTATION AND APPLIANCES: cross-shaped pendulum, seconds counter, set of loads

1. Put down the data of measurement in table 15.11.

 

Table 15.11

t,s , s m0. kg r, m h, m R, m , kg l, m R1, m I, kgm2 I, kgm2
                     
 
 
 
 

 

2. Determine the moment of inertia of the pendulum by with the dynamics method

 

where k is the correct coefficient, it takes into consideration the mean forces of friction and errors of the time measurement (k=.3); m is the mass of the falling load; r is radius of the sheaf; t is the time of the load fall.

3. Determine the moment of inertia of the pendulum with the theoretical method

,

where is the mean mass of four loads at the shafts; R is the distance between the axis of revolution and the centre of gravity loads; m is the mass of one shaft; m is the mass of the hoop; R1 is the radius of the hoop.

4. Calculate . Calculate the half width of the confidence interval for r and h using the main errors of measurements and m and g are the table values. Calculate the half-width of the trust interval for t using five measurements.

5. Put down the results of calculation in table 15.12.

 

 

Table 15.12

ti, s Δti. s Δti2, s2 Half-width of the trust interval
Δt, s Δr, m Δm0, kg Δg, m/s2 Δ I, kgm2
                 
     
     
     
     

∑ ti = ∑Δ ti2= =

 

6. Compare I and I′. Make analysis of the experiment results.

 

Control questions

1. Obtain a relationship between linear and angular velocity.

2. Obtain a kinetic energy of rotation.

3. Obtain a formula for moment of inertia if the entire disk.

4. How is a moment of force determined?

5. Obtain the fundamental equation of rotation.

6. Make a comparison between the formulas of motion of a particle and derive the laws for the rotation of a solid body.

7. Obtain the law of conservation of momentum for the rotation motion.

8. What is a gyroscope?

 

Author: S.V. Loskutov, professor, doctor of physical and mathematical sciences.

Reviewer: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.

 

Approved by the chair of physics. Protocol 3 from 01.12.2008 .

 

16 6.

 

: , , .

: ii , , , .

:

1. ii I.

2. ii I1, i .

3. ii i i .

4., .

 

16.1

 

ii , ii i. ii , ii , i , i . , ii . (2), (6). (5) (8), , , (7). (4) , i .

. , ii . , . 㳿, , ii :

, (16.1)

m ; g - ; R- , ; r - ; L ii; .

, (16.2)

; k .

(16.1) ii .

, (16.3)

I0 ii i 1 ( 16.2); I- ii i 2 ( 16.2); m - ; a - ( 16.2).

16.1

 

ii i 2 i :

, (16.4)

I1 ii , ii, a i ( 16.2); I ii ( 16.2). ii I0 :

, (16.5)

r . ϳ I0 (16.3) i , , ,

. (16.6)

 

 

 

 

 

16.2

 

 

1. ³ 5 (1).

2. t 10 .

3. .

4. a i .

5. t10 .1 ÷ 3.

6. i a d.

7. T (16.2), t t.

8. ii I (16.1),

: m = 2 , R = 0,13 , r =5,5×10-2, L= 2,2 .

9. ii I1 (16.1), m m + 2 . m .

10. ii i 2 (16.4).

11. ii I i 2 (16.6). 16.1.

 

16.1

  t, t, T, , a, d, I, ×2 I1, ×2 ×2 I ×2
 
.
                                 

12. i .

 

1. : , ii, , , .

2. i 㳿 .

3. .

4. .

5. ii 㳿 .

6. .

 

 

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3. .. . / , 2004.- 442 .

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