Characterization of the frictional properties
of the motion of magnets over
paramagnetic surfaces.
Measurement of the kinetic friction coefficient and the magnetic viscosity between a
neodymium magnet and an aluminium paramagnetic surface.
Data:5th March 2012
Word Count: 3998
Physics Extended Essay
Andrea Fernández Buitrago
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Andrea Fernández Buitrago
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1. ABSTRACT
The purpose of this investigation is, as stated in the title:
Characterization of the frictional properties of the motion of magnets over
paramagnetic surfaces.
Measurement of the kinetic friction coefficient and the magnetic viscosity between a
neodymium magnet and an aluminium paramagnetic surface.
The aim of this experiment is to measure experimentally the kinetic friction and the magnetic
viscosity which determine the motion of a neodymium magnet over an aluminium surface,
finding out the kinetic friction coefficient and the magnetic viscosity .
The experiment that is executed in this Extended Essay is a simple electromagnetic brake, and
the method carried out is constructed upon it. There is a magnet of mass m falling down a
paramagnetic inclined sheet, and there is a magnetic force which stops the magnet from
accelerating. The slope is what we have modified during the experiment, leaving the rest
constant.
The fall of the magnet will be compared to a body falling through a viscous fluid in order to
study and characterize the friction between them. From this analogy with a viscous fluid we
define , the magnetic viscosity.
The key aspect of this experiment is that in order to find out all of this there is only one thing
that needs to be found out experimentally: the terminal velocity ( ) for each angle .
We verify that our method is correct when the result of plotting
straight line whose slope is
and its independent term is
against
is a
. Along the way other
characteristics of the magnet and the slope are found out.
Also, in the plots x versus t we appreciate the typical behavior of a body falling down within a
viscous fluid.
Abstract Word count: 284
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TABLE OF CONTENTS
1.Abstract……………………………..………………………………………………………………..page 2
2.Introduction……………………..…………………………..……………………………………..page 5
2.1 Aim………………………………………………………………………………………..……page 5
2.2 Theoretical background………………………………………………………….……page 5
2.2.1 Electromagnetic field……………………………………………………………………page 5
2.2.2 Electromotive force and induced current……………………………………..page 6
2.2.3 Faraday and Lenz’s Laws……………………….………………………………………page 8
2.2.4 Simple electromagnetic brake. Eddy and Foucault currents………...page 8
2.2.5 Paramagnetic surfaces……………………….………………………………………..page 10
2.2.6 Terminal velocity………………………………………………………………………….page 10
2.2.7 Sheet resistance……………………………………………………………………………page 10
3.Experiment…….…………………….……………………………………………………………..page 11
3.1 Materials………………………………………………………………………………….page 11
3.2 Variables …………………………………………………………………………………page 11
3.3 Method ……………………………………………………………………………………page 12
3.4 Assumptions ………………………………………………………………………...….page 13
3.5 Analogy with the movement in a viscous fluid……………………….….page 16
3.6 Results……………………………………………………………………………………….page 20
3.6.1
3.6.2
Data collection……………………………………………………………………..page 20
Data analysis…………………………………………………………………….....page 21
4. Conclusions……………………..…………………………….………………………………….page 25
4.1 Conclusions…………………………………………………………………………………………….page 25
4.2 Evaluation………………………………………………………………………………………….……page 25
4.3 Possible improvements………………………………………………………………….………page 26
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5. Appendix……………………………………………………………………………………………page 27
5.1 Tables 4-13…………………………………………………………………………………page 27
5.2 Graphs 2-5…………………………………………………………………………………..page 30
5.3 Errors………………………………………………………………………………………….page 32
6. Bibliography………………….……………………………………………………………..…….page 34
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2. INTRODUCTION
Magnetism hasn’t been a topic I could easily understand. At first, I thought it was because I
lacked the theoretical knowledge to be able to fully understand it, but as we tackled the topic
in Physics HL class I realized that it wasn’t a matter of knowing Lorentz’s law or that the
electromotive force is a potential difference. It was a matter of visualization. What I found
most difficult about magnetism was predicting and imagining the direction of the forces and of
the induced currents.
Nonetheless, magnetism remains one of the topics that most fascinates me in Physics.
Although one may say that there are many forces we cannot see, like gravity, magnetism
defied everything I’d learnt until that moment. When a falling body should accelerate, then I
am proven wrong by the electromagnetic brake. This is what drew me to choose this topic, to
be able to see magnetism at work.
2.1 Aim.
We aim to measure experimentally the kinetic friction and the magnetic viscosity which
determine the motion of a neodymium magnet over an aluminium surface. To this end, we
develop a simple and efficient method based on the fundamentals of the magnetic brake. As a
byproduct we argue that, once the magnetic properties of the magnet are known, our method
can be applied to determine the surface resistivity of any paramagnetic material.
The parameters to be measured are then the coefficient of kinetic friction
viscosity caused by the neodymium magnet.
and the magnetic
2.2 Theoretical background
2.2.1 Electromagnetic fields.
An electromagnetic field is a physical field that is produced by the movement of electrically
charged bodies. These bodies then affect the behavior of other charged bodies which are in
the vicinity of the field.1 The range of effect that a charged particle has on others depends on
the strength of the magnetic field, which is denoted by B.
Lorentz’s law is used to define magnetic forces:
(i)
Where q is the charge of the particle and v is their velocity.
1
AGUILAR, César A. & CHIMBO, José A. Descripción del funcionamiento de los discos de frenado magnético. Escuela Superior
Politécnica del Litoral, Guayaquil, 2010.
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In a circuit:
(ii)
where I is the intensity of the circuit and is the length of the object where the current is
created.2
2.2.2 Electromagnetic induction and electromotive force.3
As a conductor moves through a magnetic field, the following applies
As the electrons move downwards they will experience a force, because every moving charged
particle moving through a magnetic field experiences a force.
As the electrons move left, positive charges remain at the right and there is now a potential
difference. If we connect the two ends to a resistor, there is now a circuit to which the
conductor acts as a battery.
2
3
ANONYMOUS. General principle of electromagnetic brakes. ETD 02112000, 2009
HAMPER, Chris. Higher Level Physics developed specifically for the IB Diploma. Pearson Education Limited, Italy, 2009.
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There is now an induced current in the conductor. As the electrons move through the resistor,
more electrons move left and so the direction of the current is the opposite, right.
As a result of the movement of charged particles in a magnetic field, a magnetic force acts
upon the moving particles. Its direction depends on the direction of the current.
“The electromotive force is the amount of mechanical energy converted into electrical energy
per unit charge.” Its unit is the volt.
(iii)
Where l is the length of the conductor and v its velocity.
2.2.3 Faraday and Lenz’s Laws.
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The magnetic flux is the product of the density of field lines by the area they are acting in, in
order to find out the actual number of lines.4
(Tm2) (iv)
We can define the electromotive force as the rate of change of the magnetic flux:
(v)
Lenz’s contribution to Faraday’s laws lies in the direction of the electromotive force. The
electromotive force is such that it opposes the movement that causes it, to stop the variation
of the magnetic flux.5 Then:
(vi)
2.2.4 Simple electromagnetic brake. Eddy currents and Foucault currents.
The simplest form of an electromagnetic brake is of a magnet falling down a slope made from
a paramagnet such as aluminium.
As the magnet falls, there is a change in the magnetic flux of the slope, which entails an
electromotive force. As the magnet moves, the magnetic flux of the place it has just left
decreases, and the magnetic flux of the point it is moving towards increases.
(vi)
4
5
IES Doña Jimena de Gijón. Concepto de flujo. Spain, 2009.
LÓPEZ, Luciano Federico. Ley de Faraday-Lenz. Instituto Senderos Azules, Buenos Aires, 2011 .
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An electromotive force is a potential difference, and so a current is generated. The movement
of charged particles originates a magnetic field. In order to oppose this change in magnetic
flux, the current will be so that it causes a magnetic force opposite to the movement of the
magnet6:
The induced currents that happen in this example are called Eddy or Foucault currents. They
are “currents induced in conductors to oppose the change in flux that generated them”7. They
occur in very small loops, and are circular or elliptical, as ‘Eddy’ literally means a ‘circular’
current.
They may produce undesirable effects, like an increase in the internal temperature of the
material.8
2.2.5 Paramagnetic surfaces.
Paramagnetism is a type of magnetism in which the body that possesses it is only attracted to
a magnet when an external magnetic field is applied. It’s the tendency of the paramagnets’
electrons to align with a magnetic field.9
Unlike ferromagnets, paramagnets don’t undergo hysteresis. They aren’t magnetized by a
magnetic field and stop being aligned with it in its absence.
2.2.6 Terminal velocity.
6
7
8
9
Depfisicayquimica.blogspot.com, Freno electromagnético. IES “Antonio Maria Calero”, Córdoba, 2010.
PAVLIC, Ted. What is an Eddy Current? Physlink.com, 2012.
TutorVista.com. What Is Eddy Current. Youtube.com, 2010.
POLYAKOV, Val. Resistivity of Aluminium. The Physics Factbook, 2004.
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It’s defined as the maximum velocity that a body submerged in a fluid reaches when a constant
force acts upon it. 10
In the case of a body in free fall, this terminal velocity is reached when the friction and the
weight of the fluid displaced become the same as its own weight.
2.2.7 Sheet resistance.
Sheet resistance or resistivity is a measure of the resistance present in thin films, and it
depends on their thickness. It’s commonly used to define materials such as semiconductors. 11
12
(vii)
where is resistivityand t is thickness.
10
11
NAVE, C. R. Terminal velocity. HyperPhysics, Georgia State University, 2012.
University of New South Wales. Sheet resistance. Australia, Sydney, 2010.
12
VAN ZEGHBROECK, Bart. Resistivity – sheet resistance. University of Colorado, Department of Electrical, Computer and Energy
Engineering, 1997.
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3. EXPERIMENT
3.1 Materials.
The two essential materials for this experiment are a neodymium magnet and an aluminium
sheet. A tool to measure distances is also required, such as a metre or a ruler. A chronometre
is necessary in order to measure times.
Aluminium sheet.
Neodymium magnet.
3.2 Variables.
The independent variable in this experiment is the angle that the aluminium sheet has on
respect to the horizontal, which we modify by changing the height and base it forms when it is
inclined. It is measured with either a ruler or a metre.
The dependent variable is the time it takes for the magnet to move down the aluminium
sheet, which we use to later calculate the terminal velocity for each angle. In order to measure
time a chronometre will be used.
The controlled variables are:




Temperature and pressure.
The magnet’s characteristics (mass, shape, B…). The same magnet will be use in all the
experiments.
The aluminium sheet’s characteristics (mass, shape, sheet resistance…). The same
aluminium sheet will be used in all the experiments.
Initial velocity of the magnet (initially at rest). It’s controlled by placing the magnet
without any force acting upon it and then releasing it.
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3.3 Method.
We weigh the magnet used in this experiment. We then measure the radius and depth of the
magnet; and the length, cross-section and thickness of the aluminium sheet. We draw marks
on the aluminium sheet every 5 centimetres with an indelible marker. We will measure the
time taken for the magnet to cover the first 5 centimetres (0-5cm), the time taken for it to
cover the next 5 centimetres (5-10cm), and so on until we reach the 45cm mark.
We place the aluminium sheet so that it forms an angle with the horizontal. We measure the
angle created by using the height and the base of the triangle this creates. Placing the
neodymium magnet at the top of the aluminium sheet with an initial velocity of 0, we release
the magnet.
The magnet will fall down the slope and, according to Faraday-Lenz’s Law this will generate a
force opposite to the movement. This will cause the magnet to decelerate. There will be a
point in which the opposing forces will be equal because of the effect of the kinetic friction, at
which the acceleration of the magnet will be zero. The velocity of the magnet at this point is
the terminal velocity.
Using the chronometer, we measure the time it takes for the magnet to cover every five
centimetres drawn on the aluminium sheet. We release the magnet at this angle a minimum of
six times. We then proceed to repeat the procedure with a different angle. The terminal
velocity is reached when the time it takes for it to cover each 5cm interval remains constant.
As
and the space interval is constant (5cm), then if the time is also constant the velocity
is as well.
When this happens there is no acceleration acting on the magnet, because, following
Newton’s formula
, the sum of the external forces acting on the magnet is zero.
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3.4 Assumptions.
As the magnet moves down the aluminium sheet there is a change in the magnetic flux upon a
given area of the aluminium surface. As defined by Faraday-Lenz’s Law, a change in the
magnetic flux over time generates an electromotive force with the formula:
(vi)
Where
.
We assume then that the magnetic flux density of the magnet remains constant during the
experiment. Then the only element that changes is the space covered by the magnetic field,
which changes with time:
(viii)
The expression
is proportional to the radius of the magnet and its velocity, since the area
covered by the magnet depends on its velocity.
(ix)
Thus, the electromotive force depends on the angle of the ramp, as velocity does.
The movement of the magnet as it goes down the ramp induces a closed current that must
have some component transverse to the aluminium sheet (an Eddy current). The shape of
these closed currents is unknown to us, but their location must be where the magnet is at that
moment. They don’t have to reach the ends of the aluminium sheet.
Because of the resistivity of aluminium these currents are dissipated very quickly, so readings
for their intensity are very difficult to make.
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Our second assumption is that this induced current follows Ohm’s laws:
(x)
As the electromotive force is a potential difference, then the potential difference (V) of Ohm’s
law is the same as the electromotive force.
(xi)
Substituting:
(xii)
Where is the sheet resistance of aluminium, which is characteristic to our aluminium sheet.
This resistance we can find out by:
(vii)
If
, and I are proportional:
(xiii)
As was already proportional to –v:
(xiv)
This proves that there is a relationship between the velocity of the magnet, the electromotive
force and the intensity.
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3.5 Analogy with the movement in a viscous fluid.
The forces that act upon the magnet are the following:
The weight of the object can be divided into two components, one that is parallel to the ramp
and another that is perpendicular to it.
The component perpendicular to the ramp, Wy, has another force equal and opposite to it, the
normal force Fn, which is a product of the force that the surface acts on the magnet. If they
were not equal, either the ramp would break or the magnet would fly in mid-air. As neither of
them is the case, they are equal to each other.
(Sum of forces in the y axis) (xv)
On x axis there are three forces that are acting. The first of them is one of the components of
the weight, Wx, which is moving the magnet down the ramp. The other two have opposite
direction to this one, opposing it. One of them is a magnetic force,
that is created by the
induced current and the change in magnetic flux as the magnet goes down the aluminium
sheet. The other one is friction, in this case kinetic friction as it occurs while the magnet is
moving.
This balance of forces doesn’t necessarily have to be zero, since the magnet could be
accelerating or not, depending on the angle.
(sum of forces in the x axis) (xvi)
We say it depends on the angle because the component x of the weight is:
(xvii)
This magnetic force we are going to define as a Lorentz force that is acting upon the magnet,
as it is “a force that acts on moving particles in an electromagnetic field”.
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(xviii)
The charge
that is taken into account in this formula we do not know. It depends on the
characteristics of the aluminium sheet, such as its mass, its number of electrons and its
resistivity; and depends as well on the geometric and magnetic properties of the magnet that
originates the movement of the charges.
This
can be deduced by a dimensional analysis – see below.
The x axis in the force diagram is the one that interests us:
;
;
;
(xix)
Knowing from before that
, then:
(xx)
Finally:
; (xxi)
(xxii)
This force diagram is very similar to that of the forces that act upon the movement of a body in
a viscous fluid.
In the fall of a body in a viscous fluid there are three forces:



Weight
Weight of the fluid displaced
Force caused by the viscosity of the fluid, which depends on the velocity of the body
moving through the fluid.
; (xxiii)
In a diagram:
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Where Wb is the weight of the body, Wf the weight of the fluid displaced and Fv the force
created by the viscosity of the fluid.
So:
(xxiv)
; (xxv)
Where
If we compare the formulae for the body in a viscous fluid and the magnet moving along the
aluminium sheet:
(xxv)
–
(xxii)
The height in the fluid is equivalent to the distance in the neodymium magnet example, and
the three forces acting upon the objects of both examples are equivalent. The only difference
is that in the example of the magnet there is an angle to be taken into account, which in the
example of the fluid doesn’t appear because it is completely horizontal.
Fluid and body
Weight of the
body
Mg
Weight of the fluid
displaced
Force caused by
viscosity
Aluminium sheet and
magnet
Using this analogy, then the coefficient of kinetic friction is the equivalent to the density of the
fluid, and the viscosity of the fluid is equivalent to the product of the effective charge of the
aluminium and B.
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Viscosity of the fluid
The viscosity
is then in our case a magnetic viscosity.
In this experiment what we know is the terminal velocity at certain angles. Terminal velocity is
achieved when the acceleration caused by the sum of all forces is zero. As we already know
the formula for the forces acting on the magnet, then:
–
; (xxvi)
Where vt is terminal velocity.
We proceed with the calculations:
(xxvii)
(xxviii)
(xxix)
;
As
(xxx)
is equal to
(xxxi)
Where the unknowns are
.
This can be then expressed as a linear function, in which:
y = mx + n
Y
X
M
N
Drawing a graph with these characteristics will make us be able to know the values for the
magnetic viscosity and the coefficient of kinetic friction.
The functional dependence of the magnetic viscosity on the conductivity properties of the
aluminium sheet and the magnetic and geometrical properties of the magnet can be worked
out by dimensional analysis.
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The electromotive force can also be defined by
. In the surface circuits there is no
such thing as l, but the effective length must be of the order of the radius of the magnet, r, and
depends on its circular shape :
(xxxii)
Where C1 is a constant that we take into account since we don’t know how the circuit behaves.
Following the relationship between and I established before:
(xxxiii)
(xxxiv)
The magnetic force that is originated by a circuit is given by the formula
, and
given the expression for the intensity that we have deduced from the potential difference:
(xxxv)
(xxxvi)
Where
are the characteristics of the magnet.
Being that
has been defined by us as:
(xviii)
(xxxvii)
(xxxviii)
Then:
(xxxix)
Knowing
and
we can find out these characteristics:
(xl)
Furthermore, the displacement x(t) of the magnet can be found out from the forces acting
upon it, since the result of integrating acceleration is velocity, and the result of integrating
velocity is the displacement. So, if we integrate this formula, the result is the equation that
shows the movement of the magnet:
13
13
()
M. Donaire, private communication.
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3.7 Results.
3.6.1 Data collection
In the following results the value of gravity will be assumed to be 9,8 m/s2. All units are in the
International System. The calculation of errors in the measurements will be expressed at the
end.
These are the characteristics of the neodymium magnet used:
Table 1. Characteristics of the neodymium magnet.
Height (m) ±0,001
0,010
Radius (m) ±0,001
0,014
Mass (g) ±0,01
44,49
The direction of the neodymium magnet’s magnetic field is perpendicular to it, meaning,
parallel to its surface vector.
The characteristics of the aluminium sheet are the following:
Table 2. Characteristics of the aluminium sheet.
Length (m) ±0,001
0,489
Sheet resistance (Ω) ±
Thickness (m) ±0,001
0,010
2,8
Resistivity (Ω·m)
2,82
The angles studied are the following:
Table 3.Angles the terminal velocity was measured for.
Base (m) ±0,001 Height (m) ±0,001 Angle (º) ±0,5 Angle (rad) ±0,009
0,100
0,236
0,380
0,150
0,163
0,180
0,200
0,430
0,250
0,109
0,225
0,225
0,280
0,109
0,109
0,109
0,109
0,219
0,109
47,5
43,6
40,5
36,4
36,0
33,8
31,2
28,6
27,0
23,6
0,829
0,761
0,707
0,635
0,628
0,59
0,545
0,499
0,471
0,412
Every few experiments the aluminium sheet would become magnetized for a few minutes.
While the aluminium was magnetized, the magnet would go slower, thus rendering any results
taken while this happened useless.
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After approximately two hours of experiments the magnet would stop moving down the slope
completely and experiments would have to be finished for an interval of time.
The data obtained for each angle is found in the Appendix.
The terminal velocities for each angle are then:
Table 14.Terminal velocity for all angles studied.
Angle (º) ±0,5
Angle
(rad) ±0,009
47,5
43,6
40,5
36,4
36,0
33,8
31,2
28,6
27,0
23,6
0,829
Space covered (m)
±0,001
0,761
0,707
Time (s) ±0,02
0,24
0,27
0,31
0,35
0,42
0,48
0,50
0,53
0,56
0,67
0,005
0,635
0,628
0,590
0,545
0,499
0,471
0,412
Terminal velocity (m/s)
±0,003
0,021
0,019
0,016
0,014
0,012
0,010
0,010
0,009
0,009
0,007
3.6.2 Data analysis.
Knowing this data, we can draw a linear graph in order to find out the values of
each angle.
Y
���� cos∝
(
±7 *
7,1 *
6,0 *
4,8 *
4,0 *
3,4 *
2,8 *
2,7 *
2,4 *
2,3 *
1,8 *
and
Table 15.Characteristics of Graph 1.
y = mx + n
X
M
±0,01
(
)
for
N
(
1,09
0,95
0,85
0,74
0,73
0,67
0,61
0,55
0,51
0,44
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The maximum slope for the graph is in red and the minimum slope is in blue.
All the data fall inside the tendency line because of their error lines, though there are some
that stray a bit, such as the data for angle 36,0º, data #6 in the graph. This seems to be a
random error. The tendency line adjusts fairly well to the data, with a R² of 0,958, being the
maximum R²=1. The adjustment could be better nonetheless.
At a first look the graph seems to fit our expectative, since with this tendency line both
are positive.
and
According to the characteristics from before:
Table 16.Figuring out the slope and n.
y = mx + n
M
N
y
X
���� cos∝
(
±0,01
(
)
(
±3 *
)
±7 *
±2 *
8,0 *
-2,0 *
We can now calculate
and
:
Table 17.Figuring out the magnetic viscosity and the kinetic friction coefficient.
y = mx + n
(
±3 *
)
(
±2 *
8*
-2 *
)
(
)
±0,04
±500
12500
0,25
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The value of seems to be a possible value for a kinetic friction coefficient, but there is no
way to know whether this is so for . For neither of them we can check whether the values are
correct, for they depend on the aluminium and magnet used and are characteristic of my
materials. What we can state is that the data is precise, because the tendency remained the
same as we added more data to the graph, though we cannot know if it is accurate or if there
are any systematic errors in the way of applying the method.
The geometrical and magnetic properties of the magnet are then:
Table 18. Geometrical and magnetic properties of the magnet.
Sheet resistance (Ω)
(
±
±
)
(
±500
12500
0,035
2,8
We can now represent the movement of the magnet:
0,05
0,045
Distance covered (m)
0,04
0,035
0,03
0,025
27,0º
0,02
23,6º
0,015
0,01
0,005
0
0
1
2
3
4
5
6
7
Time (s)
Graph 6
Errors cannot be appreciated in the graph. The rest of the graphs are in the Appendix.
We can observe that as the angle decreases, the time taken for the magnet to cover the entire
distance increases as well, which makes sense since the weight in the x axis is given by the
angle.
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The slope of the graphs is positive as it should be, because distance increases with time and
the magnet doesn’t go backwards.
The data doesn’t follow a linear tendency, but in a curve. This makes sense as the movement
equation is supposed to be:
()
You can see the effect of the viscosity in the concavity of the graphs, which proves our theory
about the magnetic viscosity.
This doesn’t hold true for angle 36,4º, because this one shows a linear tendency. Another
observation for this same angle is that it takes less time for it to cover the distance than for
40,5º, when the force should be greater at 40,5º. This shows definitely that these
measurements indeed hold some type of error. This shows as well in Table 7 (36,4º), as all data
show the same value from the beginning, which cannot be possible.
As the angles grow smaller, the difference in movement between them grows smaller as well.
There isn’t much difference between the movement at angles of 27º and 23,6º in Graph 6.
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4. DISCUSSION
4.1 Conclusions.
We have met our objective of making a characterization for the friction between a magnet and
a paramagnetic surface by means of a simple method. This consists of measuring the terminal
velocity of the magnet for each angle of the inclined plane.
By comparing the equation of motion of the magnet with those of an object in free fall within a
viscous fluid, we have identified the magnetic the magnetic viscosity with the product of qeffB
as highlighted on page 18. Graph 6 shows the presence of this viscosity because of their
concavity.
From the slope and the independent term of the linear regression curve of Graph 1 we have
found out the values of and , though we can only make an approximation of their error
since they were calculated by the graphing program.
By means of a dimensional analysis we have found out the functional dependence of on the
aluminium sheet resistivity and on the magnetic field and geometry of the magnet –Eq. Once
the aluminium resistivity is known, the properties of the magnet are all cast in the value of
.
In this method, there was no need to find out the characteristics of the induced circuits (like
potential difference or intensity), or to know the magnetic properties of the magnet such as its
B. The only things needed were an aluminium sheet and a magnet with a magnetic field
perpendicular to it.
4.2 Evaluation.
As the method is very simple, it doesn’t imply a huge amount of error in its measurements. We
are only measuring the time to cross a distance. As for some of the angles this time is very
small, this makes it more difficult to measure it. This may be a cause for error in the biggest
angles.
As well, as was stated in the results, the aluminium would become magnetized. As aluminium
is a paramagnet, this should not occur, but an explanation for this could be that it is not
completely pure and may have some traces of a ferromagnet in its composition. If that were
so, then the value of the resistivity of aluminium that we have researched may not hold true
for the aluminium sheet used in this experiment. As well, this resistivity of aluminium was a
tabulated result, which probably means it was an average between different results and may
have some error in itself.
All in all, the error doesn’t appear to be too great, since the tendency line adjusts pretty well
and all the data follow a clearly established pattern. The adjustment could be better
nonetheless. There is also to take into account the error of the graphing program in calculating
the tendency line.
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An important issue is that there is only one value for both and , as the experiments were
all using the same magnet. Any new data would just be added to Graph 1. There isn’t any
research data available to compare, either, and so we have no idea whether the final values
are correct or if there was a big amount of error we have not accounted for.
Other possible causes for error are:



An incorrect angle because the surface we are doing the experiment on isn’t perfectly
horizontal to the ground.
Controlled variables didn’t remain constant during the experiment.
The magnet wasn’t initially at rest; we were inadvertently providing it with an initial
force.
4.3 Possible improvements.
Further characteristics of the magnet and the aluminium sheet could be found out, as well as
the static friction between them and . As we now know the magnet’s geometrical/magnetic
characteristics, we could repeat this same experiment with other paramagnetic materials of
which we know the resistivity. We would then find out the two unknowns of
,
and
2
B.
Using this same method we can also find out the sheet resistance of other surfaces of which
we do not know the composition, such as alloys.
It would also be relevant to verify our dimensional analysis by repeating this same experiment
with another neodymium magnet, by for example carrying the experiment out with two
exactly equal magnets so that B is double.
Other possible improvements are:



Measuring the potential difference or intensity of the induced currents.
Doing all the experiments in one sitting.
Drawing the movement equation we have deduced and comparing it with the actual
results.
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5. APPENDIX
5.1 Tables 4-13
Table 4.Time taken to cover the distance for an angle of 47,5º
Distance(m) ±0,001
Time (s)
Average time (s) ±0,02
±0,01
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,29
0,33
0,28
0,29
0,33
0,31
0,29
0,27
0,32
0,32
0,34
0,32
0,24
0,24
0,19
0,26
0,21
0,22
0,33
0,30
0,33
0,30
0,30
0,36
0,34
0,32
0,29
0,30
0,29
0,26
0,26
0,23
0,24
0,26
0,24
0,26
0,33
0,30
0,31
0,33
0,33
0,25
0,34
0,34
0,32
0,26
0,34
0,20
0,25
0,22
0,25
0,26
0,26
0,26
0,33
0,31
0,31
0,30
0,30
0,29
0,24
0,24
0,24
Table 5.Time taken to cover the distance for an angle of 43,6º
Distance(m) ±0,001
Time (s)
Average time (s) ±0,02
±0,01
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,30
0,44
0,33
0,38
0,38
0,38
0,35
0,36
0,33
0,33
0,29
0,31
0,27
0,33
0,28
0,29
0,27
0,30
0,35
0,42
0,37
0,37
0,38
0,35
0,35
0,35
0,33
0,32
0,36
0,31
0,31
0,27
0,27
0,27
0,30
0,27
0,43
0,43
0,41
0,43
0,40
0,40
0,33
0,35
0,33
0,33
0,31
0,30
0,26
0,19
0,27
0,26
0,22
0,27
0,40
0,38
0,38
0,35
0,33
0,31
0,27
0,27
0,27
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Table 6.Time taken to cover the distance for an angle of 40,5º
Distance(m) ±0,001
Time (s)
Average time (s) ±0,02
±0,01
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,50
0,58
0,61
0,65
0,55
0,52
0,38
0,43
0,33
0,29
0,31
0,31
0,31
0,31
0,33
0,31
0,31
0,31
0,67
0,35
0,57
0,32
0,50
0,28
0,39
0,39
0,33
0,35
0,32
0,29
0,35
0,28
0,31
0,3
0,31
0,31
0,44
0,48
0,39
0,34
0,44
0,36
0,27
0,37
0,33
0,34
0,30
0,33
0,30
0,33
0,33
0,29
0,31
0,28
0,50
0,48
0,44
0,37
0,33
0,31
0,31
0,31
0,31
Table 7. Time taken to cover the distance for an angle of 36,4º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,31
0,30
0,32
0,37
0,34
0,39
0,36
0,35
0,31
0,47
0,35
0,41
0,38
0,40
0,37
0,40
0,35
0,53
0,33
0,40
0,37
0,42
0,31
0,42
0,30
0,41
0,30
0,39
0,29
0,36
0,36
0,34
0,35
0,30
0,30
0,30
0,40
0,34
0,34
0,30
0,35
0,33
0,34
0,36
0,30
0,35
0,30
0,33
0,29
0,34
0,36
0,30
0,30
0,31
0,35
0,35
0,36
0,35
0,35
0,35
0,35
0,35
0,35
Table 8. Time taken to cover the distance for an angle of 36,0º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,44
0,44
0,40
0,50
0,41
0,41
0,38
0,44
0,44
0,38
0,44
0,42
0,43
0,41
0,43
0,41
0,44
0,44
0,44
0,40
0,41
0,41
0,44
0,46
0,47
0,44
0,38
0,44
0,37
0,41
0,39
0,44
0,44
0,41
0,37
0,41
0,43
0,40
0,42
0,43
0,43
0,44
0,42
0,44
0,44
0,41
0,44
0,41
0,41
0,44
0,41
0,44
0,44
0,41
0,43
0,43
0,43
0,43
0,42
0,42
0,42
0,42
0,42
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Table 9. Time taken to cover the distance for an angle of 33,8º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,64
0,55
0,54
0,68
0,49
0,56
0,45
0,47
0,50
0,50
0,53
0,46
0,39
0,55
0,44
0,42
0,50
0,48
0,73
0,62
0,44
0,64
0,38
0,58
0,50
0,50
0,52
0,45
0,47
0,48
0,45
0,50
0,48
0,57
0,50
0,45
0,54
0,69
0,52
0,52
0,43
0,58
0,45
0,48
0,43
0,50
0,40
0,55
0,45
0,52
0,42
0,52
0,45
0,49
0,63
0,56
0,50
0,48
0,48
0,48
0,48
0,48
0,48
Table 10. Time taken to cover the distance for an angle of 31,2º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,66
0,66
0,69
0,56
0,53
0,6
0,54
0,55
0,53
0,55
0,50
0,52
0,47
0,50
0,47
0,50
0,53
0,50
0,60
0,53
0,55
0,56
0,57
0,55
0,60
0,50
0,53
0,50
0,47
0,55
0,47
0,50
0,54
0,50
0,47
0,50
0,54
0,69
0,55
0,6
0,56
0,58
0,50
0,58
0,47
0,50
0,47
0,50
0,50
0,55
0,50
0,47
0,47
0,50
0,61
0,59
0,57
0,55
0,51
0,52
0,50
0,50
0,50
Table 11. Time taken to cover the distance for an angle of 28,6º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,68
0,69
0,70
0,76
0,61
0,61
0,57
0,55
0,55
0,60
0,54
0,54
0,52
0,52
0,52
0,53
0,54
0,53
0,71
0,75
0,65
0,65
0,68
0,66
0,60
0,57
0,54
0,45
0,54
0,46
0,54
0,53
0,53
0,54
0,53
0,53
0,69
0,62
0,60
0,61
0,55
0,57
0,57
0,53
0,55
0,50
0,56
0,54
0,54
0,52
0,53
0,54
0,53
0,53
0,69
0,66
0,61
0,57
0,53
0,53
0,53
0,53
0,53
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Table 12. Time taken to cover the distance for an angle of 27,0º
Distance(m) ±0,001 Time (s) ±0,01 Average time (s) ±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,88
0,60
0,74
0,65
0,69
0,91
0,72
0,70
0,57
0,52
0,52
0,50
0,59
0,45
0,59
0,51
0,60
0,78
0,56
1,07
1,04
0,67
0,69
68
0,53
0,63
0,64
0,59
0,61
0,62
0,64
0,63
0,58
0,42
0,39
0,52
0,81
1,08
0,77
0,62
0,63
0,71
0,56
0,75
0,54
0,77
0,59
0,73
0,53
0,54
0,48
0,60
0,61
0,48
0,83
0,75
0,72
0,65
0,60
0,60
0,56
0,56
0,56
Table 13. Time taken to cover the distance for an angle of 23,6º
Distance(m)
Time (s) ±0,01
Average time (s)
±0,001
±0,02
0,000 to 0,005
0,005 to 0,010
0,010 to 0,015
0,015 to 0,020
0,020 to 0,025
0,025 to 0,030
0,030 to 0,035
0,035 to 0,040
0,040 to 0,045
0,70
0,78
0,78
0,78
0,65
0,82
0,70
0,86
0,74
0,82
0,69
0,66
0,73
0,73
0,78
0,67
0,69
0,75
0,77
0,80
0,76
0,86
0,65
0,74
0,78
0,52
0,60
0,57
0,74
0,69
0,65
0,67
0,63
0,65
0,67
0,67
0,79
0,85
0,74
0,69
0,70
0,64
0,y8
0,57
0,61
0,67
0,51
0,70
0,65
0,61
0,65
0,61
0,63
0,61
0,78
0,77
0,70
0,69
0,67
0,66
0,67
0,67
0,67
5.2 Graphs 2-5
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0,055
Distance covered (m)
0,045
0,035
0,025
47,5º
43,6º
0,015
0,005
-0,5-0,005 0
0,5
1
1,5
Time (s)
Graph 2
2
2,5
3
3,5
0,05
0,045
Distance covered
0,04
0,035
0,03
0,025
0,02
40,5º
0,015
36,4º
0,01
0,005
0
0
0,5
1
1,5
2
2,5
3
3,5
4
Time
Graph 3
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0,05
0,045
Distance covered (m)
0,04
0,035
0,03
0,025
0,02
36,0º
0,015
33,8º
0,01
0,005
0
0
1
2
3
4
5
Time (s)
Graph 4
0,05
0,045
Distance covered (m)
0,04
0,035
0,03
0,025
31,2º
0,02
28,6º
0,015
0,01
0,005
0
0
1
2
3
4
5
6
Time (s)
Graph 5
5.3 Errors.
There are two types of errors: absolute errors and relative errors. The relative error is the
result of diving the absolute error of a measurement by the measurement.
E(a) is the absolute error and
is the relative error.
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The errors in the direct measurements are given by the lowest measurement they can provide.
In the case of the distance, for example, if the minimum measurement of the metre is 0,001m,
this is then the error of the measurement. The same applies to the timings measured.
If the operation that is done with the data is a simple operation, then the absolute errors of all
the data used is added.
In the specific case of a sum, then the errors are calculated in the following way:
If a division or multiplication, then what is added is the relative error of all the data used in the
operation. The result of this is multiplied by the result of the division or multiplication.
(a · b)
If a more difficult operation takes place, like a sine or cosine, then the following applies:
Where f is any function.
An example of this calculation of errors would be in the error of
And with this we can find the errors of
:
for any angle .
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6. BIBLIOGRAPHY
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LÓPEZ, Luciano Federico. Ley de Faraday-Lenz. Instituto Senderos Azules, Buenos Aires, 2011 .
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PAVLIC, Ted. What is an Eddy Current? Physlink.com, 2012. Available online at:
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POLYAKOV, Val. Resistivity of Aluminium. The Physics Factbook, 2004. Avaiable online at:
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http://www.youtube.com/watch?v=zJ23gmS3KHY
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Electrical, Computer and Energy Engineering, 1997. Available online at:
http://ecee.colorado.edu/~bart/book/mobility.htm#sheetres
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