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Computer simulations enhance qualitative meaning of the Newton’s second law
1
Silvana Mico [email protected]
1
Jorgo Mandili,1 [email protected]
Valbona Tahiri, [email protected]
2
Rezart Muco [email protected]
1
Department of Physics, University of Vlora, Albania
2Department of Computer Science and Electrical Engineering, University of Vlora, Albania
Abstract
Newton’s second law of motion is the most important and useful equation in mechanics.
This law gives the relationship between force and motion. Researchers have increasingly showed
that students don’t have a clear idea of Newton’s second law. To many students the force is
cause of motion. Their “misconceptions” produced by the common sense evidence are highly
resistant to change. It is important for students to know what the connection between force and
motion is. The Newton’s second law in many textbooks for undergraduate level is treated so
abstractly and students can’t reach a Newtonian view of the connection between force and
motion. If the study of all kind of motions is done over the second law and not in separate parts,
student’s conceptual understanding of mechanics will be increased.
The main task of mechanics is the study of the motion’s state of the mechanic system
through determination of coordinate versus time. The solution of this task is determined when is
known initial state of motion of the particle and when is recognized the specific character of the
forces as function of coordinates of the particle. Setting force and initial conditions for the
position of the particle in differential equation of Newton’s second law and solving this equation
we are able to describe how the state changes in time.
Computing and communication technology continue to make an increasing impact on all aspects
of education. Easy Java Simulations are powerful didactical resources that give us possibility to
focus our student’s attention on the physics principles. Using Easy Java Simulations we can
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create our simulations through which will be studied the motion of a particle under the action of
a specific force. In this paper we present the effectiveness when are combined web-based
computer simulations and lectures in understanding of logical and conceptual aspects of the
Newton’s Second Law of Motion.
Keywords: Newton’s second law, motion, force, change of the state, conceptual understanding,
simulations, computational modeling.
INTRODUCTION
The major goal of physics education researchers is to identify the student difficulties in
learning of conceptual basis of physics. Many students have difficulties in conceptual
understanding of physics. They like to see the physics as a collection of equations and think that
their task is to concentrate on calculation procedures. Researchers have shown that often is
ignored the direct experiences and discussion of physics concepts which can be used in different
situates [8]. To avoid this tendency of students, are required rational forms of knowledge
transmission, which will allow organizing it within a reasonable volume while maintaining the
deep of arguments and harmonization of all knowledge in different areas of physics. There are
many ways to reform the process of learning, but all require changing of the conception of
learning by integrating thinking and doing.
“Science courses rarely reflect the practice of science. In most courses, students do not
"do" science. Instead they only hear lectures about already validated theories. Not only do they
not have an opportunity to form their own ideas, they rarely get a chance to work in any
substantial way at applying the ideas of others to the world around them.” Ronald K. Thornton.
The new generation is very flexible in using new technologies. Information technology offers
the potential for a rapid and radical change, but the technology supports learning when activities
that they include have clear objectives and criteria. One of the best practices that integrate the
students in the use of new technologies where they themselves control learning is dynamic
learning using simulations of various tasks. They are modeling tools such as Easy Java
Simulations [2] which leave space the phenomenological thinking and can produce the best
learning experience. Easy Java Simulations are powerful and effectiveness didactical resources
that give us possibility to focus our student's attention on the understanding of fundamental
concepts. According to a Chinese proverb "A picture is worth a thousand words", this fact is
crucial for the learning because it provides the opportunity to manage complex information.
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A. METHODOLOGY
We have chosen the Newton’s Second Law of motion because it gives life all classical
mechanics having extremely wide range of applications in nature and is an interesting topic to be
treated by simulations. If students fail to study classical mechanics this occurs due to the not
right understood of the Laws of motion.
Firstly we present some ideas about student’s conceptual difficulties on Second Law and a
short theory context of the power of Newton’s Second Law to describe different motions of
particles depending of character of forces.
After that, we present an example of web-based Easy Java Simulations in terms of Law of
motion which develop understanding of relation between force and motion. These simulations
are facilitator leading students to identify forces, build free body diagram and describe the
motion.
Finally we present the data from student’s conceptual understanding of Newton Second Law.
These data are given from student’s ideas before introducing Newton’s law, after traditional
lectures and after students working with simulations. Here are included 50 students at University
of Vlora from various backgrounds such as physics, mathematics, engineering, chemistry, etc.
The students attended to the class PHY 151(Introduction to Physics 1) Fall 2009, in the first year
to undergraduate level. We have evaluated the impact of simulations on understanding of logical
and conceptual aspects of Newton Second Law.
B.STUDENT’S MISCONCEPTIONS ON NEWTON’S SECOND LAW
Students are familiar with Newton’s Laws probably from Middle School or High School. The
most of them are able in memorizing of laws and can say each word of Newton Laws. Indeed,
r
r
there is no difficulties in formulating of motion law and applying the simple equation: F = ma .
But, they don’t have a clear meaning and moreover don’t believe Newton’s Laws. If they know
what the laws say will not have a clear understanding of them.
This is due that students have their common sense concepts about motion and in most of
cases they are Aristotelian’s. It took about 2000 years to move from Aristotelian concept of
motion up to Galileo to believe that force is changed because of motion, for example, that a net
force is required to keep an object in motion at a constant velocity. We should not be surprised to
find that it is a problem for ordinary students today. Accordingly, common sense beliefs should
be treated with genuine respect by instructors [7]. These common sense beliefs are difficult to
change and they come from daily experience. For example, although the words position, velocity
and acceleration are different they have the same meaning for the most of the students and are
precisely these misconceptions that difficult the understanding of the laws of motion [10].
These misconceptions reinforced by way of Newton's laws are treated in most textbooks of high
r
r
school and university undergraduate level. They leave the students with formulas: F = ma , but
virtually no understanding of the content and meaning of the second law [9].
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C. ON NEWTON’S SECOND LAW
The state of the dynamic system is completely specified by the coordinates as functions of
time. If the force on a particle is known then the Newton's second law, called law of causality,
determines the acceleration of a particle and from this, the position of the particle at any time.
So, in principle, the motion of particle is completely predictable. The differential equation of
motion is:
r
r
r
d (v )
d 2r
F =m
=m 2
dt
dt
(1)
which state that: “changes in the amount of movement (evolution of the situation) is proportional
to the force applied to the particle that moves ". Differential equation (1) characterizes a process
because it connects the relations of defined variable sizes required by the other. Solution of
r
Newton's differential equation takes concrete meaning only when is known the form of F as a
r
function of the particle coordinates. If the character of F is known, then the equation of motion
(1) in principle allows to determine the dependence of the particle coordinates. Really, while the
equation of motion determines changing the speed of the material point for each time interval dt
r
r r
r F
( dv = dt ), it derives the determination of the material point’s position in space ( dr = v dt ).
m
Here we can see that the mechanical state of the material point is given by its coordinates and
speed. To obtain the general solution of differential equation of Newton, a special resolution
(which corresponds to the given process), help us initial conditions: position and velocity of
material point in a moment accepted as the initial time, i.e the initial state of movement of the
particle (prehistory of movement) when we study it. It is understood that the initial conditions
(initial state), make that the situation further movement comes fully defined. Precisely this is the
fundamental task of classical mechanics based on space, time and absolute motion (the classical
point of view), can be determined in this way: The study of the state of motion of dynamical
system (particles, the particles system, absolutely rigid body) through determination of their
coordinates as functions of time. Solution of this task is completely defined if: (1) Are known
initial conditions of mechanical system (prehistory of motion). (2) Are known concrete laws of
forces as functions of particles coordinates. Equations of motion that can be taken further link
their states of mechanical system between each-other, in different moments of time (history of the
motion). When the law of forces are uncertain or the force acts during a very short interval of
time, this task is resolved by other general laws- Conservation laws of energy and momentum.
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Achieving a Newtonian point of view of the connection between force and motion, after
introducing of Second law, all kind of motions should be studied over the Second law. The
following concept map illustrates this idea:
Newton’s Second Law
r
r
r
d (v )
d 2r
F =m
=m 2
dt
dt
Non equilibrium:
Equilibrium
F ≠ 0:a ≠ 0
F = 0 : v = 0 or v = const
Constant force:
Motion
under the
influence
of:
F = const
-weight force
-tension force
-normal force
-frictional force
r
r
dv
F
=
dt
m
Differential
equations
of motion:
Motion with
constant
acceleration
Resistive force
proportional to
object velocity:
r
r
Fr = −b ⋅ v
-drag forces
dv
= mg − b ⋅ v
dt
The acceleration
becomes zero when
object reaches
terminal speed
Restoring force:
r
r
F = − kx
-restoring force
on the spring
r
dv
kr
=− x
dt m
Harmonic motion
Central force:
k
r2
-gravitational force
-Coulomb’s force
F =−
dv
M
=G 2
dt
r
Circular motion
Figura 1. The concept map of Newton Second Law, that helps students to identify the connection between
force and motion.
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D. SIMULATIONS
Firstly, students are asked to save on own computer simulations by browser materials from
Open Source Physics(OSP) [1,3], available internet resources. Easy Java Simulations (EJS)
expands the OSP tools by providing a free open-source program. Users can examine, modify,
and redistribute the model with minimal effort. There are many forms to use simulations in
education. But the most effective way is when students have access to the programming of the
simulation [5]. With EJS materials students aren’t limited to manipulating the variables into the
graphics and animations of simulations. After carefully instructional support, (the most of
students have difficulty to learn without continuous support) students tend to be familiar in
using simulations [12].
Simulation: Block sliding down an incline plane
The original example explores the role of angle of plane and frictional force in the motion of a
block sliding down an incline plane. The modified simulation shows corresponding of the
graphs of the position, velocity and acceleration versus time
A screen shot for this modified simulation is given in figure 2.
The students can observe forces
on the block sliding down
simultaneously
with
the
graphical
presentation
of
motion. Initially the component
of gravity along the plane, Ft, is
compensated by static friction
force, Fsf, and the block stay at
rest. But, static friction force
can’t exceed the limit value
µ•N. When the slope of plane is
increased by dragging up the
double arrow at the plane top, Ft
force is increased and is being
larger then this limit and the
block sliding down. Then the
static friction force is replaced
by a smaller force, force of
dynamic friction Fdf and so the
net force is being non-zero. Figure 2. A screen shot of a simulation that represents a block sliding down
inclined plane and graphical presentation of motion.
This simulation leads the
students to the logical and
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conceptual meaning of the statement: the motion does not need a cause but the change of
motion need a cause.
Integrating differential equation of Newton for motion of the block under constant net force can
derive equations of motion:
F =m
d 2x
dt 2
F 2

 x = x 0 + v 0 t + 2m t

F

→ v = v 0 + t
m

F

a = m

So, if initial variables, net force and object mass are known, parameters of state-position,
velocity and acceleration-are determined at any given instant of time. Students can explore the
motion of the block along the incline, with different initial variables changing slope of the plane
and coefficient of static and dynamic friction. Although initial position and velocity can change
the form of graphs don’t change. This is always a motion with constant acceleration, under a
constant net force. Working with similar examples of simulations, explores the motion under a
specific force (constant force, restoring force, resistive force or central force) helping students to
correct mental model about Newton’s Laws and other physics phenomena [10,11,13].
E.
EVALUATION
OF
STUDENT’S
UNDERSTANDING
OF
NEWTON’S SECOND LAW
As an example on understanding
of force and motion concepts,
students are asked in class-group
work (all they had already
completed a study of kinematics
in their physics classes) to this
conceptual question: How the
position,
velocity
and
acceleration change versus time
when the block sliding down
inclined plane? What’s the
connection between motion and
forces acting on the block?
80
70
60
50
40
30
20
10
0
Before
instructions
After traditional
instructions
After working
with simulations
% of students
with Newtonian
point of views
Figure 3. Data to evaluate understanding of Newton Second Law.
It be seen that only 6% of students change their mental model
after traditional instructions and 69% of students change their model
after working with simulations.
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Students have written their predictions and have discussed with each-other. For many students
all quantities- position, velocity, acceleration and force- increased linearly respect time. In their
replies forces have nothing to do with change of the speed; the force (only gravity force) needed
to keep the block moving. The most of students replie: the block is free to go down, gravity
gives its an initial velocity and it accelerates because the slope of the plane. Students believe
that a constant force is required to keep an object in motion at a constant velocity and when the
velocity is increased the acceleration must increase and the force also must increase. The motion
of the block sliding down inclined plane is not motion with constant force and that’s why the
velocity increased. Only 7% of the student’s opinions before instructions were from a
Newtonian point of view. After traditional instructions only 13% of students give an exact
answer. After working with simulations 76% of students have a force and motion conceptual
understanding according to Newtonian point of view.
F.
CONCLUSIONS
Understanding Newton's second law is the key of understanding mechanics. Before the
introduction of Newton’s Second Law the students should know what we mean by the term
motion, force and mass. Rushing to solving problems due their ability to manipulate only the
equations of motion, not results in Newtonian conceptual understanding. It is essential to give
the law of motion a context, showing how this law is to be used. Our instruction should
concentrates in interpretations haw second law determines motion of a particle under a specific
force and development of individual experience such as simulations. The students learn better
when they faced with two sources of information, one source from authority (instructors and
books) and the other from direct experience.
Using EJS is an effectiveness educational practice that supports constructing of conceptual
understanding of Newton Second Law of motion. The possibility to change variables and
exploring in the same time forces and graphs of motion provides students in a very short time
with ability to correct their own mental model and develop a clear sense of the relations
between force and motion.
REFERENCES
[1]. The Open Source Physics (OSP) http://www.compadre.org/osp.
[2]. Wolfgang Christian & Mario Beloni, Benjamin Cummings, Physlet Physics Interactive Illustrations,
Explorations and problems for Introductory Physics, August, 2003.
[3]. Esquembre F (2009) http://www.um.es/fem/EjsWiki/Main/Documentation, accessed 1/27/2010.
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conceptual evaluation, Am. J. Phys. 66(4), 228–351,1998.
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collection, 2009 MPTL 14, Udine Italy.
[6]. Steve Stonebraker, Dedra Demaree, and Lei Bao, Using an interactive simulation to teach centripetal force,
AAPT 129th National Meeting, August 2004.
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[7]. Ibrahim Abou Hallouna and David Hestenes, Common sense concepts about motion, Am. J. Phys. 53 (11),
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[8]. E. F. Redish, Teaching Physics with the Physics Suite, (Wiley, 2003).
[9]. A. B. Arons, Teaching Introductory Physics, (Wiley, 1997).
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physics: Examples from kinematics, American Journal of Physics, Volume 55, Issue 6, pp. 503-513,1987.
[11] Gokhale, A., Effectiveness of computer simulation for enhancing higher order thinking. Journal of Industrial
Teacher Education 33, 36-46 (1996).
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