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