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ALTERNATIVE CONCEPTIONS ON THE TEACHING
OF THE CONCEPT OF MASS
J. Valadares
Universidade Aberta
R. da Escola Politecnica, 141-147, 1200 Lisboa, Portugal
From an educational perspective, one of the claims of constructivism is this: learning
is an activity of personal exploration that implies an active reorganisation of a
framework of meanings about the world. The pre-conceptions in the mind of a
student about some segment of science have an important role in the way of learning
that segment. This is the main reason why there have been many investigations trying
to detect the students' ideas in a great range of subjects. On the other hand, teachers
and schoolbooks' conceptions have been poorly investigated, although the social
influence ofa teacher on his students is well known. Some of the students
misconceptions are induced by their teachers.
Consequently, research based on the teachers' ideas is very important. There are
teachers' alternative conceptions in fundamental subjects such as mass, weight,
energy, heat, and so on. Some of them influence decisively the students ' learning.
The author of this communication has made an exhaustive research into conceptions
of mass both in the history of Physics and in Physics teaching. h.; found some
historical and epistemological reasons for the multiple answers to questions like
these:
What is the scientific meaning of mass?
Does the mass ofa particle depend on its velocity?
Is the mass of a particle equivalent to its total energy?
[n an isolated system, could there be conversion of mass into energy or energy
into mass?
• Is the total mass of a system the sum of their particles' mass?
" Is the energy of a particle zero when its mass is zero?
• Etc.
•
•
•
•
The objective of this communication is to give the results of the research developed
on the teachers' ideas about mass, and to discuss the historical and epistemological
reasons why some of them are smely misconceptions, using a methodology based on
heuristic tools.
Bibliography:
(I) C. Adler, Does mass really depends on velocity, dad, American Journal of
Physics, 55, 8, Aug. 1987
(2) R. Baierlein, Teaching E = m c' , The Physics Teacher, March 1991
(3) R. Bauman, Mass and Energy: The Low-energy Limit, The Physics Teacher, 32,
6, Sept. 1994
(4) M. Jammer, Concepts of Mass in classical and modern Physics, Harvard
University Press, Cambridge, 1961
170
ALTERNATIVE CONCEPTIONS ON THE TEAClDNG
OF THE CONCEPT OF MASS
J. Valadares
Universidade Aberta
R. da Escola Po litecnica, J41-147, J200 Lisboa, Portugal
Abstract
From an educational perspective, one of the claims of constructivism is this : learning is an activity of
personal exploration that implies an active reorganization of a framework of meanings about the world.
The pre-conceptions in the mind of a student about some segment of science have an important role in the
way of learning that segment. This is the main reason why there have been many investigations trying to
detect the students' ideas in a great range of sUbjects. On the other hand, teachers and schoolbooks'
conceptions have been poorly investigated, although the social influence of a teacher on his students is
well knO\Am. Some of the students misconceptions are induced by their teachers.
Consequently, research based on the teachers' ideas is very important. There are teachers' alternative
conceptions in fundamental subjects such as mass, weight, energy, heat, and so on. Some of them influence
decisively the students' learning.
The author of this communication has made an exhaustive research into conceptions of mass both in the
history of Physics and Physics teaching. He found some historical and epistemological reasons for the
multiple answers to questions like these:
What is the scientific meaning of mass?
Does the mass of a particle depend on its velocity?
Is the mass of a particle equivalente to its total energy?
In an isolated system, could there be conversion of mas sinto energy or energy into mass?
Is the total mass ofa system the sum of their particles ' mass?
Is the energy of a particle zero when its mass is zero?
Etc.
The objective of this communication is to give the results of the research developed on the teachers' ideas
about mass, and to discuss the historical and epistemological reasons why some of them are surely
misconceptions, using a methodology based on heuristic tools.
Bibliography:
(I) C. Adler, Does mass really depends on velocity, dad, American Journal of Physics, 55, 8, Aug. 1987
(2) R. Baierlein, Teaching E = m c2, The Physics Teacher, March 1991
(3) R . Bauman, Mass and Energy: The Low-energy Limit, The Physics Teacher, 32, 6, Sept. 1994
(4) M. Jammer, Concepts of Mass in classical and modern Physics, Harvard University Press, Cambridge,
1961
1. T\vo different conceptions about th rdati, istic
mo~nt1.l
A well known current university Physics book! writes:
"As a result, if the conservation of linear momentum is to be valid in any inertial frame
(as demanded by the principle of relativity), a new definition of linear momentum is
required. The relativistic definition of linear momentum is
p=mv
where the relativistic mass, m, is
The quantity mo is the rest mass of the particle - the mass measured in its rest frame"
Another book2, about the definition of momentum, writes:
v
"Consider a particle of mass m with velocity
in some reference frame. The modified
definition of the momentum
of the particle is
p
p
=
mv
.Jl- v 2 I c 2
= pnv
With this definition, momentum is conserved in every inertial frame."
These diferente conceptions are disseminated in many other books. After all, what is the
relativistic momentum, m v or }111v?
:2. \\'h at is the mass () r a purt i c Ie '?
One concept is a «construct». It depends on a structure of other concepts. The concept of mass
in the early Newtonian Physics is different of the concept of mass in the Physics of the last
century. Mach's concept of mass is certainly different of mine.
I'm going to speak of «my concepts». Certainly they are not mine, because they have resu Ited
from a «negotiation» with the concepts of others.
When I speak of mass of a particle, I'm thinking of a property of the particle. If it is a property,
it depends on the particle itself, on its internal state (if it's important to consider the structure
of the particle). It must be an invariant if the particle is isolated, when the internal state of the
particle doesn't change. The mass of a particle is the magnitude of its momenergy 4-vector.
It is a magnitude characteristic of the particle and totally independent of its state of motion.
The only concept of mass that I know with these characteristics corresponds to the following
definition:
momenergy
= mass x
-~
spacetime displacement
proper time for that displacement
Adopting the same units for time and space (whether meter or second or year) and a given
inertial frame, the momenergy components of a particle are:
E=m~
dr
p =m
Y
The square of the magnitude of momenergy is, then,
dy
dr
dz
p= = m d-r
For two inertial
Sand S' we
units (for
In
In conventional units
the
invariant is
in SI units),
mass ofa
2
can be defined
2 2) 1/2
pC
As
energy of the
we see
dt
r
as
equation
Qm
We know that the proper time between two events is
=
pV'PC~rI
than the laboratory time:
=
is, in conventional units,
when v == 0,
Vf"~<;/J')
c, is a
are different only
The
in a magnitude that
an isolated
a property
system, a
collisions
decay or annihilation the
unaffected by any
a value independent from the
of reference
The only
magnitude I know with
the total momenergy. So, I
characteristics is the magnitude
consider as mass
an isolated
the magnitude of its total momenergy.
4-vector
is unaffected by collisions
the parts
system. It is
unaffected by any
system may undergo.
transformation, decay or annihilation
same units
time and
magnitude
system is
=
4-vector
of the
In conventional units we have
An important frame is the zero-total momentum frame (where the momentum of the system
is zero) . Then, it is, in a zero-total momentum frame and with the same units for mass and
energy:
ms
n
n
;= 1
;=1
n
= Es = IE; = I
m; +
I
K;
;=1
As we see, the total mass of a system is not the sum of their particles' mass. The kinetic
energy of the particles also contributes to the mass of the system.
When the particles interact, as well as move, the energies of interaction have to be taken into
account. They therefore contribute to the total energy of the system, Es , which gives the
mass, in conventional units:
ms
4. Sut ... is ther e suc h thing
=
::lS I11::lSS?
Recent work by Bernhard Haisch, Alfonso Rueda, H. Puthoff and others appears to offer a
radically different insight into the idea of mass. Their work suggests that inertia is a property
arising out of a "vast, all-pervasive electromagnetic field" - the zero point field, which exists
in the vacuum even at the temperature of absolute zero. The interaction between this
background «sea» of electromagnetic radiation, unifonn and isotropic, and the massless
electric charges immersed in it creates the appearance of mass. Mass does not exist. Only
massless electric charges and energy do. A stone is heavy because its great number of electric
charges is embedded in the zero point field and is being acted by it4.
5.
Mi~conepts
or mass
abou t the concep t
5. 1 The mass ofa body is "the amount of mat1er of the b dy"'.
It's an old misconception. It comes from the first definition ofNewton in the Principia. Today
we know that the amount of matter (better, the amount of substance) may be expressed in a
unit named mole (mol). If we have a given amount of a substance (0,5 mol of NaCl, for
example) its mass changes with its temperature and with the physical state (atoms tightly
bonded in a solid are less massive than the same atoms free).
5.2 The mass o f a parti cle depends on the ve locity of the particle
As we see, the mass of a particle can be defined by the equation
(~
r-(r
~
(in conventional units)
The energy of the particle depends on the inertial frame (then on the velocity). The momentum
also depends on the inertial frame. But the mass of the particle is an invariant, doesn't depend
on the inertial frame , that is, doesn't depend on its velocity.
Let's translate this into space-time language. A particle is at rest in an inertial frame. Its 4vector of energy and momentum points to the pure timelike direction, it has energy, no
momentum. Mass equals the energy. In an accelerator, the particle is put in motion at high
speed . The space component of the 4-vector, originally zero, grows to a great value (it
corresponds to momentum). The direction of the momenergy 4-vector changes (it tilts from
the vertical, the purely timelike direction). Its time component, that is, the energy of the
particle, also changes. Nevertheless, the magnitude of themomenergy 4-vector, that is, the
mass of particle doesn 't change.
5.3 A particle h& a re lativi ti mass: it measures the inertia of the particle wh en it mo es
at high ve locity.
The acceleration of a particle is
a=
F-(F.f3 ) /J
rm
-
Where F is the force acting on the particle and j3
v
=- .
c
We see that in the general case the acceleration doesn 't have the direction of the force and we
1 - When
F
= -=-
that gives the inertial mass. There are two exceptions:
F is perpendicular to
v , we can consider a "transverse mass" mt = y m
cannot get the expression m
2 - When F is parallel to
a
v, we can consider a "longitudinal mass" ml = I
m
We have two, and not only one expression where m depends on v. We cannot speak about a
property of the particle.
The so called "relativistic mass" is equal to the "transverse mass". This mass cannot measure
the inertia or resistance to the increase of velocity, because the force perpendicular to v
doesn't increase this velocity .
When we apply a force on a moving particle of a given rest mass, it appears to have more
resi stance to acceleration when its speed v increases because it takes more time to get a given
increase Cv. But it is an effect of time dilatation, not of mass increase. Measured by a clock
instantaneously travelling with the particle, the time (proper time) is always the same for the
same effect of the force. But, from our point of view, the time is dilated (laboratory time).
This is a kinematic effect, not a dynamic effect. The rest energy of the particle is the same.
Then, the mass is the same. 5
5.4 The ma s of a system is the sum of the masse o f the ir partic lc:s
A body is an agglomerate of particles. The mass of the body as being the sum of the mass of
its particles it' s a spontaneous thinking. All the classical Physics is based on this thinking.
In a weJl known paper «Does the inertia ofa body depend upon its energy-content?»6, Einstein
wrote these famous words:
"The mass of a body is a measure of its energy-content". A consequence of this sentence is:
if we have a system, a microwave oven, for example, with some food, we must consider, to
the mass of the oven, not only the masses of the particles (of the food and of the oven), but
also all the energy it contents, including the electromagnetic radiation used to cook or heat
the food. Then, the mass of the oven is more than the sum of the particles mass.
A consequence of our definitions presented before is : energy is additive; momentum is
additive; but mass is not additive. An example:
Let's consider two photons with the same energy, hv, moving in opposite directions: one of
them has the momentum p and the other has the momentum - p.
For each photon we have:
E (energy) = hv
p (magnitude of photon momentum)
=
hv Ie
As we see, photon is a massless particle, but it has energy.
For the system oltwo photons we have:
E (energy)
= hv + hv = 2 hv
p (magnitude of photon momentum) = hv Ie - hv Ie = 0
Interpretation: a photon has no rest energy, that is, no mass . However, a photon can contribute
with energy and momentum to a system of particles, then contribute to the mass oftbe system.
A system of zero-mass photons itself can have nonzero mass 6 .
5.5. The equivalence mass-energy means that the mass of a palti Ie and its energy are
esse ntially the same property.
The statement that mass and energy are equivalent doesn ' t mean that energy and mass are the
same. The energy of a particle (or an isolated system) is only the time component of a
momenergy 4-vector. Then it depends on the inertial frame from which the particle (or the
isolated system) is regarded . In contrast, the mass measures entire magnitude of that 4-vector.
Then is an invariant: it is independent of the inertial frame .
The equivalence of mass and energy for a particle refers to the rest energy of the particle, not
to its total energy. For a system refers to the energy of the system in the zero-total-momentum
frame. The correct relation between mass and energy for a particle is
Eo == me
2
and not
E == mc 2 or E 0 = moc 2 or E
= moc 2 7
5.6. In an iso lated system where the slim of the proper masses o f its particles decrease (or
increa e) there is conv r ion of mass into en rgy (or energy into mass)
It is not true! The mass of an isolated system remains unchanged by interactions between the
constituents of the system. As a matter of fact, the system mass is the magnitude of the total
momenergy. This is unaffected by collisions among the parts of the system or any
transformations these parts may undergo. As the mass of an isolated system is constant, it is
impossible to be converted in energy. On the other hand, energy of an isolated system is
constant. If it is constant, it is impossible to be converted in mass.
5.7 When a system transfers e nergy to the exterior, its mas is converted into energy
In the paper of Einstein we have refered before, we can read: "ifthe energy changes by L, the
mass changes in the same sense by Ll9 x 1020 "When a system transfers energy to the exterior,
its energy decrease, then its mass also decrease. Any changes in mass and energy occur in
parallel. There is no conversion of one into the other8 . If the energy of a body changes by
M o, then its mass (measures the inertia) changes in the same sense by flm = flEoIc 2•
When, for example, there is an expiosion, the total mass of the rest ofthe bomb, the expanding
gases, the fragments ofexplosion and the radiation has the same value as before the explosion .
What happens is merely a change in the makeup of mass or energy contents of the system:
less rest energy in the individual constituents - sum of their individual masses has decreased;
more kinetic energy, including kinetic energy of the photons and neutrinos produced. In a the
zero-total momentum frame and with the same units for mass and energy, given the
n
expression ms
n
n
= E s = L: E, = L: m + L: K; , the first term of the sum bas decreased and
i
,=1
i=1
the second telTI1 has increased but ms is constant.
5.8 The grav itationa l rna s is equal t 111 = E / c 2
An argument in favour of equation m = E / c 2 is this: it defines the gravitational mass, then
we need it to explain the gravitational attraction. It is not true. The gravitational attraction
between two relativistic bodies is determined by their energy-momentum tensors, not just by
their energies. On the other hand, the existence of a gravitational mass implies the existence
of a gravitational force with the sense of the origin of the gravitational field , what is not valid
in general relativity. As a matter of fact, when a relativistic particle such as a photon or an
electron travels with energy E and velocity v =fJc in the gravitational field created by a heavy
body of mass M, the force acting on the particle is given by the equation
F = -GM(E lc2)[F(1+1i 2)_p
(p .F)]
r3
g
When Ii «1 , this equation implies
_ -GM(EoIc )r
2
Fg =
As m = Eo I
c? we
r
3
obtain the classical equation.
However, when Ii == 1, the force is not directed to the origin of gravitational field. It has
a component along the velocity.
If, for example, we have a photon falling vertically towards Earth, we have
p ( iJ .F ) = F and er can think in E I c
2
Ii
=1 and
as m, but this is not anymore than a mathematical
variable necessary to use a classical expression. When the photon travels horizontally we
have to think in 2 x E I c2 as m. There is not a property of the photon called gravitational
mass, exclusively dependent on it.
References:
I. H. Benson, University Physics, John Wiley &Sons, Inc., 1991, p. 809
2. W. Gettys; F. Keller; M. Skove, Physics, McGraw-Hill book Company, 1989, p. 894
3. E. Taylor; J. Wheeler, Spacetime Physics, W.H . Freeman and Company, 1992, p. 203
4. B. Haisch; A. Rueda; H. Puthoff, Beyond E
=
mc2, The Sciences, november/december 1994
5. C. Adler, Does mass really depends on velocity, dad, American Journal of Physics, 55, 8,
Aug. 1987, p. 741
6. The same as 3., p. 232
7. Okun, The concept ofmass, Physics Today, June 1989, p. 31
8. R. Baierlein, Teaching E = m c 2 , The Physics Teacher, March 1991
Other bibliography recommended:
Einstein, Lorentz, Weyl, Minkowsky, The principle of Relativity, a Collection of Original
papers on the theory of Relativity, Notes by A. Sommerfeld, Dover Publications, Inc., first
publish. in 1952
R. Brehme, The Advantage of Teaching Relativity with Four- Vectors , American Journal of
Physics, Vol. 36, N°] 0, October 1968
M. Jammer, Concepts of Mass in classical and modern Physics, Harvard University Press,
Cambridge, 1961
A. Miller, On Einstein, light quanta, radiation, and relativity in 1905, American, Journal of
Physics, Volume 44, n° 10, October 1976
J . Cushing, Electromagnetic mass, relativity, and the Kaufmann experiments, vol. 49, n° 12,
december 1981
1. Lapidus, The Falling Body Problem in General Relativity, American, Journal of Physics,
Volume 40, October 1972
Jorge Valadares