ZANDERS, Juris
Riga - 10, a.k.412, LV-1010, LATVIA, Latvian University of Agriculture, azandere@hotmail.com
Abstract: Strategy of engineering education is to become a specialist capable to compete in world's labour market. For achieving this aim prevails declared pragmatism - spend a minimum of time and get a maximum of efficiency. It compels the teaching stuff to change educational tactics from profound exposition of material to telling about ruling central ideas and laws developing logical thinking as much as possible.
To realize ideas above is developed a new structure of physics' exposition-fundamental structure of subject and methodical scheme for training and achieving skills to solve physics' problems, using detailed logical consistency of flight of thoughts.
Keywords: physics, methods, teaching, postulates, terms
My abilities are highly middling
in comparison to ones of many
others. If I've done more it is
thanks only to methods.
R. Descartes
The highest level on engineering education strategy for students may be bachelor's, master's or doctor's degree in the chosen specialty and capability to compete in world's labour market obtaining in the social and professional scale the state conformable their wishes and abilities.
In the age of developed technologies and global information net Internet it goes out of significance the criteria of XIX and XX centuries about an educated man as "vessel full of erudition" or "torch-bearer longed for new knowledge". Instead of them now comes on declared pragmatism - how much and what I have to do in order to get entry in marks' booklet, to receive diploma, certificate, fat job etc., because curricula are heavy scheduled, amount of new information increases and students have no time and force to do all honestly.
Such a situation compels lecturers to change the manner of educational tactics from profound exposition of material [1, 2, 3] to explanation of ruling central ideas and laws of subject developing logical thinking of students as much as possible [4]. This approach corresponds with quickly changing labour market's conditions, according to them expert may have to change specialty several times in his life-time. Such activity is more successful for person, who's brain is not overcrowded with fact, but who knows ruling ideas and laws of different subjects and has high logical thinking.
In this aspect is elaborated a new original structure of physics` exposition - fundamental structure, which allows in separate chapters of physics` course clearly to separate postulates, central ideas (general conceptions, terms), laws of physics` and equations corresponds them, rules gotten on base of laws of physics, problems' examples, illustrating application of physics' laws and rules in solving engineering problems, using detailed logical consistency of flight of thoughts.
According to thoughts above is gotten a compact and obvious formation of course of physics, making it tangibly thinner and more accessible for user, and developed an integrated notion about physics and ability to use it in praxis.
The idea of fundamental structure of physics' exposition comes from a language. In order to speak a foreign tongue first of all you must learn meanings of words (in physics them corresponds postulates and central ideas). After them you must know principles according to them words are connected in sentences expressing thoughts and feelings (in physics them corresponds objective laws of universe gotten through observations). Knowledge of grammar rules helps you to orient specific questions of formation of words and sentences (them corresponds in physics a great number of rules and computations' formulae which lightens application of physics' laws in praxis).
It goes without saying - a person can not speak some language without knowing meanings of words and principles of construction of sentences, but for speaking without mistakes you must know also the rules of grammar. Unfortunately students usually find themselves in a situation, in which they ought "to speak correctly" (in physics - to solve some engineering problem) without understanding meanings of words and ability to make sentences of them.
Rest part of this report is dedicated to coming into existence idea about fundamental structure of physics' subject concerning to chapter of physics - classic mechanics of material point and rigid body:
postulates: pass of time, idea of space;
central ideas (general conceptions; terms): material point,
mechanical motion, pure translational motion, pure circular motion,
trajectory, path, displacement, average and instantaneous linear
velocity, average and instantaneous angular velocity, average and
instantaneous linear acceleration,
average and instantaneous angular acceleration, inertia, mass,
force, completely rigid body, centre of mass, completely elastic
collision, completely inelastic collision, momentum, impulse, lever
arm (moment arms), torque (moment), moment of inertia, angular
momentum, kinetic energy, potential energy, mechanical work, power,
tensile stress, conservative and nonconservative mechanical
systems;
laws of physics and equations corresponds them: Newton's laws, Newton's law of gravitation, law of centre-of-mass motion, kinetic and rolling friction laws, Hooke's law, principle of conservation of momentum, principle of conservation of angular momentum, law of conservation of total mechanical energy, law of conservation of energy.
Before solving physics' problems a student have to learn methodic scheme for compiling a system of equations by following sequence:
Read the problem thoughtfully (many times if necessary) till you understand the physical content of the problem and can find laws for use in solving it.
If you can't understand the physical essence of the problem, find the chapter (or chapters) of your physics' textbook concerning the example and read them once more.
Establish qualities using traditional letters or other symbols, writing in a column all qualities and their numerical values. If values aren't given in SI units, change them to this system.
Clear up main (original) unknown qualities, establish notations for them and write these notations under horizontal line, which separates given qualities from original unknowns.
Draw vertical line, which sections off the qualities and unknowns, and on the right of it make a graphical scheme of the physical problem we have. In the scheme write all notations used in the problem. The scheme must correspond in principle with content of problem. The making of graphical schemes detects for the student imperfections in his understanding of the problem's formulation as well as selected formulae of laws. Precise graphical scheme is about half of the battle in finding the solution.
Compile system of independent equations according to the problem. (Equations are independent if each additional equation cannot be gotten through mathematical transformations of equations written above). When compiling it is useful to start by taking the equation (or equations), which includes the main unknown (~s) we are searching for. After them write other equations for finding additional unknowns. (Additional unknowns are such ones, which appear in the formulas, but are not written in column of original unknowns). When writing every equation, apply notations used earlier. To the right of each equation in the list write the underlined notation for the unknown which is to be found through the solving of the equation. Next to that write the additional unknowns of the equation. (Unknowns are all qualities used in written equations and not possible to find in column of used qualities or are not universal physical constants). While writing every new independent equation, check it for new additional unknown (~s). If they have appeared fix them on the right side of list after the first unknown. If the new equation contains once again unknowns from above equations, unknowns should not be mentioned again in this line. The work of compiling system is done, if the number of written independent equations equals the number of underlined unknowns in the system. Such control must be used all the time as you compile the system. Superfluous (in comparison with number of unknowns) equations as well as insufficiency of them shows mistakes made in the compiling the system. If the system of equations is complete (number of equations equals number of unknowns), then in physical meaning the problem is solved. Further activities are more a question of mathematical skill than of physical skill.
Solve the system of independent equations by getting formulas, in which on the left of the equals sign are written notations of original unknown qualities, but on the right - letters and symbols of given qualities and constants. By solving the system of equations we often use the methods: from more simple equations get expressions for additional unknowns and put them in more complicated ones or main equations, containing main unknowns.
Check the correctness of the dimension of formulas for main unknowns on the right side. Doing this remember, please, that signs of plus or minus between parts of right sides' expressions don't change dimension in this part of formulas (these signs change only the numerical multiplier, but it's not significant in checking dimensions).
Put in the acquired formulas for original unknown qualities the numerical values of given qualities and constants in system SI units and calculate the results.
If you have gotten the functional connection of two or more qualities, then you can draw on graph or other paper a diagram of the connection.
Practical use of this methodic scheme for compiling a system of equations is illustrated in following physical problem:
On an inclined plane, the height of which is 1 m and the length 2 m, is placed a 2 kg block having a sliding friction factor to the plane of 0.2. To the block is fastened an untensible weightless thread, which is thrown over an immovable cylindrical pulley on top of the inclined plane, and to the other end of thread freely hangs a body of 5 kg mass. Radius of cylindrical pulley is 5 cm and it's mass is 0.5 kg. Calculate the acceleration of the 2 kg and 5 kg masses and the spring force in thread on both sides of pulley. Ignore friction and sliding of thread in pulley.
Figure 1. Inclined plane and forces
Given qualities:
Original unknowns:
For convenience in showing manipulations with equations and unknowns, first of all is written a complete system of equations, but commentaries are given further.
- spring force in thread, fastened to the first
block;
m1 - mass of first block;
- sliding friction force to 1st block;
- component of force of gravity acting on mass
parallel to inclined plane;
a - acceleration of motion;
k - sliding friction factor;
- component of force of gravity acting on mass
perpendicular to inclined plane;
- force of gravity on 1st body;
- acclivity angle of inclined plane;
h - height of inclined plane;
l - length of inclined plane;
- force of gravity on 3rd body;
- mass of 3rd body;
- spring force in thread, fastened to the 3rd
body;
- mass of immovable pulley;
I - momentum of inertia of pulley against axis of cylinder;
- angular acceleration of pulley rotation;
N - prop reaction force to force of normal pressure;
q - gravity acceleration.
In compiling a system of 12 complete equations we expand in the
following way. As we must find acceleration of two bodies and spring
forces in thread, this is a dynamics' problem in mechanics.
Acceleration, mass and force are united in Newton's 2nd low. Equation
(1) is written for mass 
, projecting vectors on
universalized curved line axis q, which coincides with
configuration of thread and positive direction of which is from mass
to mass 
. Correlation
between masses 
and 
(
> 
) helps to find
direction of acceleration - it concurs with positive course of axis
q. If it's not possible to make a decision about positive direction
of acceleration, it can be chosen at will. If a sign of acceleration
after solving system of equations and
calculation is the same as the one chosen, solution is correct. If the
signs are opposite, solution of system must be repeated using contrary
directions for vectors 
and 
.
Equation (1) contains four unknown qualities: 
and . Presuming that this equation (1) is
written for finding unknown 
(therefore we
underline this symbol), we ascertain need of three new independent
equations for additional unknowns 
, 
and main unknown .
For finding additional unknown 
is written
equation (2), which gives relation between friction force 
, sliding friction factor 
and
force of normal pressure 
. As equation (2) is
written for finding , we write this symbol
underlined on line (2). Checking symbols from (2) and given qualities
from column, we find new additional unknown ,
write it after in line (2) and equation (3) for
solving 
. We write 
underlined on the right of equation (3) as well as two new additional
unknowns 
and 
.
Equation (4) is written for 
and units force of
gravity , mass of 1st body 
and gravity acceleration g, containing no new
additional unknowns. Therefore on the right side of line (4) is written
only underlined symbol . For finding is written equation (5), using popular trigonometric
formula and expressing [underlined symbol on
the right side of line (5)] through 
[new
additional unknown, written on right side of line (5) after ].
Equation (6) is written by expanding through
cathetus h opposite to angle and
hypotenuse 
in triangle 
(it follows from similarity of triangles 
and ). On the right side of
line (6) we write only underlined symbol
because there are no new additional unknowns. Therefore we return to
equation (1), in which we have two unknowns 
and without expressions for them till now.
From triangle we write equation (7) for 
noting, that 
is opposite
to angle cathetus and 
- hypotenuse. Underlined 
is
written on the right of line (7). There is no new additional unknown.
For finding 
we utilize Newton's 2nd law
according to accelerated motion of mass 
along
the axis 
: on the end of line (8) is written
underlined main unknown , adding two new
unknowns: 
and original unknown 
. For writing equation (9) use the same ideas as for (4)
and commentaries also are similar: on the end of line (9) we write
underlined symbol 
.
Independent equation (10) for 
is gotten by
using main equation for rotation dynamics for body with immovable
rotational axis. On the end of line (10) are written underlined main
unknown 
and two new additional unknowns 
and . As pulley is in
the form of a cylinder, equation (11) is taken from textbook - momentum
of inertia for cylindrical body against the
axis of cylinder. On the end of line (11) we write underlined symbol
.
The last independent equation (12) for is
found as kinematic coherence between angular and linear accelerations
and accordingly
and pulley radius r. On the right of line (12) we write only
underlined symbol .
Thus we have put together a complete 12 equation system (12 unknowns),
solving of which is done by taking expressions from equations (2), (3),
(4), (5), (6), (7), (9), (11) and (12) for all additional unknowns and
inserting them in equations (1), (8) and (10) for original unknowns
, 
and 
. Doing this we have three independent equations for
three unknowns. Solution of this three equations' system gives us the
final expressions for original unknowns:
, (13)
, (14)
. (15)
Checking of dimensions for final expressions gives:
, (16)
, (17)
. (18)
Now we have plausible three formulas (13), (14) and (15) for
calculating numerical values of , 
and 
, using numerical values of
given qualities: a = 4.94 
, = 23.1 N and = 24.3 N.
Inspecting attentively equations (13), (14) and (15) together we can exclude radius of cylindrical pulley r. It means correctness of physical problem also without given quality r. If so, then 12 equation system will have thirteenth unknown r, but, as we have seen, system can be solved, because r cancels.
If you are not searching for forces, solving physical problems, especially in mechanics, is more convenient by using laws of conservation, in physical problem above - law of conservation of energy.
Applying the concept of energetics to the problem above, we presume a
start position for block at point A with
zero velocity, further moving with acceleration
and covering distance 
. At the same time the
same distance is covered by mass 
vertically downward. Equation (20) expands law of
conservation of energy: 
- lost potential
energy for mass 
; 
-
obtained potential energy for mass 
; 
and 
- acquired kinetic
energies for masses 
and accordingly; 
- acquired
kinetic energy of rotation for pulley; 
- work
of friction forces:
|
(19) |
|
|
(20) |
|
|
(21) |
|
|
(22) |
|
|
(23) |
|
|
(24) |
|
|
(25) |
|
|
(26) |
|
|
(27) |
|
|
(28) |
|
Most of 10 equations in the system (19) 
(28) are the same as in the system (1)
(12), except equations (19), (21) and (28),
easily found in a textbook of physics.
Solving system (19) (28) for
acceleration 
, we get the same formula (13).
If, solving mechanics' problems, it is possible to use either forces',
or energetic conception, better use the last one which usually is more
simple.
The expounded logical system to solving physical problems is a system of consistent operations, which can help to acquire the skill of solving physical problems with certainty of the solution's accuracy.
The author is ready to co-operate with everyone who shares his views about strategy and tactics of engineering education .
1. YOUNG, H.D. & FREEDMAN, R.A. University Physics. U.S.A.: Addison-Wesley Publishing Company, Inc., 1996. 1484 p.
2. VOGEL, H. Gerthsen Physik. Berlin Heidelberg New York: Springer Verlag, 20. Auflage, 1999. 1262 S.
3. VALTERS, A., APINIS, A., OGRINS, M., DANEBERGS, A., LUSIS, DZ., OKMANIS, A. & CUDARS, J. Fizika. Riga: Zvaigzne, 1992. 733 lpp. (in Latvian).
4. ZANDERS J. Fizikas uzdevumu risinasana. Jelgava: LLU, 1996. 22 lpp. (in Latvian).