An inclined plane problem solution
Val Voroshilov
The first introduction into the course
Study
physics without problem solving is the same thing as learning swimming and
never enter into water.
To be
learned to swim is necessary to swim, i.e. to lie down on water, to try move
hands and legs... At the first place, certainly, the one will fail; the one
will drink some water and do not move, but gradually, time after time the one
will be doing better and better. And at some day the first success will be
achieved!
The same
situation is occurred when we want to learn how to solve physics problems (and,
actually, any everyday problems). To be learned to solve problems is necessary
to solve problems: to read a text and conditions, to imagine as clearer as
possible a circumscribed situation, to draw a sketch, to try to make out the
necessary formulas. The wrong attempt is not mistake! If you will fail at first
time, you will have the opportunity to look on what was done and try again.
So, the
recipe is very simple. To find the solution of a problem, it is necessary 1) to
want to find the solution of this problem; 2) to believe that the problem can
be solved; 3) to believe that you are able to find the solution; and 4) begin
solving the problem, and trying and trying and trying....
The very
first question, which is usually occurring while reading the text of a problem,
is “how to start finding the solution of this problem?” Usually I answer the
following. Let's assume, that you are invited by the friend on a party. You
come, and there are a lot of unfamiliar people. What do you usually do at first
minutes? Naturally, in such a situation
you are trying to find somebody familiar and you are looking around, trying to
recognise somebody. The same situation is occurring when you are staring
solving a problem; by reading the text, try to find out something (anything!)
already familiar to you. On scientific language it is named to bring a problem
under a model.
When
looking for a some familiar person you can use different indicators: first of
all it is a face, but also a voice, the shape of a figure, a manner to behave,
a stile to dress and etc. When a physics problem trying to solve you have to
use some indicators too. As a teacher I have
a bunch of such of indicators in my head, by using of which it is
possible to go through the problem step by step. This bunch of indicators is
called “a reference basis of acting”.
Now I
very shortly tell you how it is possible to use the theoretical knowledge to
recognise and construct solution of physical problems.
Physics
studies phenomena, i.e. processes, which happen to various objects.
The
phenomenon is first of all that we can see and feel (smell, touch or hear). We
see the motion of a bird in the sky; it is a phenomenon, we hear a sound of a
rain behind the window; this is a phenomenon, we shot the final triple; it is a
phenomenon too! We consider as a phenomenon also that we can imagine, but only
if into the event that we have imagined there is a place for the person, who in
that imaginary world can feel the same feelings, that the actual person will in
the real world. For example, anybody have never seen with the own eyes a
molecule of water. But we can imagine them as a tiny balls, which are flying
and rotating as a cloud of mosquitoes. We can imagine that we are putting into
this mosquitoes cloud a small seed of fern, we are taking a microscope and
watching what is happening to the seed.
In this imaginary world we can see the seed moving up and down, to the left and
to the right because of small balls - the molecules – which are flying on the
seed from the different directions and hitting it, forcing to make those
diverse movements. But after have this imagined we can actually take the seed
of fern and put it in water and we can watch the seed under a microscope. We
will not see any small balls - molecules. But the chaotic movement of the seed
will be seen and it will be exact the same as we saw it in our
imagination! This results us to an
idea, that there is really such a phenomenon as movement of molecules (hence,
there are molecules!).
We call
as an object everything that there is in a nature, everything, that we can see,
hear, touch, experience: a home, molecules, light, stars, etc. That object,
which we can feel, which we can point on, which has a certain sizes and form is
frequently named as a body (machine, stone, molecule are bodies). There is an
object of other kind, which is named a field. It is usual something broaden in
different directions and varying from one point to other (as a sinusoidal surface
of a sea).
The
processes is a name for everything that can happen to objects: they can move,
or be at rest, be heated, be broken up,
be shining, be attracted to other objects and etc.
As a
matter of fact, physics studies just with what, what and why happens.
These are three main questions, the answers on which physicists keep searching,
searching and will continue to search (it is their job). And from these three
questions we always have to start when we are at the beginning of any physics
problem.
It is
not required from us opening the laws of a nature, which were open by the
scientists some days-years-ages ago. We need only to know those laws and to be
learned to recognise those physics situations, for which the application of
those laws is necessary. For example, if I need to find a speed of a sputnik,
which rotates about the Earth, I shall recognize in this problem the following
situation: there are two objects (Earth and satellite), they interact with
another (Earth attracts the satellite), the satellite is moving circularly
about the Earth. Key concepts circumscribing/describing the given physics
situation and permitting to me to recognise it are “interaction of objects by a
gravitational attraction” and “movement around, making a circle” (it is the
same as finding a familiar face and voice among the crowd of the visitors). I
know, that if the situation of interaction of bodies is considered, for certain
it is necessary to use the Newton’s laws (compare: I know, if I have met the
person with such a face and such a voice, I should name the person with that
name). I know, that if the motion is happening to be in a circle, hence, most
likely, the formulas for the centripetal acceleration be considered. I know
also, that if we have to deal with objects which are attracted to each other,
almost for certain it is necessary to apply the law of gravity. OK. So. I have
found in the text of the problem some indicators (interacting objects, circular
motion, gravitation). I have written some formulae. Attention, please! I do not know, whether all these
formulae are really necessary to solve the problem, and I do not know, whether
will be all of them enough to find the solution of my problem, but, at least,
now I have
something to begin with!
Pay attention to such details: to find out
what a physical situation do I have to analyse, it was not important to me to
know the exact formulating of physics laws, it was important to me only to
recall, what kind of lows can be necessary for finding the solution of a problem.
Now, after I have recognised what laws take place, I automatically have
determined, what knowledge will be necessary to me for further calculations,
and at the first place I need to find out the exact formulating of physics laws
which I have just recognised. Same situation sometimes happens when we are
trying to recognise our friends: at first there comes reliance to us on that we
have seen this person before, we have met him/her earlier, we knew the name of
the person, hence in our memory there is this name, and only then, in a several
seconds, the name is popped up in the memory.
From the short consideration above is visible
now, that to recognise a specific physics situation we have to deal with, it is
necessary to know beforehand:
- What
kind of physical phenomena exist in Nature, that is what can being happened
(using our analogy with the familiar person, we should know beforehand with
whom and in what situations we can meet);
- By
using of what indicators we can distinguish one phenomenon from other (as well
as one person from other);
- What
main laws and the concepts describe the phenomena (it is like addresses and
names of ours friends);
- and
finally, what are the exact formulating of those laws and concepts (for
ours friends it is like we know their profession, what they can do).
To know
all these premises, we have to investigate the different physics theories. It
is necessary to learn/teach the theory, but for what only that during problem
solving we are able to answer the following questions:
- What
kind of objects there are considered in a problem (about what objects is spoken
in a condition, we need to have names for all those objects);
- What
is happening to those objects (are there in a motion, at rest, what kind of
motion do they have, do they have any influence on each other, what is a nature
of their interaction and etc);
- What
is the difference between the objects, what parameters, concepts and physical
quantities are necessary to describe the objects and the processes, whether
these physical quantities are changing
with time or not;
- What
laws of a nature and main definitions or concepts are needed to be used for the
describing of the considered phenomena, i.e. objects, and processes which are
happening;
- What
are the exact formulas and statements for these laws and definitions.
Certainly,
the list of such of questions could be continued, but it would not be very
useful. To start solving a problem the given list is quite enough.
The last
note of a general character, which is necessary to make before beginning a
problem solving, concerns the formulas used for. Obviously, without a formulae
and equations we cannot solve any physics problem. We can spend a long time
analysing the problem, but sooner or later we will need to write the formula ¹
1, then formula ¹ 2 and etc.
To
understand correctly essence of a physics problem solving, it is necessary to
understand, that all physics formulae are divided into three main groups: the
first one is the laws of a nature, and the second one is the definitions of
physical quantities. There is one more group of the formulae which are just
consequences derived from the laws and definitions.
A
definition usually answers a question “what is it” or “what is the name of it”
(velocity is..., an acceleration is...). Definitions are the result of an
agreement between physicists on how to name something. As a matter of fact, a
definition is just an indication (we point our index finger to something and
say “it is …”). We meet definitions almost every day during a childhood. For
example, a dad points his index finger to a car and makes a word “a car” and
gives thereby the definition of this word and the name of this object.
The
physics law answers a question “how to calculate/find/derive”, “what is the
relation between this and that” (for example, how can be found the force of
attraction between two bodies). A physics law connects one physical quantity to
the other one (which we know already from their definitions), links them one to
another. The main difference between the laws and the definitions that the laws
occur not because of the agreement between people, all laws are result of
carefully carried out experiments, in which the magnitudes of various physical
quantities were measured.
At the starting
the problem solving, before writing the main laws necessary for the solution of the problem, we should
know first all processes which could be described by those laws. To write all
the necessary definitions we need to know first by which of them different
objects and processes can by described. Understanding Physics means recognizing
physical processes and correct using corresponded definitions and laws. This
what we will study how to do it.
An inclined plane problem solution