# Energy implications of a load depending on geometrical configurations in an
oscillator

Elena Campillo Abarca  Jimena de Hita Fernández  Almudena Martínez Cedillo
Miguel León Pérez  Laura Morón Conde  Andrei Sipos  Daniel Heredia Doval
Javier Domingo Serrano  Rubén González Martínez  BioCoRe, Biological
Cooperative Research S.Coop, Madrid, Spain

###### Abstract

This paper studies, for a specific oscillatory system composed by a pendulum
connected to a seesaw, how the geometry of the different mechanisms of energy
introduction conditions the resulting movement, to achieve both a greater
amplitude of oscillation due to a change of velocity and an acceleration in
its movement. The different configurations that give rise to the acceleration
of motion are therefore analyzed. The study is carried out from a kinematic
point of view, theoretically simulating an energy increase in the system and
analyzing its response in terms of angular velocity and of modification of
apparent weight. Subsequently, the force diagram necessary to generate the
accelerated motion is analyzed. The magnitude of the external force to be
applied and its dependence on the direction and angular instant in which it
are exerted is evaluated. It is observed how for some specific configurations
this magnitude is negative, implying that the system is capable of
accelerating when subjected to a brake or load on it.

##  1 Introduction

An oscillatory system can illustrate a variety of physical behaviors with the
advantage that the starting point is more accessible [1]. The pendulum system
is not a recent discovery, it has been extensively analyzed since the study of
pendulum isochronism by Galileo [2] who gave account of the dependence
relation between the period and the amplitude of a pendulum’s oscillation [3].
The present work is part of a series of investigations that provide a clear
construction, through the study of an oscillator, of fundamental conceptual
objects that can be extrapolated to more complex rotors.

The starting point is a seesaw system with a simple pendulum at one end. First
it will be explained how the application of forces to this pendulum can adopt
different geometrical configurations. Subsequently, the complete system is
considered, showing that the limit given by the oscillation amplitude will
mark the apparent weight variations.

The study then proceeds to analyze the response of the system’s oscillatory
movement when exerting on it modifications of the environment that provoke the
acceleration of the rotation [3, 4]. Two acceleration mechanisms are taken
into account: the direct supply of energy and the application of an external
force. It is shown that, for some specific geometrical configurations, the
oscillatory motion of the system accelerates even if a brake is applied.

##  2 Methods

To begin with, two possible ways of applying a thrust force on a pendulum to
increase the amplitude of oscillation will be presented. On the one hand, it
is possible to apply the thrust force in a tangential direction with regard to
motion. On the other hand, it is possible to obtain the same resultant by
applying this force in a normal direction. This second option implies that it
is possible to produce such an increase in amplitude through a displacement of
the center of mass.

Next, the response of the oscillator, consisting of a seesaw with a pendulum
at one end (Fig.1), is simulated in different dynamic rotational
configurations. The investigation is therefore approached through two
procedures.

First, the response to the direct introduction of energy into the oscillator
will be studied. The relationship between the angular velocity acquired by
introducing energy into the system and the dynamic balance necessary to
maintain the accelerated rotation of the oscillator will be analyzed. In both
cases its expression in terms of apparent weight will be considered. On the
other hand, the different geometries of direction and point of application of
the force necessary for the acceleration of the system, as well as the
possible geometric results of this acceleration, will also be analyzed.

To perform the calculations, the rotation of the pendulum is fixed on the
plane perpendicular to the ground and the zero value of potential is set as
the value corresponding to the position of the oscillator $\theta=0^{\circ}$.
Likewise, due to its lack of quantitative and qualitative impact, the
dispersion effects due air and rotation axis friction are disregarded.

##  3 Result and discussion

###  3.1 System description

A system that combines a seesaw and a simple pendulum of mass $m$, with a
variable mass $M$ to keep the system in dynamic equilibrium at all times
(Fig.1), is supposed.

Figure 1: Description of the oscillating pendulum system. Where $m$
corresponds to the mass of the oscillating pendulum and $M$ to the variable
mass.

In addition, the position at which the pendulum is in stable equilibrium,
i.e., where its kinetic energy due to motion is maximum ($\theta=0^{\circ}$),
is set as zero potential. The forces involved in the motion of the oscillator
are the weight of the oscillating mass $m$, the centrifugal force it acquires
due to its inertia [5] and the weight of the variable mass $M$. In the present
analysis air and rotational axis friction are disregarded. As a consequence,
the oscillator will have a uniform angular velocity which will change only if
some force is introduced.

###  3.2 Introduction of forces in the system

In the first stage of the analysis, only the part of the oscillator comprising
the simple pendulum will be studied (Fig.2). The forces involved in its motion
are the weight of the mass $m$ rotating at $\theta(t)$ and the centrifugal
force it acquires due to its inertia [5]. Along the movement, the weight
remains in normal direction with regard to the ground while of the centrifugal
force always has a radial direction [1]. The work that these forces will have
to do in order to introduce a given energy increase in the system depending on
the point and direction in which they are applied will now be analyzed. The
different dynamic configurations to be considered will be the following: on
the one hand, the application of a force in a tangential direction (Eq.1) with
regard to motion by means of an external thrust; and, on the other, the
appllication of a force in a normal direction (Fig.2) with regard to motion
that modifies the position of its center of mass.

Figure 2: Description of the oscillatory system with motion given by
$\theta(t)$. Where $m$ is the mass of pendulum and $l$ is the pendulum length,
$ml\vec{\omega^{2}}$ corresponds to the centrifugal force and $m\vec{g}$ to
the weight. Visualization of the normal and tangential forces $\vec{F_{n}}$
and $\vec{F_{t}}$.

$\vec{F_{t}}=m\vec{a_{t}}=-mgsen\theta(t)\widehat{t}$ (1)

$\vec{F_{n}}=m\vec{a_{n}}=(mgcos\theta(t)+mlw^{2})\widehat{n}$ (2)

To obtain the angular velocity of the system, $\omega$, a kinematic analysis
of the motion is required: the energy balance at two points of the path, those
corresponding to the maximum and minimum potential is evaluated (Ec.3) [1, 6].
There is a quadratic dependence between the angular velocity of the
oscillation and the point from which its motion starts, represented by
$\theta_{0}$ (Ec.4) or, what is the same, between the angular velocity and the
angular amplitude of the motion (Fig.3).

$\displaystyle\left.\begin{matrix}E_{0}=mlg(1-cos{\theta}_{0})\\\
E_{1}=\frac{1}{2}ml^{2}\omega^{2}\\\ \end{matrix}\right\\}E_{0}=E_{1}$ (3)

${\omega^{2}}=\frac{2g}{l}(1-cos{\theta}_{0})$ (4)

Figure 3: Dependence between the angular velocity $\omega$ and the angular
amplitude $\theta_{0}$ from which the pendulum motion starts.

In Fig.4 the scheme of the transformation when applying a tangential force
with regard to motion when the oscillator is at the point of its maximum
kinetic energy is showed. In this first scenario, in order to increase the
amplitude of the oscillation by an angle $\Delta\theta$ the force must
overcome the tangential component of the pendulum’s weight (Ec.5).

Figure 4: Representation of the transformation in which an increment $h$ is
reached with an increase in the amplitude of oscillation $\Delta\theta$, due
to the application of a tangential force with regard to the motion
$\vec{{F}_{e}}$ on the mass $m$.

$\vec{F_{e}}=(mgsin(\theta_{0})+mgsin(\theta_{0}+\Delta\theta))\widehat{t}$
(5)

The potential energy variation between the height achieved after the angular
increment $\Delta\theta$ (Ec.6) and the height the pendulum would achieve if
it naturally oscillated from $\theta_{0}$ [7] would be equal to the work done
by applying a thrust force tangentialto the motion (Ec.7).

$h=h_{1}-h_{0}=l(cos(\theta_{0})-cos(\theta_{0}+\Delta\theta))$ (6)

$W_{t}(\theta_{0},\Delta\theta)=mgh(\theta_{0},\Delta\theta)=mgl(cos(\theta_{0})-cos(\theta_{0}+\Delta\theta))$
(7)

As mentioned above, a second way to increase the amplitude of the pendulum’s
movement would be applying a force in a normal direction with regard to
motion, displacing the center of mass of the system [8, 9, 10]. Again, the
force would be applied at the point of minimum potential energy i.e., at
$\theta=0^{\circ}$ (Fig.5).

Figure 5: Displacement $\delta$ of the center of mass by the action of an
external force $\vec{F}_{CM}$ normal to the pendulum’s motion. It is
visualized how the translation implies an increase in the angular velocity
$\omega$ with a consequent increase in the amplitude of oscillation
$\Delta\theta$.

Like energy, the magnitude of angular momentum $\vec{L}$ is conserved
throughout movement (Eq.8). Considering this symmetry at the point where the
displacement takes place, the increase in angular velocity (Eq.9) that results
from such a transformation is obtained [9, 11].

$\displaystyle m{l^{2}}{\omega_{1}}=m(l-\delta)^{2}\omega_{2}$ (8)

$\displaystyle\omega_{2}=\frac{l^{2}}{(l-\delta)^{2}}\omega_{1}$ (9)

The displacement of the center of mass of the system, $\delta$, implies an
increase in its angular velocity from $\omega_{1}$ to $\omega_{2}$ and,
consequently, an increase in the oscillation amplitude, $\Delta\theta$. The
relationship between the two values, $\delta$ and $\Delta\theta$ [8, 11], is
calculated by taking advantage of the other symmetry of the system and the
conservation of energy (Ec.10).

$\delta=l-l{\left(\frac{sin\left(\frac{\theta_{0}}{2}\right)}{sin\left(\frac{\theta_{0}+\Delta\theta}{2}\right)}\right)}^{2/3}$
(10)

The external force that will be necessary to carry out this transformation
(Ec.11), and the work to be done in the same direction of displacement of the
center of mass overcoming both the inertia component and weight (Ec.12) are
thus calculated.

$\vec{F}_{CM}=(m(l-\delta)w^{2}+mgcos\theta)\widehat{n}$ (11)

$W_{n}(\theta_{0},\Delta\theta)=\int_{l}^{l-\delta}-F_{CM}\cdot
dx=\int_{l-\delta}^{l}(mw^{2}x+mg)\cdot dx=\\\
=m\omega^{2}(\theta_{0})(l\delta(\theta_{0},\Delta\theta)-\frac{\delta^{2}}{2}(\theta_{0},\Delta\theta))+mg\delta(\theta_{0},\Delta\theta)\hskip
56.9055pt$ (12)

###  3.3 Direct energy input into the system

Once the geometrical determination of the forces and energy increments in a
simple pendulum are known, the direct introduction of energy in the case of
the oscillating system can be analyzed. Two different options will be
considered: first, a constant increase of angular velocity; then, the
introduction of a constant angular acceleration.

####  3.3.1 Angular velocity regimes

The first scenario to be considered is the introduction of an energy increment
in the system’s motion at its potential energy maximum. The natural
oscillation travels through $360^{\circ}$ with a given angular velocity
$\omega$ (Eq.4) which depends on the gravity $g$, the pendulum length $l$ and
the initial angle $\theta_{0}$. The relation between the change in the
pendulum’s angular velocity and the magnitude of the energy increment that has
been introduced can be obtained from the movement’s energy balance (Eq.13).
That is, by analyzing the kinematic scheme of the system at the points of
maximum and minimum potential it is observed that the external energy input
will result in a change of the angular velocity (Eq.14).

$\left.\begin{matrix}E_{0}=2mlg+\Delta E\\\
E_{1}=\frac{1}{2}ml^{2}{\omega_{f}}^{2}\\\ \end{matrix}\right\\}E_{0}=E_{1}$
(13)

$\omega_{f}^{2}=\frac{4g}{l}+\frac{2}{ml^{2}}\Delta
E=\omega_{0}^{2}+\frac{2}{ml^{2}}\Delta E$ (14)

As can be seen in Eq.14 and can be visualized in Fig.6, there is a quadratic
relationship between the magnitude of the energy increment introduced and the
resulting angular velocity. Two categories of response can therefore be
established depending on the range of angular velocity with which the
oscillator rotates. In the environment close to the natural angular velocity
the quadratic component will govern the response but, at high angular
velocities, the behavior becomes linear.

Figure 6:  Ratio between the angular velocity $\omega$ of rotation of the
pendulum and the energy increment $\Delta E$ introduced into the system.

The fact that the system consists of a pendulum connected to a seesaw (Fig.7),
leads to a closer look at the kinetic phenomenon that is being described. As
the seesaw remains in equilibrium condition, the mass $M$, with only vertical
motion, must adapt to the variations of apparent weight undergone by the
oscillator along its path (Eq.15). That is to say, the torque $\vec{\tau}$
experienced by both ends of the system must be compensated, obtaining the
expression for the variation of the apparent weight in the system (Eq.16).
Thus, the increase of the angular velocity of the system due to the
introduction of energy into it will modify the apparent weight along the
oscillation.

Figure 7: Seesaw system. Where $m$ corresponds to the mass of the pendulum
swinging due to $\theta(t)$ with length $l$ on one arm $r_{1}$ with weight
$\vec{F_{m}}$ and inertia $ml\vec{w^{2}}$. On the other side is the arm
$r_{2}$ in equilibrium depending on its mass $M$ variable, with weight
$\vec{F_{M}}$.

$r_{1}(ml\omega^{2}cos\theta+mg)=r_{2}Mg$ (15)

$M(t)=\frac{r_{1}}{r_{2}}\left(\frac{l\omega_{f}^{2}}{g}cos\theta(t)+1\right)m$
(16)

If one considers the apparent weight variation corresponding to oscillations
of $60^{\circ}$, $180^{\circ}$or $360^{\circ}$ amplitude and with $\omega_{f}$
given by $\omega$ of Ec.4, one can visualize the fact that at the point of
minimum potential energy, the oscillator has an apparent weight of its own
that depends on the amplitude of the oscillation. Taking into account the
amplitude limit, this apparent weight can become 2 times its own weight for a
$60^{\circ}$ oscillation, 3 times for a $180^{\circ}$ oscillation, and 5 times
for a full $360^{\circ}$ oscillation (Fig.8).

Figure 8: Variation of apparent weight along the motion for oscillations of
amplitudes of $\theta$ = $60^{\circ}$, $180^{\circ}$ and $360^{\circ}$. That
is, for initial angles of oscillation $\theta_{0}$ = $30^{\circ}$,
$90^{\circ}$ and $180^{\circ}$.

It can be visualized how the existing variation in the expression of the
pendulum’s apparent weight due to the introduction of energy evolves for
different regimes of angular velocities $w_{f}$ from $100\,\mathrm{rpm}$ to
$200\,\mathrm{rpm}$ (Fig.9) and also for a higher velocity range such as
$1000\,\mathrm{rpm}$ to $2000\,\mathrm{rpm}$ (Fig.10).

The existence of a tendency in the high velocity range towards a harmonic
behavior of the apparent weight can thus be verified. This is because, given
its magnitude, the inertia component governs the motion, shielding the
difference due to the oscillator’s weight between the point of maximum and
minimum potential, leading to an isotropic response throughout the
oscillation.

Figure 9: Variation of the apparent weight on the gravity axis with respect to
the initial angular velocity, $100\,\mathrm{rpm}$, $150\,\mathrm{rpm}$ and
$200\,\mathrm{rpm}$. Figure 10: Variation of the apparent weight on the
gravity axis with respect to the initial angular velocity $1000\,\mathrm{rpm}$
and $2000\,\mathrm{rpm}$.

Table 1 shows the different angular velocities and energy increments, as well
as the relative variation of the apparent weight between the points of maximum
and minimum potential energy. The tendency towards such symmetry in the motion
due to the increase of the angular velocities in the system can be clearly
observed.

$\omega\,\mathrm{(rpm)}$ | $\Delta E\,\mathrm{(J)}$ | M/m max | M/m min | $\Delta max-min(\%)$
---|---|---|---|---
59,78 | 0 | 5 | -3 | 40
100 | 35,23 | 12,19 | -10,19 | 16,40
150 | 103,17 | 26,17 | -24,17 | 7,64
200 | 199,72 | 45,76 | -43,76 | 4,37
1000 | 5463,51 | 1120,00 | -1118,00 | 0,17
2000 | 21912,85 | 4477.01 | -4475.01 | 0,04
Table 1: Relationship between the magnitude of the energy increment applied,
the resulting angular velocity and the relative variation between of maximum
and minimum apparent weight points.

####  3.3.2 Regimes of uniform acceleration

The variation of the pendulum’s apparent weight that occurs when it has a
given constant acceleration will now be analyzed.. That is, the angular
velocity will increase after each traveled degree.. In this case the angular
velocity of the oscillator responds to Eq.17 for an accelerated circular
motion [1, 6].

$\omega_{f}^{2}=\omega_{0}^{2}+2\theta\alpha$ (17)

Where $\theta$ corresponds to the angular variation that the oscillator has
traveled and $\alpha$ the angular acceleration with which it rotates. Taking
into account the seesaw system (Fig.7) the variation of the apparent weight
along the oscillation (Ec.18) can be found through inserting Ec.17 in Ec.16.

${M(t)}=\frac{r_{1}}{r_{2}}(l\omega_{0}^{2}cos\theta+2l\theta\alpha
cos\theta+g)m$ (18)

It is possible to visualize in Fig.11 the behavior of the apparent weight. An
angular acceleration of $1000\,\mathrm{rpm^{2}}$ has been chosen to perform
the calculation, with the aim of accentuating the representation of the
features raised. Starting with $l=1\,\mathrm{m}$ and an initial angle of
$180^{\circ}$ it can be observed that once the pendulum travels the
$360^{\circ}$, the apparent weight at the end of the oscillation is $7\%$
higher than at the beginning, as visualized in the discontinuity of Fig.11.
From this relationship it follows that, after fulfilling an oscillation with
an accelerated motion, the apparent weight of the pendulum will have changed
with respect to the initial one even though the angular position coincides.
The analyzed behavior will be amplified as the angular travel of the
oscillator increases, i.e. during the successive cycles it completes, thus
producing positive and negative modifications of the apparent weight with
respect to its value at rest.

Figure 11: Variation of the apparent weight of the pendulum over a complete
oscillation of $360^{\circ}$ when the system has a constant angular
acceleration $\alpha=1000\,\mathrm{rpm^{2}}$.

###  3.4 Geometry of dynamic configuration: accelerated motion by application
of external force

An accelerated motion of the oscillator by applying an external force on the
system is now analyzed. With this aim, different scenarios, where distinct
geometrical configurations of the point and direction of application of the
said external force are evaluated, are going to be proposed.

####  3.4.1 Angles and direction of application

It is proposed to find the magnitude of the external force necessary for the
oscillator to rotate with a certain value of angular acceleration starting
from its natural angular velocity $\omega_{0}$. This force will depend on the
direction in which it is applied on the oscillator, as well as on the angular
instant in which it is applied. To illustrate the response of the system, two
directions of application are chosen: normal and tangential with regard to the
motion. This choice is explained by two factors: it simplifies the calculation
and it shows the magnitude range in which the dynamic scheme is developed,
since these directions correspond to the maximum and minimum projections of
the inertia component respectively. In the first case, the magnitude of the
external force necessary to accelerate the oscillator when this force is
applied normal to the motion on the mass $M$, located at the other end of the
seesaw, is analyzed. The motion considered in the analysis has a uniformly
accelerated character, so the angular velocity at each angular instant will
correspond to the relation referred to in Ec.17. To visualize the magnitude of
the force that will be necessary to apply as a function of the angular instant
in which the oscillator is located, the condition of equilibrium of the seesaw
is maintained, imposing that the torque $\vec{\tau}$ at both ends must be
compensated. The value of $M$ along the oscillation is thus found for a motion
without external acceleration (Eq.16), considering that
$\theta_{0}=-180^{\circ}$ and a complete oscillation of $360^{\circ}$.
Substituting the Ec.16 in Eq.19 it is found that, in the case where the
external force is applied on $M$ in the direction of gravity, in order to
obtain a motion with acceleration $alpha$, the variation of the apparent
weight has to compensate the normal force (Eq.20).

Therefore, the thrust or normal force to be applied will have to be equal in
magnitude to the desired apparent weight change (Eq.20), i.e. equalizing both
forces and imposing the torque condition.

$r_{1}(ml\omega_{f}^{2}cos\theta(t)+mg)=r_{2}(Mg+F_{normal})\\\ $ (19)

$\frac{\overrightarrow{F}_{normal}}{m}=\left(\frac{r_{1}}{r_{2}}(l\omega_{0}^{2}+2l\theta\alpha-4g)cos\theta\right)\widehat{n}$
(20)

In the second case, the magnitude of the force to be supplied if applied on
the mass $m$ of the oscillator in the direction tangential to its motion is
studied (Eq.21). The dynamic balance must include the contribution of the
tangential projection of the oscillator’s forces, considering that the angular
acceleration is the ratio of the tangential acceleration to the pendulum
length.

$\frac{\vec{F}_{tangencial}}{m}=(gsen\theta+\alpha l)\widehat{t}$ (21)

The results of this analysis for different values of angular acceleration are
displayed in Fig.12 and Fig.13. Specifically, the force that needs to be
applied in the tangential and normal directions to the oscillator’s motion in
order to obtain angular accelerations of 100, 1000 and
$2000\,\mathrm{rpm^{2}}$ is shown.

Figure 12: Application of the external force as a function of the angular
instant required to generate the accelerated motion of the oscilator up to
$\alpha=100\,\mathrm{rpm^{2}}$, $1000\,\mathrm{rpm^{2}}$ and
$2000\,\mathrm{rpm^{2}}$ applied in the normal and tangential directions up to
$750^{\circ}$. Figure 13: Application of the external force as a function of
the angular instant required to generate the accelerated motion of the
oscilator up to $\alpha=100\,\mathrm{rpm^{2}}$, $1000\,\mathrm{rpm^{2}}$ and
$2000\,\mathrm{rpm^{2}}$ applied in the normal and tangential directions up to
$300^{\circ}$.

The main feature observed in these relations Eq.20 and Eq.21, which can be
visualized in Fig.12 and Fig.13, is that the external force modifies its sign
synchronously with the period of the oscillator. That is to say, in order to
achieve an accelerated motion in the oscillator it will be necessary to
perform on it a pushing or braking force depending on the angular interval in
which it is applied. The fundamental implication of this result is the
interpretation of that negative component of the apparent force or weight.
There are geometries for which, depending on the angular velocity, the angular
instant and the magnitude of the load, the application of the latter on the
mass $M$ accelerates the oscillator instead of slowing it down. That is to
say, the meaning of the load varies depending on the geometry and the instant
in which the motion of the system is found.

This leads to scenarios where, under the adequate conditions, not only there
are not any losses due to energy applications but also it is possible to
extract energy from the system. The difference in the application of the
thrust force, whether tangential or normal, highlights the importance of the
direction of application as well as the angular instant at which such force is
applied. Therefore a normal thrust force will always be greater in magnitude
than a tangential one due to the inertial weight component of the mass $m$ of
the oscillator. Observing Eq.20 and Eq.21 it becomes clear that, for the
tangential thrust force, the projection of the rotational inertia component is
zero. That is, the resultant will not reflect this change because the system
will not increase its angular velocity along its path. These results can be
observed in Fig.13, where for a tangential thrust the amplitude does not vary
along its path. On the contrary, for a normal thrust its amplitude will be
increased along its path. These changes are more significant for high angular
accelerations, since they entail larger magnitude increases of the system’s
angular velocity.

####  3.4.2 Resultant acceleration vector

So far it has been considered in the analysis that the resultant angular
acceleration has the direction of the spin axis of the oscillator above $M$,
but other conceptual scenarios, in which this resultant is forced to take
another direction can also be opened, evaluating the magnitude of external
force that will be necessary in order to cause the acceleration of the system
above $m$. Two possible scenarios in which the acceleration is fixed in a
different direction will now be analyzed.

In the first case, for the sake of simplicity, it is proposed that the
resultant is restricted to the axis of gravity, in order to evaluate the
maximum contribution that the motion may have in that direction. The apparent
weight variation that will be necessary to contribute along the oscillation is
calculated (Eq.22).

It can be visualized in Fig.14 how forcing a resultant behavior in the normal
direction with regard to mass $m$ makes a negligible difference compared to
the normal direction with regard to mass $M$. In particular, for the case
where the length of the pendulum is $l=1\,\mathrm{m}$, the translation with
respect to the acceleration in the direction of the spin axis is less than
$2N$.

$\frac{\overrightarrow{F}_{normal}}{m}=\left(\frac{r_{1}}{r_{2}}((l\omega_{f}^{2}-4g)cos\theta+\alpha)\right)\widehat{n}$
(22)

Figure 14: Comparison of the magnitude required to produce the resultant
acceleration $\alpha=1000\,\mathrm{rpm^{2}}$ in the axis of rotation described
in Fig.5 and in the case where it is restricted to the gravity or normal axis.

In the second case the direction of the resultant acceleration is set in a
tangential direction with regard to the pendulum’s motion (Eq.23).

$\frac{\vec{F}_{tangencial}}{m}=(gsen\theta+\alpha)\widehat{t}$ (23)

It is implied that the difference between the normal and tangential resultant
will be proportional to the pendulum’s length. In this paper it has been
assumed that the pendulum’s length $l=1\,\mathrm{m}$ in order to make the
conceptual implications of the calculations clearer. For this case, the
behaviors in both projections coincide in magnitude.

These two assumptions are particular cases of the infinite directions that can
be theoretically posed. It is thus shown that the possibility of extracting a
force from the system, when it is accelerated by the application of a charge
under certain geometrical configurations, is available for all the possible
directions the resultant force vector can adopt.

##  4 Conclusions

Since it does not have a rectilinear motion, the system undergoes a
centrifugal force due to its rotational inertia in the direction of the rod
and towards the outside of the axis of rotation. On the other hand, the
velocity of the mass $m$ is perpendicular to the inertia component along the
oscillation, i.e. tangential to the motion.

Considering these directions the necessary forces, both normal and tangential,
to be applied in order to produce a given angular increase in the amplitude of
the oscillation have been analyzed. A feature in the present investigation is
the difference in apparent weight corresponding to the amplitude of the
oscillator, which makes it possible to observe that for higher amplitude
ranges, a greater force must be applied to move the center of mass. This
variation of apparent weight can be up to 5 times greater for a complete
oscillation of $360^{\circ}$.

The accelerated motion of the pendulum can be achieved through the direct
introduction of energy into the system or applying some force on it. The
magnitude of force is reflected in the different geometries of application,
depending on direction sense and point of application. Therefore, the effect
of a normal thrust will be much greater in magnitude than the effect of a
tangential thrust due to the inertia component of the oscillator mass itself.

It is demonstrated in the present work that the magnitude of the external
force necessary to cause the acceleration of the oscillator takes negative
values for certain angular regions. That is to say, an external load or brake
applied synchronously to the period of the pendulum may result, under the
right conditions, in an increase of the angular acceleration of the motion
and, therefore, in a possible extraction of energy from the system.

It is thus exposed the behavior in which certain configurations can be
extracted from the system, such as the ability to exert a force resulting in
an increase in the acceleration of the oscillator, and that is extensible to
all geometric results that may occur.

##  5 Acknowledgments

We would like to thank our colleagues of Zoocánica S.Coop. and the cooperative
entities that have set it up, because with their work they make possible this
and other scientific and technological research projects carried out in a
social economy framework, betting on a transfer model that seeks the public
interest and the transformation of the productive framework.

## References

  * [1] J. W. D. Halliday, R. Resnick, Fundamentals of Physics, Wiley, 2007, 10th edicion.
  * [2] F. D. Céspedes, El primer experimento de galileo galilei., Revista Latinoamericana de Ensayo Fundada en Santiago de Chile.
  * [3] F. A. Buyse, Galileo galilei, holland and the pendulum clock, O que nos faz pensar 26 (41).
  * [4] S. M. Modino, Emmy noether y su impacto en la fısica teórica.
  * [5] G. L. Baker, J. A. Blackburn, The pendulum a case study in physics, Oxford University Press, 2005.
  * [6] P. A. Tipler, G. Mosca, Física para la ciencia y la tecnología. Volumen 1A: Mecánica, Vol. 1, Reverte, 2021.
  * [7] P. Binder, J. C. Bragg, The elastic ballistic pendulum, Physics Education 54 (5) (2019) 053003.
  * [8] W. B. Case, The pumping of a swing from the standing position, American Journal of Physics 64 (3) (1996) 215–220.
  * [9] C. P. y Julio Gratton, El columpio, ANALES AFA 16 (1).
URL https://anales.fisica.org.ar/journal/index.php/analesafa/article/view/264

  * [10] S. Wirkus, R. Rand, A. Ruina, How to pump a swing, The College Mathematics Journal 29 (4) (1998) 266–275.
  * [11] P. Tea, H. Falk, Pumping on a swing, American Journal of Physics 36 (1968) 1165–1166.