Weight as a contact force %\LaTeXe %
Jos\'{e} M. L. Figueiredo
In 1901 the third Conf\'{e}rence G\'{e}n\'{e}rale des Poids et Mesures (CGPM) defined the weight of a body as ``the product of its mass and the acceleration due to gravity". In practical terms the weight force corresponds to the gravitational force. However, this gravitational definition of the weight lacks logic from the perspective of present knowledge and can be misleading. In a space traveling time, weight associated concepts such as ``true weight", ``apparent weight", ``weightlessness", ``zero gravity", ``microgravity", are ambiguous and strong deceptive. The superfluousness or ambiguousness of these concept would be removed if an operational definition of the weight of a body as the force the body exerts on its support or suspender is adopted. In this definition the weight force does not act on the body and it is not necessary to describe the body's sate of motion. This avoids the imprecision of the concepts such as ``true weight" and ``apparent weight" of a body, for example. This paper discusses the need of the weight force of a body concept in Physics, and asks for a re-exame of the CGPM definition to improve the physics teaching and learning. %Furthermore, because the concept of weight is not fundamental in Physics as opposed to the %gravitational force it is believed its omission will improve the physics teaching and learning. %adopting the above-mentioned %operational definition. It is believed this will boost student's understanding of the weight force %associated subjects.
FigueiredoJ-4-en.tex
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\title{Do we really need the weight force?}
% Force line breaks with \\

\author{Jos\'{e} M. L. Figueiredo}
 \email{jlongras@ualg.pt}
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%\altaffiliation[Also at ]{Physics Department}%Lines break automatically or can be forced with \\
\affiliation{Departamento de F\'{\i}sica da Faculdade de Ci\^{e}ncias e Tecnologia da Universidade
do Algarve, Campus de Gambelas, 8005-139 FARO, Portugal
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\date{\today}% It is always \today, today,  but any date may be explicitly specified

\begin{abstract}
In 1901 the third Conf\'{e}rence G\'{e}n\'{e}rale des Poids et Mesures (CGPM) defined the weight of
a body as ``the product of its mass and the acceleration due to gravity". In practical terms the
weight force corresponds to the gravitational force. However, this gravitational definition of the
weight lacks logic from the perspective of present knowledge and can be misleading. In a space
traveling time, weight associated concepts such as ``true weight", ``apparent weight",
``weightlessness", ``zero gravity", ``microgravity", are ambiguous and strong deceptive. The
superfluousness or ambiguousness of these concept would be removed if an operational definition of
the weight of a body as \textbf{the force the body exerts on its support or suspender} is adopted.
In this definition the weight force does not act on the body and it is not necessary to describe
the body's sate of motion. This avoids the imprecision of the concepts such as ``true weight" and
``apparent weight" of a body, for example. This paper discusses the need of the weight force of a
body concept in Physics, and asks for a re-exame of the CGPM definition to improve the physics
teaching and learning.

%Furthermore, because the concept of weight is not fundamental in Physics as opposed to the
%gravitational force it is believed its omission will improve the physics teaching and learning.

%adopting the above-mentioned
%operational definition. It is believed this will boost student's understanding of the weight force
%associated subjects.
\end{abstract}

\pacs{01.40.-d, 01.40.Fk, 01.40.Gm}% PACS, the Physics and Astronomy Classification Scheme.

%\keywords{Suggested keywords}%Use showkeys class option if keyword display desired

\maketitle

\section{Introduction}

Everyone who teaches introductory physics is used to deal with the confusion students experience
when first confronted with the differences between concepts of mass and weight, and with the
ambiguities of the notions related to the weight definition based on the gravitational force such
as ``true weight", ``apparent weight", ``weightlessness", ``zero-gravity", and ``vertical". Several
studies have shown college and first year university students still have some misconception about
the basic physics related to the meaning of weight and have difficulties applying it in
imponderability or accelerated environments \cite{Galili}\cite{GurelAcar}\cite{Sharma}. This is
evident when they are asked to explain what happens if some everyday events such as walking were to
take place in imponderability. The students often think ``microgravity", ``zero gravity", or
``weightlessness" situations refer to events occurring outside the Earth's or other celestial
body's gravitational influence. They are surprised to find out that during a typical shuttle flight
mission in an orbit at an altitude of 400 km the gravitational force of the Earth is only 12\% less
than at the Earth's surface.
%Particularly, few students do recognize that it is the free fall movement of the spacecrafts that
%creates the seemingly weightlessness environment in the vehicle and any body orbiting the Earth or
%another celestial body is in state of continuous free fall.

In the literature there are several definitions of weight based on the gravitational force: ``the
weight is the Earth gravitational force" \cite{Serway}; ``the force exerted by the Earth on an
object is called the weight of the object" \cite{Serway}; ``the weight of a body is the total
gravitational force exerted on the body by all other bodies in the universe" \cite{YoungFreedman}.
Frequently, the weight is considered a fundamental property of matter under the influence of a
gravitational field. Furthermore, in 1901 the Conf\'{e}rence G\'{e}n\'{e}rale des Poids et Mesures
declared ``the weight of a body is the product of its mass and the acceleration due to gravity"
\cite{3CGPM}.

These gravitational based definitions of weight are widely used in despite of the fact they are not
entirely satisfactory at the present knowledge and the number of ambiguities associated to the
gravitational definition and on the meaning of weight and weight-related concepts are even
misleading, leading to several misconceptions which can contribute to widening the gap between what
is taught and what is learned by the students.

Very few authors adopt the alternative operational definition of weight of a body as ``the force
which a body exerts on its support or suspender that prevents its free fall" \cite{MarionHornyak}.
This definition is in agreement with our daily experience and with the present knowledge. Following
this definition the weight of a body is a force that results always from the direct contact of the
body with other body, i.e., the weight force is a contact force.

This paper discusses weight, microgravity, weightlessness, vertical, up and down concepts, and the
necessity of reviewing the weight concept/definition together with the advantages of the adoption
of the operational definition of weight or its abandonment.

\section{Acceleration due to the gravity}

In the frame of the classical physics, the force of gravity is a long-range force, and, as far we
know, cannot be shielded \cite{gravitationalshielding}. In practical situations it is independent
of the state of motion of the objects. The acceleration due to the gravity corresponds to the
acceleration of the motion of a body as a result of the gravitational force, and in a given instant
equals the ratio of the gravitational force and the body amount of matter.

Accordingly the General Theory of Relativity the gravity corresponds to a modification (curvature)
in the space-time continuum caused by a concentration of mass or energy, that is, the space-time
geodesics surrounding substantial mass or energy are curved lines and the bodies go through some
form of curved orbital path.

\subsection{Gravity under newtonian physics}\label{gravity-newton}

Since Isaac Newton presented the law of Universal Gravitation it is well accepted that the
gravitational interaction is universal and depends only on the body's quantity of matter and the
distance between their centers of mass. Following the works of Kepler and Galileu, Newton concluded
that the Earth's force of gravity \(\vec{F}_{g}\) exerted on our bodies or other mass \(m\) owing
to their gravitational interaction is given by
\begin{equation}\label{FgTm}
\vec{F}_{g}=-\frac{GMm}{|\vec{r}|^{3}}\vec{r}=\vec{\Gamma}m,
\end{equation}
where \(G\) is the gravitational constant (6.67\(\times10^{-11}\) N m\(^{2}\)kg\(^{-2}\)), \(M\) is
the Earth's mass, \(\vec{r}\) is the position vector of the body center of mass relatively to the
Earth center of mass. The gravitational force the Earth exerts on the body \(\vec{F}_{g}\) can be
written as the product of the body's mass and the local acceleration due to the gravity,
\(\vec{F}_{g}=m\vec{g}\); the vector \(\vec{g}=-\frac{GM}{|\vec{r}|^{3}}\vec{r}\) corresponds to
the body's acceleration due to the Earth gravitational field \(\vec{\Gamma}\), and its magnitude is
approximately equal to 9,8 m s\(^{-2}\) at sea level. The intensity of the force of gravity can be
measured with the aid of a dynamometer (or a spring scale), provided that the body and the
dynamometer are at rest relatively to the Earth.

Let us consider a body at rest on the surface of the Earth at a given latitude, Fig. \ref{Fig1}.
The body is acted upon by two forces: the force of gravity \(\vec{F}_{g}\) pointing towards the
center of the Earth and the force of the reaction of the Earth's surface (the support reaction
force) \(\vec{N}\), whose direction is determined not only by the force of gravity, but also by the
spinning of the Earth around its axis. Accordingly the second law of dynamics
(\(\vec{F}=d(m\vec{v})/dt\), where \(\vec{v}\) is the velocity of the mass \(m\)), the resultant
force \(\vec{F}\) of these two forces ensures the daily rotation of the body along the local
parallel. As a consequence the directions of \(\vec{N}\) and \(\vec{F}_{g}\) (given by equation
\ref{FgTm}) do not coincide. The direction of the measured \(\vec{F}_{g}\) and \(\vec{g}\) (so the
direction of \(\vec{N}\)) differs from the direction towards the center of the Earth given by
equation \ref{FgTm}, except at the poles and equator, by an angle whose the maximum amplitude is
less than 0.1\(^{0}\). In addition with the exception at the poles, a scale or a dynamometer
measures less than the gravitational force given by equation \ref{FgTm} because the net force
needed to provide the centripetal acceleration necessary to ensures the body keeps up with the
daily rotation of the Earth: assuming a spherically symmetric and homogenous Earth the sensed
acceleration of gravity is about 0.03 m s\(^{-2}\) (0.35\% of \(g\) given by
\(\vec{g}=-\frac{GM}{|\vec{r}|^{3}}\vec{r}\)) less at the equator than at the poles. Furthermore,
the variation of density and the surface irregularities of the Earth give rise to local changes in
the gravitacional field and to the vector \(\vec{g}\).

Nevertheless, throughout the rest of the text we will consider the Earth as an homogenous sphere
and the effects of its rotation around its axis and the translation around the Sun or other motions
will be neglected because the values of the linear and the angular accelerations acquire by a body
due to these effect are very small when compared with the acceleration due to the gravity. For
simplicity, the Earth will be considered a frame of reference at rest during the characteristic
time of the phenomena analyzed here. The effect of the atmosphere and the gravitational influences
of other celestial bodies are also neglected.

\begin{figure}[hbt]
\begin{center}
\includegraphics{Fig1.eps}
%\epsfbox{Fig1.eps}
\end{center}
\caption{\label{Fig1} Forces acting on a body at the Earth's surface for a given latitude:
\(\vec{F}\) is the net force due to the force of gravity \(\vec{F}_{g}\) acting on the body and to
the reaction of the Earth's surface \(\vec{N}\) acting on the body. \(\vec{N}\) and \(\vec{F}_{g}\)
do not constitute an action-reaction pair because they are applied to the some body.}
\end{figure}

\subsection{Gravity under the theory of general relativity}

The General Theory of Relativity addresses the problems of gravity and that of nonuniform or
accelerated motion. In one of his famous conceptual experiments Einstein concluded that it is not
possible to distinguish between a frame of reference at rest in a gravitational field and an
accelerated frame of reference in the absence of a significant gravitational field:
%An observer in a closed space capsule who found himself pressing down on his seat could not tell
%whether he and the capsule were at rest in a gravitational field, or whether they (he and the
%capsule) were undergoing acceleration far enough from a significant gravitational field -
Einstein's principle of equivalence. From this principle of equivalence Einstein moved to a
geometric interpretation of gravitation: the presence of very large mass or a high concentration of
energy causes a local curvature in the space-time continuum. The space-time geodesics become curved
lines, that is, the space-time curvature is such that the body paths are no longer straight lines
but some form of curved orbital paths. Maintaining the classical view of the gravitation we can
associate to the body's curved path motion a \emph{centripetal acceleration} that is referred as
the acceleration due to gravity.

\section{Operational definition of the force weight of a body}

What humans and matter experience as weight is not the force of gravity. What they experience as
weight is actually the consequence of the reaction of the ground (or whatever surface they are in
contact with or hang up) pushing ``upwards" against them to counteract the force they are exerting
on the surface - \emph{the weight force of the body}. A good evidence of this is given by the fact
that a person standing on a scale moving up and down on his toes does see the indicator moving,
telling that the measured force is changing while the gravity force, that depends only on the
person's and the Earth' masses and the distance between their centers of mass, does not vary to
induce such clear observable changes on the scale meter. Another evidence happens when going
towards the Earth surface in an elevator: one experiences a greater strain in the legs and feet
when the elevator is stopping than when it is stationary or moving with constant velocity because
the floor is pushing up harder on the feet.

\subsection{Weight of a body at rest}

Consider a body at rest on the surface of the Earth, Fig. \ref{Fig2}a \cite{MarionHornyak}. In this
situation the body experiences a force \(\vec{F}_{g}\) due to gravitational pull of the Earth. The
reaction force to this force is \(-\vec{F}_{g}\) and corresponds to the gravitational force exerted
on the Earth by the body. The force pair \(\vec{F}_{g}\) and \(-\vec{F}_{g}\) constitutes an
action-reaction pair, and consists of the force \(\vec{F}_{g}\) that acts on the body and the force
\(-\vec{F}_{g}\) that acts on the Earth.

The tendency of the body to accelerate towards the center of the Earth due to \(\vec{F}_{g}\) must
give rise to a force \(\vec{P}\) acting on the Earth surface: \(\vec{P}\) is the force exerted by
the body on the Earth surface, Fig. \ref{Fig2}b. If the body exerts on the Earth surface a force
\(\vec{P}\), the Earth solid surface reacts exerting a force \(\vec{N}\) on the body that balances
the force \(\vec{P}\), Fig. \ref{Fig2}b. The force \(\vec{N}\) is called the normal force and is
the reaction to \(\vec{P}\) and we have \(|\vec{N}|=|\vec{P}|\). The action \(\vec{P}\) the body
exerts on the Earth (or other body) surface corresponds to the force \textbf{weight of the body}
(that is the operational definition of weight).

Hence the body experiences no acceleration (it is at rest), the net force due to the two forces
acting on the body, \(\vec{F}_{g}\) towards the center of the Earth and \(\vec{N}\) outwards, is
null (Newton's second law of dynamics). Therefore, \(\vec{F}_{g}\) and \(\vec{N}\) are equal in
magnitude, opposite in orientation and have different application points. Although in this case
\(|\vec{N}|=|\vec{F}_{g}|\) the normal force \(\vec{N}\) is not the reaction to the gravitational
force \(\vec{F}_{g}\) because this two forces act on the body (as said previously \(\vec{N}\) is
the reaction to \(\vec{P}\)). Similarly, because the Earth experiences no acceleration there are
two equal and directly opposite forces acting on the Earth, \(-\vec{F}_{g}\) applied on the Earth's
center of mass and \(\vec{P}\) applied on the Earth surface in contact with the body, with
\(|\vec{P}|=|-\vec{F}_{g}|\).

\begin{figure}[hbt]
\begin{center}
\includegraphics{Fig2ab.eps}
%\epsfbox{Fig2ab.eps}
\end{center}
\caption{\label{Fig2}a) \(\vec{F}_{g}\) and \(-\vec{F}_{g}\) is the action-reaction force pair due
to the gravitational interaction between the body and the Earth. b) \(\vec{P}\) and \(\vec{N}\) is
the action-reaction force pair due to the interaction between the body and the Earth surfaces:
\(\vec{P}\) is action of the body on the support (the weight force of the body) and \(\vec{N}\) is
the reaction of the support's surface to the body's action \(\vec{P}\) on the support surface.}
\end{figure}

Consider now the body is placed on (or hung on) a dynamometer-scale. When the body is placed on the
scale platform the dynamometer spring is compressed or extended (depending on the scale) and its
deformation is communicated to a calibrated dial read out. The body exerts an action \(\vec{P}\) on
the scale platform and through it on the spring. The scale dial reads the magnitude of the force
\(\vec{P}\) exerted \textbf{by} the body surface \textbf{on} the scale platform. By Newton's third
law of dynamics the scale platform reacts exerting a opposing force \(\vec{N}\) \textbf{on} the
body surface: both forces have the same magnitude, \(|\vec{P}|=|\vec{N}|\), and direction but
opposite orientations. It is the force  \(\vec{N}\) that prevents the  body free fall towards the
center of the Earth. The weight force is applied not to the body being considered itself, but to
the scale platform.

If the scale \textbf{is at rest} relatively to the Earth then, as previously, the magnitude of the
reaction force the support exerts on the body %action of the body on its support, the weight force magnitude,
equals the magnitude of the force of gravity acting on the body. Hence \(|\vec{N}|=|\vec{F}_{g}|\)
it results that \(|\vec{P}|=|\vec{F}_{g}|\). The weight force and the force of gravity magnitudes
are also equal \textbf{in the case of uniform and rectilinear motion} of the scale and the body in
a reference frame associated with the Earth.

What happens when the scale and body are accelerating in relation to a frame of reference on the
Earth? This is the case, for example, of an elevator during stopping or starting. What is the scale
reading in these situations? During a sudden change in the support motion (on starting or braking,
for example) does remain valid the equality \(|\vec{P}|=|\vec{F}_{g}|\)?

\subsection{Weight of a body in an accelerated frame of reference}

Consider that a body of mass \(m\) is standing on a bathroom-type scale fixed in the floor of an
elevator. When the elevator moves with an acceleration \(\vec{a}\), in accordance with Newton's
second law of dynamics and as long as the body and the scale surfaces are in contact, the body
moves together with the elevator and scale with the acceleration \(\vec{a}\) under the action of
two forces: the force of gravity \(\vec{F}_{g}\) and the scale surface reaction force \(\vec{N}\)
due to the body surface action \(\vec{P}\) on the scale. From Newton's second law of dynamics,
\(\vec{F}=d(m\vec{v})/dt\), and assuming the body's mass does not vary, the resultant of the forces
acting on the body must equal the product of its mass by its acceleration, which is the elevator
acceleration \(\vec{a}=d\vec{v}/dt\), that is,
\begin{equation}\label{000}
    m\vec{a}=\vec{F}_{g}+\vec{N}.
\end{equation}
Since the body's weight \(\vec{P}\) and the reaction force of the scale platform \(\vec{N}\)
constitutes an action-reaction pair, `\(\vec{P}=-\vec{N}\)�, equation \ref{000} can be rewritten as
\begin{equation}\label{00}
    \vec{P}=m\vec{g}-m\vec{a}.
\end{equation}
It is important to bear in mind that in the operational definition the \emph{weight of the body}
\(\vec{P}\) corresponds to the body's action force on the support, that is, the weight force of a
body does not act on the body, Fig. \ref{Fig2}b. \textbf{As a consequence to describe the motion of
a body we do not have to consider the weight force of the body.}

Equation \ref{00} allows us to determine the action a body exerts (the weight force of the body) on
its support or suspender. Depending on the orientation of the acceleration \(\vec{a}\), several
situations may occur \cite{MarionHornyak}\cite{FigueiredoJ-AJP}: i) \(|\vec{P}|<mg\); ii)
\(|\vec{P}|=0\); iii) \(|\vec{P}|>mg\). A classical case of study is the movement of an elevator
with a passenger standing on a dynamometer scale (see for example
\cite{MarionHornyak})\cite{FigueiredoJ-AJP}.

\subsection{Notions of vertical, upward and downward}

Currently, the vertical is defined as the direction of the plumb line that at the Earth surface and
at rest or on uniforme and rectilineal motion coincides with the direction of the gravity force
adopting the consideration of section \ref{gravity-newton}. However, a human being or other living
creature feels equilibrated in the direction of its weight force, and the vertical direction and
downward orientation correspond to the direction and to the orientation of the weight force,
respectively. From equation \ref{00} one can conclude the vertical and the up and down orientations
depend essentially on the body state of motion characteristics. Contrary to what is many times
stated the notions of vertical and up/down are not determined uniquely by the gravitational force.
The vertical is always the direction of the weight of the body and ``down" corresponds to the
weight force orientation. In accordance equation \ref{00}, to stay in equilibrium during a bus
starting movement we stoop forward and when it starts stopping we lean backwards. In these
situations our vertical is oblique and to not lose one's balance we align with the new vertical
defined by the new direction of \(\vec{P}\).

\section{The weight force in a weak gravitational field}\label{p-verus-g}

The body's action force, the weight force, ``appears" whenever the body's surface is constrained to
interact directly with the surface of another body. The weight of the body is opposite to the
normal force (reaction force) exerted by the surface where the body stands on or is in contact
with, which prevents it from moving through or away of the other body surface. The body action
force or its absence (weightlessness) does not depende of the existence of a gravitational field in
the region of the space where it is staying. Considere a spaceship in a region of the Universe
where the gravitational field is very small. The bodies in the interior of a spaceship traveling in
this region with uniforme and rectilinear motion would experience zero weight, because they are in
a situation of effective zero-gravity  and with uniforme and rectilinear motion. Any spring-scale
in contact and moving with them measures no weight because the objects are not constrained to
contact their surfaces to originate the normal forces (zero action - zero weight - gives rise to no
normal force).

Lets now considere the spaceship turns on its engines. In the case of a spacecraft accelerating by
firing its rockets the thrust force is applied to the back end of the rocket by the gas escaping
out of the back. The rockets thrust force is transferred to each object in the spaceship through
either pressure or tension giving rise to the bodies action (weight force) on their supports or
suspenders and the bodies in the interior of the veicule do not experience weightlessness. We can
conclude that the weight force in fact does not depend on the presence of a gravitational field.
Indeed, according to Einstein's Principle of Equivalence the bodies in a space veicule with an
acceleration \(\vec{a}\) in the absence of a gravitational field behave as the spaceship was at
rest or with constant velocity in a gravitational field with acceleration due to gravity
\(\vec{g}=-\vec{a}\).

From equation \ref{00} results that if the spaceship is accelerating uniformly out of the influence
of a significant gravitational field, that is, \(|\vec{g}|<<|\vec{a}|\), the weight of a body in
the vehicle is
\begin{equation}\label{0000}
    \vec{P}=-m\vec{a},
\end{equation}
that is, the weight force is opposite to the net force acting on the body and it is equal to the
product of the body's acceleration and mass, \(m\vec{a}\). Taking in account the considerations
made and the equation \ref{0000} the weight of a body or the weightlessness state has nothing to do
whether the body is under the influence of a gravitational field or not. In conclusion, the forces
the bodies exert on their supports (weight forces) or their absence do not require the presence of
a gravitational field. In the case of the presence of a gravitational field the force the bodies
exert on their supports depends also on characteristics of their relative motion.

\section{Imponderability and microgravity}\label{imp-micro}
%\section{Weight force versus gravity force}\label{acceleratedframe}

In free fall, that is, when \(\vec{g}=\vec{a}\) all parts of an object accelerate uniformly and
thus a human or other body would experience no weight, assuming that there are no \emph{tidal
forces} \footnote{The tidal force are secondary effects of the forces of gravity due to Earth
inhomogeneities and to the other celestial objects gravity.}. The experience of no weight, by
people and objects, is known as imponderability, weightlessness or \emph{zero gravity}, although
\emph{micro-gravity} is often used to describe such a condition. Excluding spaceflight (orbital
flight), weightlessness can be experienced safely only briefly, around 30 seconds, in an airplane
following a ballistic parabolic path. In spaceships the state of imponderabilidade or
weightlessness can be experienced for extended periods of time if the ship is outside the Earth's
or other planet's atmosphere and as long as no propulsion is applied and the veicule is not
rotating about its axis because the bodies in its interior are not constrained to be in contact
with other bodies or with the station walls or floors. In particularly, the astronauts are not
pulled against the station pavement and therefore their bodies actions on the surface of the
station are null. In real free fall situations the tidal effects of the gravity on the bodies,
although small, are equivalente to a small acceleration  and the bodies are said to be in a
``microgravity" environment because the weightlessness sensation is not complete.

The sate of imponderability experienced in orbiting spacecrafts is not as consequence of the small
value of the acceleration due to the gravity because the distance from the Earth.
%In fact, during free fall all parts of an object accelerate uniformly and the human beings and
%other bodies experience the sensation of full imponderability as long as there are no tidal forces.
Weightlessness is a consequence of the body and the spaceship accelerations to be equal and only
due to gravity. The gravity acts directly on a person and other masses just like on the vehicle and
the person and the floor are not constrained toward each other. On the contrary, contact forces
like atmospheric drag and rocket thrust first act on the vehicle, and through the vehicle on the
person. As a consequence of this contact forces the person and the floor are pushed toward each
other giving rise to the weight force.

As mentioned the term microgravity is usually used instead of weightlessness to refer the
environment within orbiting spacecraft. The use of the term microgravity without specifying its
exact meaning can strengthen the misconceptions associated to weight and gravitational force
because the term ``micro" could lead to the idea that acceleration due to gravity is very small
because the distance from Earth. To the contrary, the acceleration of the gravity due to the Earth
gravitational interaction is around 8.4 m s\(^{-2}\) at 400 km of altitude (88.8\% of it value at
the Earth's surface). Even its value at the distance of the Moon orbit is 2.63\(\times10^{-3}\) m
s\(^{-2}\), although in these regions the acceleration due to Sun's gravity is near twice this
value (\(\approx5.8\times10^{-3}\) m s\(^{-2}\)). True Earth's microgravity,
\(g\simeq1\times10^{-5}\) m s\(^{-2}\), can be only experienced at locations away from the Earth as
far as almost 17 times the Earth-Moon distance.

The term microgravity is used by the scientists to characterize the residual acceleration
experienced by the bodies in the interior of the spacecraft as a consequence of forces between the
bodies within the spaceship, the body and the spacecraft, the gravitational tidal forces and the
atmosphere dragging force. These forces induce in the bodies acceleration of intensities of some
\(\mu\)m s\(^{-2}\), giving rise to the use of the term ``microgravity". For uncrewed spaceships
free falling near the Earth it is possible to obtain 1 \(\mu g\); for crewed missions is difficult
to achieve less than 100 \(\mu g\) \footnote{``\emph{Falling upwards: how to create microgravity},"
http://w\newline ww.esa.int/esaHS/ESATRRVRXLC\underline{ }research\underline{ }0.html/; ``\emph{The
Mathematics of Microgravity}," http://exploration\newline .grc.nasa.gov/combustion/pdf\underline{
}files/math\underline{ }of\underline{ }micro.pdf.}. The main reasons are: i) the morphology of the
Earth induces local gravitational variations; ii) the gravitational effects of the other celestial
bodies, especially the Moon and the Sun, which depend on their relative position to the Earth; iii)
the acceleration due to gravity decreases one part per million for every 3 m increase in altitude
(in an orbiting spaceship the required centripetal force and hence the acceleration due to gravity
is higher at the far side than at the nearest side of the ship relatively to the Earth); iv)
although very thin, at for example 400 km of altitude, the atmosphere gradually slows the
spacecraft.

\section{Living in microgravity environment }\label{imp-micro}

The term microgravity is more appropriate than ``zero weight" or ``zero-gravity" in the case of
orbiting spacecrafts because weightlessness is not perfect. The term microgravity does not mean the
acceleration due to gravity was strongly reduced but solely that its effects on the bodies within
the vehicle were substantially reduced.

As already mentioned the bodies in the interior of a spacecraft orbiting a celestial body, such as
the International Space station (ISS) around the Earth, are in a state of imponderability
experiencing weightlessness as they do not exert any contact action on the other bodies because all
the bodies are subjected only to the gravity force being pulled towards the Earth with the same
acceleration, the acceleration due to gravity. The weightlessness present several challenges to the
human organism which is prepared to live in a gravity environment and also makes several of the
mundane human actions, such as walk, virtual impossible.

Because in the interior of the station there are no upward and downward convection currents of
particles and gases that leads to several effects on the human breathing system. The weightlessness
also interferes with cardiovascular system with the heart beating faster because there is less
resistance to the blood flow. This can also lead to the muscles atrophy, blood pump system
malfunction and difficult breathing.

It is not possible to walk in weightlessness environment because the astronauts feet are not
constrained to the station pavement and their feet action (weight force) on the surface of the
station is null. Hence there is no normal reaction force therefore the friction force is zero. At
the Earth surface it is the friction force between the pavement and the astronauts feet that gives
rise to the reaction force needed to walk.

Several plans have been proposed to create ``artificial gravity" in orbiting devices. The most
popular plan to produce ``artificial gravity" in vehicles designed to remain in orbit or stay in
out space for a long period of time are to set the spaceship into rotation with an angular velocity
\(\omega\) around its central axis. The bodies at any point distant by \(r\) from the rotation axis
will experience a centripetal acceleration \(a=\omega^{2}r\). The weight of the bodies on the outer
rim of the spaceship with radius \(r\) opposes the centripetal force.  The weight force intensity
is then given by \(P=m\omega^{2}r\).

Although this process could be used to simulate gravity it wouldn't be exactly the same. One
problem is that across the radius of a spaceship, \(g\) levels change rapidly, and different parts
of a human body will feel considerable distinct acceleration levels. That is not quite what we
experience in Earth's gravitational field.

\section{Conclusion}

The identification of weight force as the force of gravity is misleading and lacks logic from the
perspective of the present knowledge. In the operational definition discussed the weight of a body
is the action force the body exerts on the surface of another body that it is in contact with, and
depends on their relative motion. The current meaning of weight is in fact the gravitational force
and to describe the motion of a body we do not need to consider the weight force if operational
definition is adopted. In fact, having in mind  the concept of weight is not fundamental the
Physics teaching and learning would benefit if the use of the vocable weight is avoided. One
advantage would be to get rid of the common sense identification between mass and weight force
concepts in the class room.
%Bearing in mind that this does not seem executable in the near future
It is expected that this analysis will motivate physics scientific community, as well as
instructors and authors, to contribute to the review of the concept of weight either adopting the
operational definition or considering to lay it aside.

\section*{Acknowledgments}

The author is grateful to Professor Robertus Potting, Dr. Paulo S\'{a}, Dr. Jos\'{e} Rodrigues, and
Dr. Alexandre Laugier for their comments and manuscript revision. The author acknowledges the
improvements that resulted from further discussions with other colleagues.

\begin{thebibliography}{}\label{sec:TeXbooks}

\bibitem[IG(2001)]{Galili} I. Galili, ``\emph{Weight versus gravitational force: historical and
educational perspectives}," Int. J. Sci. Educ. \textbf{23}, 1073-1093 (2001).

\bibitem[GA(2003)]{GurelAcar} Z. Gurel and H. Acar, ``\emph{Research into Students' Views About Basic Physics Principles in a Weightless Environment},"
The Astronomy Education Review \textbf{2}, 65-81 (2003).

\bibitem[Sharma(2004)]{Sharma}M. D. Sharma, R. M. Millar, A. Smith, and I. M. Sefton, ``\emph{Students' Understandings of Gravity in an Orbiting
Space-Ship}," Research in Science Education \textbf{34}, 267-289 (2004).

%\bibitem[NTetc(1992)]{Niedderer} H. Niedderer el all, ``\emph{Epistemological Beliefs of Students in High School Physics},"
%NARST Annual Meeting in Boston, March 1992 (1992).

\bibitem[RAS(2000)]{Serway} R. A. Serway, \emph{Physics for Scientists and Engineers with
modern physics} (Saunders College Publishing, Philadelphia, CA, 1996) page 113.

\bibitem[YF(2000)]{YoungFreedman} H. D. Young and R. A. Freedman, \emph{University Physics with
modern physics} (Addison-Wesley, Reading, MA, 2000) page 362.

\bibitem[3CGPM(1901)]{3CGPM} R�solution de la 3\(^{e}\) CGPM,
``\emph{D�claration relative � l'unit� de masse et � la d�finition du poids; valeur conventionnelle
de \(g_{n}\)}," Comptes rendus de la 3\(^{e}\) CGPM, 70 (1901).

\bibitem[MH(2000)]{MarionHornyak} J. B. Marion and W. F. Hornyak, \emph{General Physics with
Bioscience Essays} (John Wiley \& Sons, New York, NY, 1985).

\bibitem[GS(2001)]{gravitationalshielding} T. Van Flandern e X. S. Yang, ``\emph{Allais gravity and pendulum effects during
solar eclipses explained}," Phys. Rev D \textbf{67}, 022002 (2003); C. S. Unnikrishnan, A. K.
Mohapatra, e G. T. Gillies, ``\emph{Anomalous gravity data during the 1997 total solar eclipse do
not support the hypothesis of gravitational shielding}," Phys. Rev D \textbf{63}, 062002 (2001).

\bibitem[JF(200t)]{FigueiredoJ-AJP} J. M. L. Figueiredo,
``\emph{Advantages of adopting the operational definition of weight},"
http://arxiv.org/abs/physics/0505193.


% \bibitem[RU(2000)]{Russo} B. M. Yavorsky and Yu. A. Seleznev, \emph{Physics: A Refresher Course} (Mir Publishers, Moscow, 1979) page 65.

% \bibitem[RB(1999)]{RBrown} Ronald Brown, ``\emph{Weight � Don't use the word at all},"
% The Physics Teacher \textbf{37}, 241 (1999).

% \bibitem[RCM(1999)]{RCM} R. C. Morrison, ``\emph{Weight and gravity � the need for consistent definitions},"
% The Physics Teacher \textbf{37}, 51-52 (1999).

\end{thebibliography}

\end{document}

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\title[The concept of the weight: the operational definition as a contact force]
%{Weight as a contact force: operational versus gravitational definition}
{Precis�mos mesmo da for�a peso?}
%{O conceito de ``peso do corpo" n�o � necess�rio � F�sica}

%{Ser� o conceito de ``peso" necess�rio ao ensino de F�sica?}

% The importance of clarifying the difficulties associated with concept of weight
% Advantages of adopting the operational definition of
% Advantages of adopting the operational definition of Weight

%\title[Weight as a contact force]{Weight as a contact force %\LaTeXe %}
%\title[Weight as a contact force]{Weight as a contact force:  Concepts of weight, force of gravity, imponderability, zero gravity, and microgravity %\LaTeXe}

\author{Jos\'{e} M. L. Figueiredo\footnote{E-mail: jlongras@ualg.pt}}
%\author{Jos\'{e} M. L. Figueiredo}\footnote[]{E-mail: jlongras@ualg.pt}}
\address{Departamento de F\'{\i}sica da Faculdade de Ci\^{e}ncias e Tecnologia da Universidade do Algarve, Campus de Gambelas, 8005-139 FARO, Portugal}

\begin{abstract}

Com raras excep��es, nos manuais de F�sica para os ensinos secund�rio e universit�rio e em muitas
publica��es cient�ficas, a for�a ``peso de um corpo" � definida como a for�a da gravidade ou uma
consequ�ncia exclusiva desta. %, em conson�ncia com a defini��o da Confer�ncia Geral de ``peso"s e Medidas de 1901.
Estas defini��es carecem de l�gica do ponto de vista do conhecimento actual e podem ser mesmo
enganadoras para o estudante.  A defini��o operacional da for�a ``peso do corpo" como a for�a que o
corpo exerce no seu suporte tem a vantagem de eliminar a necessidade de v�rios conceitos associados
� defini��o gravitacional ou, pelo menos, a ambiguidade de alguns deles. Neste texto discute-se o
conceito da for�a ``peso" e apresentam-se exemplos das vantagens da sua redefini��o e, em
particular, do abandono da express�o ``peso do corpo" uma vez que a no��o de for�a ``peso do corpo"
n�o � fundamental em F�sica.

%Acredita-se que a adop��o da defini��o operacional de ``peso" permite corrigir as dificuldade
%associadas aos conceitos relacionados com o ``peso".

\end{abstract}
\date{\today}% It is always \today, today,
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%\pacs{01.40.-d, 01.40.Fk, 01.40.Gm}

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\maketitle

\section{Introdu��o}

Quem lecciona disciplinas introdut�rias de F�sica est� acostumado a lidar com a confus�o que os
estudantes fazem quando confrontados com os conceitos de massa e de for�a ``peso", e com as
ambiguidades das no��es associadas � defini��o da for�a ``peso do corpo" baseada na for�a da
gravidade. V�rios estudos indicam que os estudantes do secund�rio e da universidade mostram
dificuldades em aplicar os conceitos relacionados com a for�a ``peso" em situa��es de
imponderabilidade  e em sistemas com movimento acelerado
\cite{Galili}\cite{GurelAcar}\cite{Sharma}.
%, mesmo
%ap�s serem aprovados em cursos introdut�rios de F�sica. Isto torna-se evidente quando se lhes pede
%para descrever o que acontece se algumas das actividades di�rias comuns como caminhar
% na Esta��o Espacial Internacional} ou \emph{qual a forma da chama de uma vela},
%tiverem lugar na situa��o de
%imponderabilidade \cite{GurelAcar}\cite{Niedderer}.
Com frequ�ncia, os alunos consideram que as situa��es correntemente designadas como
``gravidade-zero", ``microgravidade" ou ``sem peso" se referem a eventos que ocorrem fora da
influ�ncia gravitacional da Terra ou de outro corpo celeste e ficam, muitas vezes, surpreendidos
quando percebem que durante uma miss�o do vaiv�m espacial, por exemplo, a acelera��o devida �
gravidade a que ficam sujeitos os astronautas � apenas 12\% inferior � que experimentam �
superf�cie da Terra.

% Em particular, os estudantes nem sempre reconhecem que � o movimento de queda livre que cria a
% sensa��o de ``aus�ncia de peso" no interior do ve�culo espacial.
%Parece que eles t�m dificuldade em constatar que uma nave espacial orbitando a Terra ou outro corpo
%celeste est� num estado de queda livre permanente.

Na literatura encontram-se v�rias defini��es da for�a ``peso" de um corpo baseadas na for�a da
gravidade: ``o peso" � a for�a gravitacional da Terra", ``a for�a exercida pela Terra num objecto �
chamada ``peso" do objecto" \cite{Serway}; ``o peso" de um corpo � a for�a gravitacional total
exercida no corpo por todos os outros corpos do Universo" \cite{YoungFreedman}. Estas defini��es
s�o compat�veis com a decis�o da terceira reuni�o da Confer�ncia Geral de Pesos e Medidas (CGPM),
em 1901, que definiu ``peso de um corpo" como o produto da sua massa pela acelera��o devida �
gravidade \cite{3CGPM}.
%; o ``peso" padr�o de um corpo � o produto da sua massa pela
% acelera��o padr�o devida � gravidade; o valor adoptado para acelera��o padr�o devida � gravidade �
%9.80665 m s\(^{-2}\)

As defini��es da for�a ``peso" baseadas na gravidade e os conceitos com ela relacionados s�o
largamente usadas nas aulas e nos livros de texto de F�sica, especialmente nas escolas secund�rias
e nas disciplinas introdut�rias de F�sica do ensino superior, apesar de serem insatisfat�rias e
amb�guas, tendo em conta o conhecimento actual, e s�o, muitas vezes, origem de confus�o.
% It is well known that in the case of a body at rest on the surface of the
%Earth the magnitude of the body weight force equals the magnitude of gravitational
%force acting on the body. Moving the body to a higher altitude on the Earth surface
%or to the Mars surface (where the gravitational force is about 38\% of Earth's)
%would result in ascribing different weights to the same body. It is also accepted
%that in the case of a body in the interior of spacecraft orbiting the Earth, out of
%the atmosphere influence and not touching the walls of the vehicle, the only force
%acting on the body is the Earth gravitational force. The body is said to be in a state of
%weightlessness %(this will be discussed in the next sections).
% These examples show that the concept of weight is of very limited utility.
%
Note-se, por exemplo, a imprecis�o e a falta de clareza das no��es de ``peso verdadeiro", de ``peso
aparente", de ``sem peso", de ``gravidade zero" e de ``microgravidade".
%Note-se, por exemplo, a imprecis�o e a falta de clareza das no��es de ``peso"
%verdadeiro", de ``peso" aparente", de ``sem ``peso", de ``gravidade zero", de ``microgravidade", de
%``para cima", ``para baixo" ou ``vertical".
% Veja-se o n�mero de ambiguidades associadas � defini��o gravitacional de ``peso" e de conceitos com
% ele relacionados como por exemplo ``peso" verdadeiro", ``peso" aparente", ``sem ``peso", e ``gravidade
% zero".
A ambiguidade d� origem a equ�vocos que podem facilmente contribuir para o alargamento do fosso
entre o que se pretende ensinar e o que � aprendido pelos estudantes.
% Para uma discuss�o detalhada
% da dicotomia respeitante � defini��o de ``peso" e as suas implica��es %no ensino de F�sica
% nas perspectivas hist�rica e educacional ver \cite{Galili} (e as respectivas refer�ncias).

Poucos autores adoptam a defini��o operacional da for�a ``peso de um corpo": ``for�a que o corpo
exerce no suporte a que est� ligado" \cite{MarionHornyak}. Nesta defini��o operacional, a for�a
``peso do corpo" � uma for�a de liga��o tal como, por exemplo, a for�a de atrito ou a tens�o numa
corda. Este artigo discute o conceito de for�a ``peso do corpo", das no��es de gravidade-zero, de
microgravidade, de vertical e de ``para baixo/cima", e as vantagens pedag�gicas da adop��o da
defini��o operacional da for�a ``peso" e/ou o abandono do termo for�a ``peso" em F�sica.
% no ensino de F�sica.

%For simplicity the gravitational influences of other celestial bodies is not considered.

\section{Acelera��o devida � gravidade}

No modelo da F�sica Cl�ssica a for�a da gravidade � uma for�a de longo alcance entre os corpos com
massa e, tanto quanto se sabe, n�o pode ser blindada \cite{gravitationalshielding}. Em situa��es
convencionais � independente do estado de movimento dos corpos. A acelera��o devida � gravidade � a
acelera��o do movimento de um dado corpo material resultante apenas da ac��o da for�a da gravidade
sobre esse corpo e, num dado instante, corresponde � raz�o entre a for�a da gravidade e quantidade
de mat�ria do corpo.

Na descri��o dada pela Teoria Geral da Relatividade um campo gravitacional corresponde a
modifica��es da geometria do espa�o-tempo provocada por uma concentra��o de massa ou de energia. As
geod�sicas do espa�o-tempo s�o linhas curvas e os corpos descrevem �rbitas correspondentes a estas
linhas curvas.

%De acordo com o principio de equival�ncia de Einstein existe uma equival�ncia entre um
%referencial em repouso num campo gravitacional e um referencial acelerado na aus�ncia de qualquer
%campo gravitacional significativo.}
% As far we know, this force cannot be shielded and in practical situations is
%independent of the state of motion of the objects.

\subsection{Gravidade segundo a F�sica Newtoniana}

Desde que Isaac Newton prop�s a lei da Gravita��o Universal que se aceita que a interac��o
gravitacional entre massas � universal e apenas depende das quantidades de mat�ria em interac��o e
da dist�ncia que separa os seus centros de massa. A partir dos trabalhos de Kepler e de Galileu,
Newton concluiu que a for�a da gravidade \(\vec{F}_{g}\) que a Terra exerce num corpo de massa
\(m\), consequ�ncia da interac��o gravitacional do corpo com o planeta Terra, � dada por
\begin{equation}\label{FgmM}
\vec{F}_{g}=-\frac{GMm}{|\vec{r}|^{3}}\vec{r}=\vec{G} m,
\end{equation}
onde \(G\) representa a constante gravitacional (6,67\(\times10^{-11}\) Nm\(^{2}\)kg\(^{-2}\)),
\(M\) � a massa da Terra, \(\vec{r}\) � o vector posi��o com origem no centro de massa da Terra e
extremidade no centro de massa do corpo.
% A for�a da gravidade \(\vec{F}_{g}\) que a Terra exerce no corpo aponta para o centro de massa da Terra.
A for�a da gravidade que a Terra exerce num corpo \(\vec{F}_{g}\) pode ser escrita como o produto
da massa \(m\) do corpo pela acelera��o
devida � gravidade \(\vec{g}\), \(\vec{F}_{g}=m \vec{g}\); %onde%\(\vec{g}=-\frac{GM}{|\vec{r}|^{3}}\vec{r}\)
a grandeza \(\vec{g}=-\frac{GM}{|\vec{r}|^{3}}\vec{r}\) corresponde � acelera��o que o corpo
adquire quando sujeito unicamente � ac��o do campo gravitacional \(\vec{G}\). Junto � superf�cie da
Terra \(\vec{g}\) � praticamente constante; a intensidade de \(\vec{g}\) ao n�vel do mar �,
aproximadamente, 9,8 m s\(^{-2}\).
%\footnote{This ignores the fact that the mass distribution of the Earth is not perfectly spherical.}.

A intensidade da for�a da gravidade pode ser medida com aux�lio de um dinam�metro ou de uma
balan�a-dinam�metro, assegurando que o corpo e o dinam�metro est�o em repouso em rela��o � Terra.
% e desprezando o efeito da atmosfera para os corpos menos densos.
Esta afirma��o requer alguns esclarecimentos.
% When considering objects in the Earth's environment this force is always acting on the object.
Considere-se um corpo em repouso relativamente � superf�cie da Terra a uma dada latitude, Fig.
\ref{Fig1}. O corpo � actuado por duas for�as: a for�a da gravidade \(\vec{F}_{g}\), apontando para
o centro da Terra, e a for�a de reac��o \(\vec{N}\) que a superf�cie da Terra (ou do suporte)
exerce sobre a superf�cie do corpo. A direc��o de \(\vec{N}\) � determinada pela for�a da gravidade
e pela rota��o da Terra em torno do seu eixo. Tendo presente a segunda lei de Newton da Din�mica (a
resultante das for�as que actuam num corpo � igual � taxa temporal de varia��o do momento linear do
corpo, \(\vec{F}=d(m\vec{v})/dt\)), a resultante \(\vec{F}\) destas duas for�as assegura a rota��o
di�ria do corpo segundo o paralelo que passa pela posi��o deste. Como consequ�ncia apenas deste
efeito, as direc��es de \(\vec{F}_{g}\) e de \(\vec{g}\) ``medidas" diferem ligeiramente da
direc��o do centro da Terra - excepto nos p�los e no equador - em um �ngulo cuja amplitude �
inferior 0,1\(^{0}\). Verifica-se tamb�m que, devido � acelera��o centr�peta do corpo, a
intensidade de \(\vec{F}_{g}\) indicada na balan�a ou no dinam�metro � inferior, excepto nos p�los,
ao valor dado pela equa��o \ref{FgmM}. Acresce ainda que as \emph{irregularidades} da superf�cie e
as varia��es de densidade nas diferentes regi�es que constituem a Terra d�o origem a um campo
gravitacional n�o exactamente central pelo menos nas proximidades da superf�cie da Terra e,
portanto, a varia��es na direc��o e na intensidade de \(\vec{g}\).
%homogeneidade e n�o esfericidade da Terra d� origem a varia��es locais adicionais no valor e na direc��o de \(\vec{g}\), em particular quando se caminha do equador para os p�los.

Ao longo do resto do texto considera-se que a Terra � uma esfera homog�nea e desprezam-se os
efeitos da rota��o em torno do seu eixo e da transla��o em torno do Sol, e de quaisquer outros
movimentos, devido aos pequenos valores das acelera��es linear e angular da Terra quando comparados
com acelera��o devida � gravidade ou a outras for�as aplicadas. Isto �, a Terra � considerada em
repouso durante os tempos caracter�sticos dos fen�menos aqui analisados. � desprezado tamb�m o
efeito da atmosfera no corpos.

% : a for�a por unidade de massa ``sentida" no equador � cerca de 0,03 m s\(^{-2}\) (0,35\% do valor
% de \(g\)) inferior ao valor medido nos p�los, assumindo que a Terra � perfeitamente esf�rica e
% homog�nea.
%. Another consequence of the Earth rotation is
\begin{figure}[hbt]
\begin{center}
\epsfbox{Fig1.eps}
\end{center}
\caption{\label{Fig1} For�as que actuam no corpo em repouso na superf�cie s�lida da Terra, a uma
dada latitude: a for�a \(\vec{F}\) � a resultante da for�a da gravidade \(\vec{F}_{g}\) que actua
no corpo e da ac��o da superf�cie da Terra \(\vec{N}\) no corpo.}
\end{figure}
% Having this in mind, when a body is on the Earth surface or on a support
%attached to the Earth, the \emph{force of gravity} equals in magnitude to
%the reacting force \(\vec{N}\) but directed oppositely to it.

\subsection{Gravidade segundo a Teoria da Relatividade Geral}

A Teoria da Relatividade Geral trata a gravidade e o movimento acelerado. Numa das suas famosas
experi�ncias conceptuais, Einstein concluiu que n�o � poss�vel distinguir entre um referencial em
repouso num campo gravitacional homog�neo e um referencial acelerado na aus�ncia de um campo
gravitacional: princ�pio de equival�ncia de Einstein.
%Um observador numa c�psula fechada que se sinta pressionado contra a
%cadeira ou contra o pavimento da nave n�o pode destrin�ar se ele e a c�psula est�o num campo
%gravitacional em repouso ou numa c�psula com movimento uniformemente acelerado suficientemente
%afastada de qualquer fonte de campo gravitacional significativo -
%princ�pio de equival�ncia de Einstein.
A partir deste princ�pio de equival�ncia, Einstein prop�e uma interpreta��o geom�trica da
gravita��o: a presen�a de massa ou de energia provoca a curvatura do cont�nuo espa�o-tempo, e as
geod�sicas deixam de ser linhas rectas e passam a ser percursos curvos.
% � a forma da curvatura do cont�nuo espa�o-tempo que define as caracter�sticas do movimento dos corpos em queda livre.
Numa descri��o cl�ssica deste efeito podemos associar ao movimento (�rbita) de um corpo nas
proximidades da Terra, ou de outro corpo celeste, uma acelera��o centr�peta que corresponder� ao
conceito cl�ssico de \emph{acelera��o devida � gravidade}.

%a gravidade corresponde � curvatura no continuo espa�o-tempo provocada por uma elevada concentra��o de massa ou de energia

\section{Defini��o operational de for�a peso do corpo}

%Na discuss�o das for�as envolvidas no caso de um corpo em repouso � superf�cie da Terra ou de outro
%corpo � importante ter presente o significado da terceira lei de Newton da Din�mica
% \cite{MarionHornyak}.

O que os seres humanos e a restante mat�ria experimentam como ``peso" n�o � a for�a da gravidade. A
sensa��o de ``peso" � devida � for�a normal que os suportes exercem nos corpos, constrangindo-os de
forma a contrariar a for�a (``peso") que eles exercem nos suportes. Uma evid�ncia deste facto
ocorre quando uma pessoa em cima de uma balan�a-dinam�metro verifica que o valor indicado por esta
(a intensidade do ``peso") varia sempre que flecte as pernas sem perder o contacto com a superf�cie
da balan�a. Outra manifesta��o ocorre quando um elevador a descer, trava para parar: uma pessoa
sente um acr�scimo de press�o nas pernas e nos p�s. Estas varia��es n�o podem ser atribu�das �
for�a da gravidade, porque a dist�ncia entre os centros de massa da Terra e da pessoa praticamente
n�o se alterou, assim como as respectivas massas.

\subsection{For�a peso de um corpo em repouso}

Consideremos um corpo em repouso � superf�cie da Terra, Fig. \ref{Fig2}a. O corpo est� sujeito �
for�a \(\vec{F}_{g}\) devido ao ``pux�o" gravitacional da Terra. A for�a de reac��o a esta for�a �
\(-\vec{F}_{g}\) e corresponde � ac��o gravitacional
exercida pelo corpo na Terra. O par de for�as \(\vec{F}_{g}\) e \(-\vec{F}_{g}\) %consiste na
% for�a \(\vec{F}_{g}\)  que actua no corpo e na for�a \(-\vec{F}_{g}\) que actua na Terra, e
 constitui um par ac��o-reac��o. A tend�ncia do corpo em acelerar no sentido do centro da Terra devido � for�a \(\vec{F}_{g}\) d�
origem � for�a \(\vec{P}\), Fig. \ref{Fig2}b, for�a que a superf�cie do corpo exerce (ac��o do
corpo) na superf�cie da Terra.
%Contudo, a for�a \(\vec{F}_{g}\) n�o pode ser a �nica for�a aplicada
%no corpo, de outro modo o corpo sujeito apenas a esta for�a aceleraria na direc��o do centro da
%Terra. Tendo em conta que o corpo est� em repouso na superf�cie da Terra, deve existir pelo menos
%outra for�a actuando no corpo que se op�e � tend�ncia deste em se deslocar na direc��o do centro da
%Terra.Assim,
Se o corpo exerce na superf�cie da Terra a for�a \(\vec{P}\), a superf�cie s�lida da Terra reage
exercendo uma for�a \(\vec{N}\) no corpo que contrabalan�a a for�a
\(\vec{P}\), Fig. \ref{Fig2}b. A for�a \(\vec{N}\) %, a for�a que a superf�cie da Terra exerce no corpo,
designa-se for�a normal e � a reac��o a \(\vec{P}\): \(\vec{P}\) e \(\vec{N}\) constituem um par
ac��o-reac��o, \(\vec{N}=-\vec{P}\). A ac��o que o corpo exerce na superf�cie da Terra (ou suporte)
corresponde � defini��o operacional da for�a \textbf{``peso do corpo"}.
%A for�a \(-\vec{F}_{g}\) actua no centro da Terra e a for�a \(\vec{P}\) actua na superf�cie da Terra em contacto com o corpo.
% As quatro for�as actuantes t�m a mesma magnitude, e
%constituem dois pares ac��o-reac��o: \(\vec{F}_{g}\) e \(-\vec{F}_{g}\), e \(\vec{P}\) e
%\(\vec{N}\).
%Ter em aten��o que embora \(|\vec{N}|=|\vec{F}_{g}|\), a for�a \(\vec{N}\) n�o � a
%reac��o � for�a \(\vec{F}_{g}\) porque as duas for�as actuam no mesmo corpo.
\begin{figure}[hbt]
\begin{center}
\epsfbox{Fig2ab.eps}
\end{center}
\caption{\label{Fig2} a) \(\vec{F}_{g}\) e \(-\vec{F}_{g}\): par ac��o-reac��o devido � interac��o
gravitacional entre o corpo e a Terra. b) \(\vec{P}\) e \(\vec{N}\): par ac��o-reac��o em resultado
da interac��o entre as superf�cies do corpo e da Terra.}
\end{figure}

Em conclus�o, no corpo actuam duas for�as, a for�a \(\vec{F}_{g}\) apontando para o centro da Terra
e a for�a \(\vec{N}\) no sentido oposto. Uma vez que o corpo est� em repouso relativamente � Terra
� nula a resultante das for�as que actuam no corpo. Em consequ�ncia, as for�as \(\vec{F}_{g}\) e
\(\vec{N}\) que actuam no corpo - em pontos de aplica��o diferentes - t�m a mesma intensidade e
orienta��es opostas. De forma similar, as duas for�a que actuam na Terra, \(-\vec{F}_{g}\) e
\(\vec{P}\), t�m tamb�m a mesma magnitude, \(|\vec{P}|=|-\vec{F}_{g}|\), sentidos opostos e pontos
de aplica��o diversos.

Consideremos agora o corpo pendurado num dinam�metro ou colocado no prato de uma
balan�a-dinam�metro. Quando o corpo � colocado na balan�a, a mola da balan�a � comprimida (no caso
do dinam�metro seria distendida) e a sua deforma��o � comunicada ao ponteiro da escala da balan�a.
O corpo exerce uma ac��o no prato da balan�a e atrav�s deste na mola. O ponteiro indica a magnitude
da for�a ``peso do corpo" \(\vec{P}\), for�a exercida \textbf{na} superf�cie do prato da balan�a
\textbf{pela} superf�cie do corpo.
%De acordo com a terceira lei de Newton da Din�mica, a for�a
%\(\vec{P}\) tem a mesma magnitude que a for�a \(\vec{N}\) que \textbf{a} superf�cie do prato da
%balan�a exerce \textbf{na} superf�cie do corpo.
% O ``peso" do corpo \(\vec{P}\) corresponde � ac��o que a sua superf�cie exerce no prato da balan�a em
% resultado da reac��o \(\vec{N}\) que a superf�cie da balan�a exerce no corpo, impedindo-o de entrar
% em queda livre. O ``peso" n�o est� aplicado no corpo em considera��o mas na balan�a.

Se o corpo e a balan�a estiverem em repouso em rela��o � Terra ent�o, e como anteriormente, a
intensidade da for�a ``peso do corpo", que actua na balan�a, iguala a magnitude da for�a da
gravidade que actua no corpo, \(|\vec{P}|=|\vec{F}_{g}|\); a igualdade verifica-se tamb�m no caso
do corpo e da balan�a estarem em movimento uniforme e rectil�neo relativamente a um referencial
ligado � Terra.
%\section{``peso" em sistemas com movimento uniformemente acelerado}

Ser� que a rela��o \(|\vec{P}|=|\vec{F}_{g}|\) permanece v�lida se o corpo e o seu suporte
estiverem em movimento acelerado em rela��o � Terra? Qual � a indica��o da balan�a nesta situa��o?
%Este � o caso, por exemplo, de um elevador ao arrancar ou ao parar.

\subsection{For�a peso de um corpo num referencial acelerado}

Imaginemos o corpo colocado no prato de uma balan�a fixa no pavimento de um elevador. Quando o
elevador acelera com uma acelera��o \(\vec{a}\), enquanto estiver em contacto com a balan�a o corpo
move-se em conjunto com a balan�a e o elevador. No corpo actuam duas for�as: a for�a da gravidade
\(\vec{F}_{g}\) e a reac��o \(\vec{N}\) da balan�a � for�a ``peso do corpo" \(\vec{P}\). De acordo
com a segunda lei de Newton da Din�mica e assumindo que a massa \(m\) do corpo n�o varia, a
resultante das for�as que actuam no corpo � igual ao produto da massa \(m\) com a acelera��o
\(\vec{a}\) do corpo, isto �,
\begin{equation}\label{000}
 m\vec{a}=\vec{F}_{g}+\vec{N}.
\end{equation}
Uma vez que a for�a ``peso do corpo" \(\vec{P}\) e a reac��o \(\vec{N}\) da superf�cie da balan�a
formam um par ac��o-reac��o, \(\vec{P}=-\vec{N}\), podemos rescrever a equa��o \ref{000} como
\begin{equation}\label{00}
  \vec{P}=m\vec{g}-m\vec{a}.
\end{equation}
Ter presente que o valor indicado na balan�a corresponde � intensidade da for�a \(\vec{P}\) que o
corpo exerce na balan�a.

Correntemente, define-se vertical de um lugar como a direc��o da acelera��o devida � gravidade.
Outras defini��es baseiam-se na direc��o do fio-de-prumo, que � superf�cie da Terra e em repouso
coincide com a direc��o da for�a da gravidade. A no��o de vertical � mais geral. Por exemplo, um
ser humano ou outro ser vivo sente-se equilibrado na direc��o da for�a que exerce no suporte e a
orienta��o ``para baixo" correspondem ao sentido dessa for�a. A vertical est� sempre segundo a
linha de ac��o da for�a ``peso do corpo" e a orienta��o ``para baixo" corresponde ao sentido dessa
for�a. Da rela��o \ref{00} pode concluir-se que tanto a direc��o da vertical de um corpo como as
orienta��es ``para baixo/cima" dependem da acelera��o do corpo e da acelera��o devida � gravidade.
%Considere-se por exemplo a experi�ncia do motociclista a rodar no ``po�o da morte".
Uma constata��o quotidiana ocorre quando um autocarro arranca ou quando trava: nesta situa��o a
``nossa" vertical � obl�qua e para n�o nos desequilibrarmos inclinamo-nos na direc��o da ``nova"
vertical.

%Dependendo da orienta��o da acelera��o \(\vec{a}\), v�rias situa��es podem ocorrer.

\section{For�a peso do corpo num campo gravitacional fraco}

A ac��o que um corpo exerce nos seus suportes, a for�a ``peso do corpo", n�o depende da exist�ncia
de um campo gravitacional na regi�o do espa�o onde este se encontra. Considere-se um ve�culo
espacial numa regi�o do Universo onde o efeito gravitacional � nulo ou pouco significativo, i.e.,
\(\vec{g}\simeq\vec{0}\). Se uma nave em movimento rectil�neo e uniforme nesta regi�o accionar os
seus motores entrar� em movimento acelerado e os objectos no seu interior ser�o ``projectados" no
sentido oposto � acelera��o \(\vec{a}\) da nave, tal como acontece quando um carro acelera,
acabando estes por exercer for�as nos seus suportes ou nas paredes da nave. Isto �, quando a nave
espacial acciona os motores passa a ser actuada pela for�a de propuls�o que � transferida a cada
objecto no seu interior. Estes, ap�s entrarem em contacto com as paredes da nave ou com outros
corpos solid�rios com a nave, deslocam-se com a acelera��o \(\vec{a}\) da nave. Nestas condi��es a
equa��o \ref{00} toma a forma
% Da equa��o \ref{00} resulta que se a nave se encontrar a acelerar numa regi�o do espa�o em que o
% campo grav�tico � pouco significativo, \(\vec{g}\simeq\vec{0}\), o ``peso" dos objectos no seu
% interior tem orienta��o oposta � da acelera��o da nave:
\begin{equation}\label{0000}
    \vec{P}=-m\vec{a},
\end{equation}
i.e., a ac��o que o corpo exerce no seu suporte tem o sentido oposto � acelera��o da nave e depende
apenas da intensidade dessa acelera��o e da massa do corpo.

Pode, portanto, concluir-se que a for�a que os corpos exercem nos seus suportes ou a ``sua
aus�ncia" n�o t�m a ver necessariamente com o facto do corpo estar ou n�o sob a influ�ncia de um
campo gravitacional. Mesmo na presen�a de um campo gravitacional significativo, esta for�a depende
essencialmente das caracter�sticas de movimento do corpo e do seu suporte. De facto, as propostas
para ``criar artificialmente" o efeito da gravidade no interior de naves interplanet�rias empregam
a rota��o da nave em torno do seu centro de massa, sendo a acelera��o devida � ``gravidade
artificial" oposta � acelera��o centr�peta da nave.

\section{Imponderabilidade e microgravidade}

%Embora a situa��o de quase imponderabilidade possa ser vivida por longos per�odos de tempo numa
%nave espacial fora da ac��o da atmosfera terrestre, enquanto a propuls�o n�o for accionada e a nave
%n�o rodar em torno do seu centro de massa, a ``aus�ncia de peso" n�o � total.

%\section{Microgravidade e for�a ``peso"}\label{impond-micro}\label{``peso"-ref-acel}

Nas situa��es ideais de queda livre todas as partes de um avi�o ou de nave espacial acelerariam
uniformemente e o ambiente no seu interior seria de ``gravidade-zero", porque os corpos no seu
interior n�o sentiriam os efeitos da gravidade. Em situa��es reais o efeito da gravidade nos
corpos, embora substancialmente reduzido, faz-se sentir, e diz-se que os corpos est�o em ambiente
de ``microgravidade" porque a ``aus�ncia de peso" n�o � total. Contudo, o emprego do termo
``microgravidade" sem especificar o seu significado real pode aumentar os equ�vocos associados �
identifica��o da for�a ``peso" com a for�a da gravidade.

A total ``aus�ncia de peso" numa nave espacial s� seria poss�vel se todos os pontos da nave
estivessem em movimento rectil�neo e uniforme, fora da ac��o de qualquer campo gravitacional. Neste
caso, os corpos dentro da nave n�o experimentariam qualquer acelera��o uns relativamente aos
outros, e a situa��o seria de gravidade-zero, desprezando � claro as interac��es gravitacionais
m�tuas e com as paredes da nave.
%Os corpos ou estariam em repouso ou deslocar-se-iam uns relativamente aos outros com movimento
%uniforme e rectil�neo, podendo dizer-se que a acelera��o dos corpos em rela��o, por exemplo, ao
%centro de massa da nave seria nula - situa��o de ``gravidade-zero".
As situa��es de imponderabilidade (``aus�ncia de peso") correntes s�o frequentemente designadas, de
forma impr�pria, como ``gravidade-zero" ou ``microgravidade". � importante ter presente que no
interior de uma nave nas proximidades da Terra a intensidade da acelera��o devida � gravidade
terrestre n�o � nula (``gravidade-zero") ou diminuta (``microgravidade"): a 400 km de altura, por
exemplo, a acelera��o devida � gravidade terrestre � 8,4 m s\(^{-2}\), e mesmo � dist�ncia da
�rbita da Lua � 2,6\(\times10^{-3}\) m s\(^{-2}\), embora nessas regi�es a acelera��o devida �
gravidade solar seja cerca de duas vezes superior, i.e., aproximadamente 5,8\(\times10^{-3}\) m
s\(^{-2}\). A verdadeira microgravidade terrestre, \(g\approx 10^{-6}\) m s\(^{-2}\), s� poderia
ser experimentada em regi�es � dist�ncia de 17 vezes a separa��o entre a Terra e a Lua.

Para os cientistas, o termo microgravidade caracteriza o facto de que a nave em queda livre e os
corpos no seu interior n�o estarem todos sujeitos exactamente � mesma acelera��o, em resultado das
diferentes interac��es gravitacionais entre os corpos interiores e exteriores � nave, da ac��o de
outras for�as, e traduz-se na acelera��o residual que os corpos no interior de um ve�culo em queda
livre experimentam relativamente, por exemplo, ao centro de massa deste. Em consequ�ncia, nas
proximidades da Terra e em naves n�o tripuladas em queda livre obt�m-se com facilidade \(1 \mu g\),
enquanto que em miss�es tripuladas dificilmente se consegue menos de \(100 \mu g\). As causas mais
relevantes s�o: i) a varia��o da acelera��o devida � gravidade em resultado da morfologia da Terra;
ii) varia��es devidas ao efeito gravitacional dos outros corpos celestes, em particular do Sol e da
Lua, dependentes das posi��es relativamente � Terra
% dos restantes planetas do sistema solar;
% - for�as de mar�, efeitos secund�rios das for�as da gravidade da Terra e dos outros corpos celestes;
iii) a varia��o da acelera��o da gravidade terrestre com a altitude que decresce aproximadamente 1
parte por milh�o por cada 3 m de aumento (numa nave em �rbita a for�a centr�peta e, portanto, a
acelera��o devida � gravidade � superior na parte da nave mais afastada da Terra do que na parte
mais pr�xima);
% - quanto mais afastado um objecto est� do centro de massa da nave mais a sua acelera��o se desvia da acelera��o do centro de massa da nave;
iv) a atmosfera, embora podendo ser muito rarefeita, por exemplo, a 400 km de altura, desacelera
gradualmente a nave.

% A este prop�sito recorda-se, por exemplo, que os astronautas se preparam para as miss�es em
% imponderabilidade, treinando as tarefas a executar em tanques de �gua usando fatos especiais que
% simulam a ``aus�ncia de peso".

%Esta situa��o ocorre mesmo que a nave se encontre fora da influ�ncia de campos gravitacionais.
% Segundo o Princ�pio de Equival�ncia de Einstein, os corpos dentro de uma nave com acelera��o
%\(\vec{a}\) comportam-se como se a nave estivesse em repouso num campo gravitacional com acelera��o
%devida � gravidade sim�trica � acelera��o do ve�culo, \(\vec{g}=-\vec{a}\). De facto, as propostas
%para ``criar" artificialmente o efeito da gravidade no interior de naves interplanet�rias
%baseiam-se nesta equival�ncia: empregam a rota��o da nave em torno do seu centro de massa, sendo a
%acelera��o devida � ``gravidade artificial" oposta � acelera��o centr�peta da nave.

%De facto, cientificamente o termo microgravidade � empregue porque a situa��o de ``aus�ncia de peso" n�o � total.

%\section{Imponderabilidade e microgravidade}\label{impond-micro}

%\subsection{``peso"}

% A for�a ``peso" surge sempre que a superf�cie do corpo interage directamente com a superf�cie de outro
%corpo ou suporte a que est� ligado, sendo oposta �  for�a normal (for�a de reac��o) exercida pela
%superf�cie do outro corpo ou suporte, que impede o corpo em quest�o de se deslocar no sentido do
%centro da Terra.
%Por outro lado, se o corpo e o suporte estiverem em queda livre, desprezando o efeito da atmosfera,
%ambos experimentam a situa��o de imponderabilidade porque s�o ``puxados" pela mesma for�a, a for�a
%da gravidade da Terra. Qualquer balan�a-dinam�metro em queda livre com eles mede intensidade do
% zero, porque os corpos n�o s�o compelidos a pressionarem as suas superf�cies umas contra as
%outras. Se a ac��o dos corpos uns nos outros � nula - intensidade do ``peso" zero - tamb�m s�o nulas
%as for�as normais entre as superf�cies.

%\subsection{Estado de imponderabilidade}

%Os seus efeitos transmitem-se atrav�s das paredes do ve�culo aos astronautas e aos objectos no seu
% interior, e estes acabam por ser ``empurrados" pela nave de forma a se deslocarem com esta. Nestes
% casos, � corrente dizer-se que dentro da nave se est� num regime de microgravidade, uma vez que as
% acelera��es induzidas nos corpos s�o muito pequenos, da ordem de \(\mu g\).

%Contudo, e na maior parte das vezes, existem, para al�m da for�a da gravidade, outras for�as
%aplicadas � nave, como por exemplo a resist�ncia da atmosfera.


%Estas for�as induzem nos objectos no interior da nave pequenas acelera��es com intensidade de \(\mu\)m s\(^{-2}\) a alguns mm s\(^{-2}\), da� o termo microgravidade.

%Por�m, quando existem outras for�as aplicadas como, por exemplo, a resist�ncia da atmosfera ou o
%impulso dos motores, os efeitos destas transmitem-se atrav�s das paredes do ve�culo aos astronautas
%e aos objectos no seu interior. Como consequ�ncia, estes s�o ``empurrados" pelo pavimento da nave
%de forma a se deslocarem com esta, dando origem � for�a ``peso" do corpo (subsec��o
%\ref{``peso"-ref-acel}).
%Contudo, a situa��o de imponderabilidade n�o � devida ao grande afastamento da Terra. A acelera��o
%da gravidade � altitude t�pica dos voos orbitais, digamos 400 km, � \(\approx\)88\% do valor
%caracter�stico � superf�cie da Terra.
% i) quando estamos sentados � superf�cie da Terra: a tend�ncia do
% corpo em cair para o centro da Terra, em resultado da for�a
% grav�tica, � cancelada pela for�a de reac��o do suporte - a for�a normal;

% ii) viajando num avi�o: a for�a da gravidade � compensada pela for�a
% de sustenta��o criada pelas asas do avi�o, com a excep��o da queda parab�lica;

% iii) a reentrada na atmosfera terrestre ou a descida com um
% p�ra-quedas: a for�a de resist�ncia da atmosfera contraria a for�a da gravidade;

% iv) manobra durante um voo orbital: o mecanismo de propuls�o da nave
% fornece o impulso necess�rio para alterar o estado de movimento do ve�culo.

%\subsection{Conceito de microgravidade} %\section{Weight, weightlessness and micro-gravity}

% e s�o respons�veis pelos
%v�rios tipos de mar�. %Quando um corpo � actuado pela for�a grav�tica a acelera��o da gravidade
%entre os lados mais pr�ximo e mais afastado do corpo, relativamente ao centro do corpo criador do
%campo grav�tico, pode variar significativamente.
%}.
%\footnote{As
%for�as de mar� s�o um efeito secund�rio da for�a grav�tica e s�o respons�veis pelas mar�s. Estas
%for�as resultam do facto do campo gravitacional n�o ser constante ao longo do di�metro da Terra.
%Quando um corpo � actuado pela for�a grav�tica a acelera��o da gravidade entre o lado mais pr�ximo
%e o lado mais afastado do corpo pode variar significativamente. O que pode ter como consequ�ncia a
%deforma��o da forma do corpo, sem alterar o seu volume; se inicialmente fosse uma esfera, as for�as
%de mar� tendem a distorce-lo para uma elips�ide, com dois bojos, um apontado para o corpo e outro
%apontando no sentido oposto.}.)
%; orbiting the earth this is the case except when rockets
%are on for orbital maneuvers, and until atmospheric re-entry.
% Ter presente que neste contexto o voc�bulo microgravidade n�o significa em si mesmo que a
% acelera��o devida � gravidade no interior da nave foi reduzida, mas apenas que os seus efeitos nos
% corpos no interior da nave n�o se sentem.

\section{Conclus�o}

A identifica��o da for�a ``peso" com a for�a da gravidade carece de l�gica do ponto de vista do
conhecimento actual e �, muitas vezes, fonte de confus�o.
%Quem ensina F�sica ao n�vel introdut�rio depara-se com frequ�ncia com as dificuldades que os
%estudantes experimentam quando confrontados com esta defini��o de ``peso" e os conceitos com ele
%relacionados como ``aus�ncia de peso", ``microgravidade" e imponderabilidade.
Na defini��o operacional discutida neste texto, a for�a ``peso do corpo" corresponde � ac��o que
este exerce no seu suporte, e depende do estado de movimento de ambos. Tendo presente que o
conceito de ``peso" n�o � fundamental em F�sica, acredita-se que a aprendizagem desta ci�ncia sair�
beneficiada se a express�o ``peso do corpo" n�o for usada. Uma vantagem �bvia � eliminar a confus�o
frequente na disciplina entre o conceito de massa do corpo e de for�a ``peso do corpo".
%, sem afectar o conceito fundamental de for�a gravitacional.
%Uma vez que tal medida n�o parece muito execut�vel nos pr�ximos tempo, o autor
% sugere que, pelo menos, a express�o ``peso" de um corpo" seja substitu�da pela ``ac��o do corpo em".
Espera-se que este texto ajude a motivar a comunidade cient�fica, os professores e os autores dos
manuais de F�sica, a adoptarem a defini��o operacional da for�a ``peso", embora o abandono do
enunciado for�a ``peso do corpo" seja desej�vel.

% However, the concept of weight is not fundamental in Physics.
% I believe the best attitude is to avoid the use of the weight as the
% gravity force \cite{MarionHornyak}.
% It also intends to motivate physics instructors and text book
% authors to replace the gravitational definition of weight by the
% operational one.
% As an example, we should not ask students questions such as ``what
% is the weight of a body with 3 kg of mass on the surfaces of the
% Earth and of the Moon? Instead "what is the gravity force acting on
% a body with 3 kg of mass on the surfaces of the Earth and of the
% Moon?"

%+++++++++++++++++++++++++++++++++++++++++++++++

\section*{Agradecimentos}

O autor agradece aos Professores Robertus Potting, Paulo S�, Jos� Rodrigues, e Paulo Silva os
coment�rios e a revis�o do manuscrito.

%\ack{Author wishes to acknowledge .}
%\begin{verbatim}
%\begin{acknowlegments}
%The authors would like to thank...Author wishes to acknowledge .
%\end{acknowlegments}
%\end{verbatim}

\begin{thebibliography}{}\label{sec:TeXbooks}

\bibitem[IG(2001)]{Galili} I. Galili, ``\emph{Weight versus gravitational force: historical and
educational perspectives}," Int. J. Sci. Educ. \textbf{23}, 1073-1093 (2001).

\bibitem[GA(2003)]{GurelAcar} Z. Gurel e H. Acar,
``\emph{Research into Students' Views About Basic Physics Principles in a Weightless Environment},"
The Astronomy Education Review \textbf{2}, 65-81 (2003).

\bibitem[Sharma(2004)]{Sharma}M. D. Sharma, R. M. Millar, A. Smith, and I. M. Sefton, ``\emph{Students' Understandings of Gravity in an Orbiting
Space-Ship}," Research in Science Education \textbf{34}, 267-289 (2004).

%\bibitem[NTetc(1992)]{Niedderer} H. Niedderer el all, ``\emph{Epistemological Beliefs of Students in High School Physics},"
%NARST Annual Meeting in Boston, March 1992 (1992).

\bibitem[RAS(2000)]{Serway} R. A. Serway, \emph{Physics for Scientists and Engineers with
modern physics} (Saunders College Publishing, Philadelphia, CA, 1996) page 113.

\bibitem[YF(2000)]{YoungFreedman} H. D. Young e R. A. Freedman, \emph{University Physics with
modern physics} (Addison-Wesley, Reading, MA, 2000) page 362.

\bibitem[3CGPM(1901)]{3CGPM} R�solution de la 3\(^{e}\) CGPM,
``\emph{D�claration relative � l'unit� de masse et � la d�finition du poids; valeur conventionnelle
de \(g_{n}\)}," Comptes rendus de la 3\(^{e}\) CGPM, 70 (1901).

\bibitem[MH(2000)]{MarionHornyak} J. B. Marion e W. F. Hornyak, \emph{General Physics with
Bioscience Essays} (John Wiley \& Sons, New York, NY, 1985).

%\bibitem[RU(2000)]{Russo} B. M. Yavorsky e Yu. A. Seleznev,
%\emph{Physics: A Refresher Course} (Mir Publishers, Moscow, 1979) page 65.

%\bibitem[RB(1999)]{RBrown} Ronald Brown, ``\emph{Weight-Don\'t use the word at all},"
%The Physics Teacher \textbf{37}, 241 (1999).

%\bibitem[RCM(1999)]{RCM} R. C. Morrison, ``\emph{Weight and gravity-the need for consistent definitions},"
%The Physics Teacher \textbf{37}, 51-52 (1999).

\bibitem[GS(2001)]{gravitationalshielding} T. Van Flandern e X. S. Yang, ``\emph{Allais gravity and pendulum effects during
solar eclipses explained}," Phys. Rev D \textbf{67}, 022002 (2003); C. S. Unnikrishnan, A. K.
Mohapatra, e G. T. Gillies, ``\emph{Anomalous gravity data during the 1997 total solar eclipse do
not support the hypothesis of gravitational shielding}," Phys. Rev D \textbf{63}, 062002 (2001).

\end{thebibliography}

\end{document}

physics0506100.extracted_bib
\begin{thebibliography}{}\label{sec:TeXbooks}

\bibitem[IG(2001)]{Galili} I. Galili, ``\emph{Weight versus gravitational force: historical and
educational perspectives}," Int. J. Sci. Educ. \textbf{23}, 1073-1093 (2001).

\bibitem[GA(2003)]{GurelAcar} Z. Gurel and H. Acar, ``\emph{Research into Students' Views About Basic Physics Principles in a Weightless Environment},"
The Astronomy Education Review \textbf{2}, 65-81 (2003).

\bibitem[Sharma(2004)]{Sharma}M. D. Sharma, R. M. Millar, A. Smith, and I. M. Sefton, ``\emph{Students' Understandings of Gravity in an Orbiting
Space-Ship}," Research in Science Education \textbf{34}, 267-289 (2004).

%\bibitem[NTetc(1992)]{Niedderer} H. Niedderer el all, ``\emph{Epistemological Beliefs of Students in High School Physics},"
%NARST Annual Meeting in Boston, March 1992 (1992).

\bibitem[RAS(2000)]{Serway} R. A. Serway, \emph{Physics for Scientists and Engineers with
modern physics} (Saunders College Publishing, Philadelphia, CA, 1996) page 113.

\bibitem[YF(2000)]{YoungFreedman} H. D. Young and R. A. Freedman, \emph{University Physics with
modern physics} (Addison-Wesley, Reading, MA, 2000) page 362.

\bibitem[3CGPM(1901)]{3CGPM} R�solution de la 3\(^{e}\) CGPM,
``\emph{D�claration relative � l'unit� de masse et � la d�finition du poids; valeur conventionnelle
de \(g_{n}\)}," Comptes rendus de la 3\(^{e}\) CGPM, 70 (1901).

\bibitem[MH(2000)]{MarionHornyak} J. B. Marion and W. F. Hornyak, \emph{General Physics with
Bioscience Essays} (John Wiley \& Sons, New York, NY, 1985).

\bibitem[GS(2001)]{gravitationalshielding} T. Van Flandern e X. S. Yang, ``\emph{Allais gravity and pendulum effects during
solar eclipses explained}," Phys. Rev D \textbf{67}, 022002 (2003); C. S. Unnikrishnan, A. K.
Mohapatra, e G. T. Gillies, ``\emph{Anomalous gravity data during the 1997 total solar eclipse do
not support the hypothesis of gravitational shielding}," Phys. Rev D \textbf{63}, 062002 (2001).

\bibitem[JF(200t)]{FigueiredoJ-AJP} J. M. L. Figueiredo,
``\emph{Advantages of adopting the operational definition of weight},"
http://arxiv.org/abs/physics/0505193.


% \bibitem[RU(2000)]{Russo} B. M. Yavorsky and Yu. A. Seleznev, \emph{Physics: A Refresher Course} (Mir Publishers, Moscow, 1979) page 65.

% \bibitem[RB(1999)]{RBrown} Ronald Brown, ``\emph{Weight � Don't use the word at all},"
% The Physics Teacher \textbf{37}, 241 (1999).

% \bibitem[RCM(1999)]{RCM} R. C. Morrison, ``\emph{Weight and gravity � the need for consistent definitions},"
% The Physics Teacher \textbf{37}, 51-52 (1999).

\end{thebibliography}