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					     EXAMPLE OF CROSS-CURRICULAR CONECTION
                           Valentin Peternel, B. Sc. El. Eng.
       Vocational School for Technical Sciences, Ljubljana, Republic of Slovenia

                            valentin.peternel1@guest.ares.si


1 Introduction
The article presents cross-curricular connections on the example of measuring the
moment of inertia. First the construction of a rolling bench and preparation of technical
documentation was included into the learning process of practicum and computer science
lessons. Later, the method of measuring was included into the lessons of the following
subjects: physics, mathematics, electrical engines, and measurements and regulations.
During lessons of Slovene and English general aims can be reached, especially by
applying materials and writing reports on exercises realization. A cross-curricular link
was planned for secondary vocational-technical programs on the electrotechnics area.


2 Procedures of measuring the rotor’s moment of inertia
The moment of inertia of a rotor can be defined in several ways: by calculating (by means
of a definition equation), mechanically (by means of a pendulum, a rolling bench,
additional mass on the rotor, etc.) or electrically (by means of a runout test).
Mathematically the moment of inertia of a rotor can be calculated only to 10 to 20%
accurately 1. That is why calculating the moment of inertia by means of a definition
equation has no practical sense. Usually, experimental establishing of the moment of
inertia is used. Measuring can be carried out in several ways, mechanical and electrical.

2.1 Defining J by means of a pendulum
Rotor, hanged onto a steel wire presents a weight pendulum. Periodical rotation of hung
rotor can be described with a simple differential equation, from which you can get the
equation Eq. (1) for the calculation of the moment of inertia. The directional moment of
force D should be known and the frequency of rolling f should be measured 1.
The problem with the calculating directional moment can be avoided if additional mass is
hanged on the rotor with known moment of inertia J ' 1. The system will oscillate with a
new frequency f ' and the moment of inertia can be calculated by equation Eq. (2).


                                                                  D
                                                         J                (1)
                                                               4 π2 f 2
                                                                  J'
                                                        J         2
                                                                            (2)
                                                             f 
                                                               1
                                                              f '
              Fig 1. Pendulum
2.2 Defining J by means of a rolling bench
The rotor has to be put on the rolling bench (Fig. 2) and let it go, so that it rolls to and
fro1. The amplitudes of the deviation from the balance point have to be small. The
frequency of rolling f, the radius of the shaft r, the curve radius  of the rolling bench and
the rotor’s mass m have to be measured. By means of these measured values the moment
of inertia J of the rotor can be calculated then by the equation Eq. (3).


                                                              m g r2
                                                     J                       m r2         (3)
                                                          4 π 2 f 2 ρ  r 

              Fig 2. Rolling bench

2.3 Defining J by means of a pendulum with two parallel wires
The rotor is fixed into the fixing bearer and hanged onto two parallel wires. The fixed
rotor has to be rotated and its own oscillation has to be observed. When the length l of the
wires is much bigger than the mutual distances 2d of the wires and if the amplitudes of
the rotating oscillation are small, then the moment of inertia J of the hung rotor can be
calculated by means of the equation Eq. (4), which is used for the weight pendulum 1.




                                                                m g d2
                                                            J                        (4)
                                                               4 π2 f 2 l


     Fig 3. Pendulum with two parallel wires

2.4 Defining J with additional mass on the rotor, its shaft or pulley
The method envisages applying additional mass, which has to be hanged onto the rotor,
its shaft or onto the pulley 1. When using this method the mass of the additional weight
m and the distance between the axis of rotation and the centre of gravity of the additional
weight r should be known. Then the oscillation frequency f should be measured and the
moment of inertia J of the rotor should be calculated by means of the equation Eq. (5).




                                                              mgr
                                                        J          m r2               (5)
                                                             4π f
                                                               2 2




          Fig 4. Rotor with extra mass
3 Dealing with the theme at some school subjects
For experimental cross-curricular connection the treatment of moment of inertia was
chosen with given point to measurements methods.

3.1 Including the theme into the practical lessons (practicum)
In lower classes the students can prepare the material, product the basic component parts,
protect the components’ surfaces, assemble and execute the functional testing of the
rolling bench (Fig. 5). In higher classes the students can dismantle the engine and prepare
the rotor for the realization of laboratory exercises at professional subjects.

         Parameter                    Value
         length                       105 cm
         width                        19 cm
         height                       36 cm
         distance between the sides   12 cm
         curve radius                 50 cm
         arc length                   105 cm
         angle                        120 o
         mass                         6,00 kg
        Table 1. Dimensional parameters                   Fig 5. Mounting the rolling bench

3.2 Dealing with the theme at computer science and documentation classes
The teacher acquaints the students with basic standard elements and with the possibilities
of drawing, designing and constructing by means of a computer. He guides them at the
preparation of the technical documentation (Fig. 6).




             Fig 6. Designing the rolling bench by means of the AutoCAD software pack

3.3 Dealing with the theme at electrical engines lessons
The teacher can acquaint the students with the so-called runout test. To determine the
moment of inertia by equation Eq. (6), three things have to be known: the rated rotation
velocity n, the power of free wheeling Pio and the imaginary runout time tn (Fig. 7).
             
           (s-1)

            n         =f(t)                                       1 Pio t n
                                                             J                         (6)
                                                                  4 π2 ν2 n



                     tn         t (s)
            Fig 7. The runout curve
3.4 Dealing with the theme at measurings and automation
The teacher can acquaint students with different methods of measuring the moment of
inertia. During the laboratory exercises the students can perform the exercise with the
rolling bench and measure the moment of inertia of a rotor (Fig. 8).




                              Fig 8. Instructions for laboratory exercises

3.5 Dealing with the theme at mathematics
In the mathematics lessons the students can determine the circle equation and the length
of the arc for the circular cut of the rolling bench. The essential part of the integration of
this theme into the mathematics lessons is the transformation of the equations, which are
related to the calculation of the moment of inertia of some frequently treated bodies (ring,
cylinder, disc, tube, sphere, pad, bar, etc.). In higher classes the teacher may inform the
students with the integral calculus for the moment of inertia of some simple geometrical
bodies (cylinder, sphere, ring, rod, etc.).

3.6 Dealing with the theme at physics
The treated topic can be included and connected with the treatment of rotation, moment
of inertia and oscillation. Experimental exercises, at which students measure the
frequency of rolling, weigh the sample and determine the moment of inertia for
cylindrical body by means of equation Eq. (7) and then compare the values they got with
the previously calculated value by Eq. (8), can be carried out in the lower level (Fig.9).


                                                                    m g r2
                                                            J                   m r2     (7)
                                                                 4 π f ρ  r 
                                                                    2 2




                                                                        m r2
                                                                     J              (8)
                                                                         2
    Fig 9. An experimental exercise at physics

Experimental exercises, at which students establish dependence the moment of inertia
upon the arc or the angle and test the validity of the equation Eq. (7), can be carried out in
the middle level. An example of such a measurement is shown in Fig. 10.
Experimental work, at which students establish dependence the moment of inertia upon
the rolling bench radius , can be carried out in the higher level. For this purpose several
benches with different radius should be made (Fig. 11).
                            1,40E-05




       ( kgm2 )
                            1,20E-05

                            1,00E-05




        Moment of inertia
                            8,00E-06

                            6,00E-06

                            4,00E-06

                            2,00E-06

                            0,00E+00
                                       0   5   10       15         20   25   30

                                                    Angle    (o)




                                Fig 10. Dependence J ()                          Fig 11. Rolling benches with different radius

3.7 Dealing with the theme at English and Slovene lessons
During Slovene classes (mother tongue) and English classes (first foreign language)
students develop their reading, speaking, listening and writing skills, which are necessary
for the comprehension and creation of different professional texts.
Reading: Students search for the suggested contents in their textbooks and professional
literature. By reading professional texts related to other subjects and real situations in
industry, they spontaneously practice different reading strategies (exact reading, reading
comprehension, searching for specific information).
Speaking and listening: Students can present the procedure of measuring the moment of
inertia; the presentation can be done in other classes, too. The topic gets more interesting
because it is the schoolmates who talk and because the measuring device is actually there,
present, constructed, thus arousing interest and stimulating motivation.
Writing: Upon listening to presentations and reading texts, students take notes and
reports. By writing they are getting qualified for creating official and professional texts.


4 Conclusion
Cross-curricular relations are successful when the set aims are reached. However, the
teacher has to know precisely which goals of individual subjects he wants to reach with
cross-curricular links. The teacher has to know the goals and contents of different
subjects, which are a part of the curriculum of a certain educational program.
Cooperation of teachers of different subjects is inevitable. The choice of contents, the
applied pedagogical methods and the organization of the realization have to be adapted to
the level in their development and to the previous knowledge of students. Every cross-
curricular link has to be planned carefully and carried out well in its contents and
organization. At the end the teachers shall analyze the realization of the set goals. The
students shall be actively included into as many as possible phases of the learning
process, bearing in mind that their autonomous work is of primary importance.


5 Sources and literature
1    Avčin F., Jereb P.; Preizkušanje električnih strojev in njihove lastnosti; Third
       edition; TZS; Ljubljana, 1983,
2    Marentič Požarnik B.; Psihologija učenja in pouka; DZS; Ljubljana, 2000,
3    Marentič Požarnik B.; Zbornik Kurikularna prenova, Cilji, izhodišča in možne
       stranpoti kurikularne prenove; MŠŠ; Ljubljana, 1997,
4    Kovač M.; Dejavnosti učencev v procesu pouka; FŠ; Ljubljana, 2003.

				
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