Download 5/06 Gravitational Energy Conservation Figure 1 A

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Gravitational Energy Conservation
Figure 1
A string passes over a pulley (assumed frictionless) to connect a cart mass on the
track with a hanging bucket mass. Since the string doers not stretch, both masses
have the same velocity and acceleration. The string tension and a small, parallel
component of the gravitational force on the cart provide acceleration along the track
to the cart mass and do work on it. The normal force has no component in the
direction of motion, and does no work on the cart; it counteracts the perpendicular
component of the gravitational force, which also does no work on the cart.
About this lab:
Physics has as an overarching goal the discovery of principles of general validity. Among
the most profound are general conservation laws, involving dynamical quantities which
remain unchanged while a system evolves. A handful of such have been discovered, some
in familiar everyday experiences, and some in the less familiar world of relativistic and
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quantum behavior.
And the constancy of the dynamical quantities has been found to correspond to a
behavioral symmetry (sameness) with regard to a particular descriptive coordinate.
Conservation of energy is such a general law, and the corresponding descriptive symmetry
coordinate is time.
You will observe visually the qualitative conservation of energy during a run, and
graphically the quantitative conservation.
References: Cutnell & Johnson: Chapter 6, Figures 6.10, 6.11,6.15-6.21
Apparatus: Pasco track, Pasco cart, Lab Pro interface and cables, motion sensor, pulley,
mass bucket & long thin string, assorted mass hanger weights, wood blocks (for incline)
Study the applet:
http://www.walter-fendt.de/ph14e/pendulum.htm
Forces acting ON the bob are: gravity (down), and tension (along the string).These both
act to produce acceleration. But only gravity does work on the bob, because the tension is
always at right angles to the direction of motion (velocity vector), i.e., the cosine factor is
always zero. If you pause the motion and switch between the kinematic variables
(displacement or "elongation", velocity, acceleration) you can see the relative phases of the
quantities in the graph pattern changes. Look also at the force graph - it should be in
phase with the acceleration.
The energy graph is interesting. Note that the kinetic and potential energies are out of
phase, and have a constant total because there is no energy loss mechanism assumed - i.e.,
no friction. What kind of potential energy is involved in these animations?
Note that you can change gravity, pendulum length etc. Measurement of the pendulum
period for known length would determine the surface gravity on the moon.
Conservation of Energy
The Work-Energy Theorem states that:
The net work (+ or -) done on an object equals its change in kinetic energy.
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(1)
Gravitational Energy Conservation
W = KE final – KE initial
where the kinetic energy of an object is
KE = ½ mv2
The work done on an object is given by
W = F cos(θ)
where s is the distance over which the force acts and  the angle between the force and
the displacement. Cos(θ) may be either + or - . For a single mass dropping to the floor
from height h, the only force doing work (causing motion) is the gravitational force on the
small mass, F = mg (down), which acts over the distance h, the height from which it drops
to the floor: the work is mgh. Lateral moves don't add any work (absent friction) – the
cos(θ) factor is 0 (θ = 90 degrees).
For certain important forces (gravity, electrostatic) the work done depends only on
initial and final positions, independent of work path. For other forces, the path
matters – different paths, different work. Our present experiment involves the former type
(gravity), which is called an (energy) conservative force.
We can now expand on equation (1):
(2)
mgh = ½ mvf 2 – ½ mvi 2
The quantity mgh is the potential energy the mass acquires as it is moved from the floor
(zero potential) to its starting point (mgh potential). (Incidentally, we could also have
assigned zero as the potential energy before the fall and -mgh the potential of the bucket
mass on the floor.)
You can therefore read equation (2) as “the potential energy of a falling mass is converted
to its kinetic energy”. Note that you can also read the equation backwards, as in “a small
mass thrown in the air has its kinetic energy converted to potential energy”. In this way
one form of energy is converted into the other, and possibly back again, as in a golf ball
flight. (But the golf ball loses energy in a floor bounce and the next bounce is not as high the elastic compression forces are not perfectly energy conserving.) For conservative
forces (forces in which the work done is independent of the path, e.g. gravity), equation
(2) can also be written as:
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KE i + PE i = KE f + PE f
(i = initial, f = final)
This states that total mechanical energy, kinetic + potential, is conserved. For nonconservative forces (forces in which the work done is dependent on the path, e.g.
friction), the above lacks meaning, as there is no defined potential energy. The nore
general statement that "Work done on a system = System change in kinetic energy"
still holds; Equation 3 is just a special case of this.
Note that work on a system can be + or -, depending on whether the net force acts in the
direction of motion, or opposite. Note also that a force always acting acting perpendicular
to motion produces acceleration but no change of kinetic energy, because the cosine fa tor
is always zero. A case in point is a charged particle bent by a static magnetic field.
Another is a pendulum swinging from a string - the string tension contributes to the
acceleration without doing work.
Procedure
The Logger Pro data taking program will receive cart displacement input signals,
and calculate the related velocity (which is the same for the falling mass, as long as
the string is taut – i.e, before the bucket mass hits the floor). Logger Pro will also
calculate the energy quantities in Eq. 3, if bucket potential energies are defined as +
when the small bucket weight rises above its initial position, and – when it falls
below, and similarly for the cart.
The small pulley rotational kinetic energy is neglected, as well as friction.
> Vary the falling bucket mass (bucket + bucket load), the cart mass (cart + cart load),
and the track inclination angle. In particular, run: level track, uphill and back down,
downhill toward sensor with cart lifting the bucket mass. You can alter run time:
“Experiment: Data Collection” .
> Observe graphically the kinematic quantities displacement, velocity and acceleration,
and the derived energy quantities Total KE, Total PE and Total E (= KE+PE), checking
energy conservation.
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Figure 2
The system consists of two masses that will share the same velocity if the string does
not stretch (the cart will move the same distance as the bucket), and the same
acceleration. For a level track, both cart and bucket will contribute to KE (same V),
but only the bucket (+ its contents) will contribute to potential energy. But, for an
inclined track, the cart will also go up or down – then it will also contribute to the
potential energy.
In the data table of the Logger Pro file, double click on V, A, Total KE, Total PE and
Total E. Note that these are all calculated quantities; only the position is a measured
quantity. Examine carefully the definitions. Determine where masses and angles are input.
You will need to change these definitions whenever you change a mass or
inclination!!
Angles should be in degrees. For the level track, the angle is 0. Instead of entering the
angle in the PE definition, you could just eliminate the “sin” and enter the numerical value
of the sine: (h/x). But if you leave "sin" in the definition, you must enter the angle
> Curves are better if there is more energy in the system - e.g., faster runs from tilting
the track, more weight, etc. (less effect of friction). A preliminary run will indicate the
valid length (run duration). Then the length of the run can be reset (Experiment:
Data Collection, length) for a re-run. But this does not reset the graph time limits
If the track is tilted so that both cart and bucket are falling, both terms in the total PE
definition should be – relative to initial PE, etc. It is best to put angles or sines in the
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PE definition as + and then deal explicitly with the signs of the two separate terms. A
falling object should have – PE sign, and a rising object, +.
> A perfectly good run can display energy curves (lower right plot) which suppress the
vertical scale in the time region of interest.
If there is a portion of a two-second graph (typically at the end) which suppresses the
good data part, there are various ways to deal with it:
a) Edit: Strike through data cells. This inactivates those data in the plot. (They can be
restored.) This is probably best.
b) Reset the plot limits to cut out the unwanted parts of the energy plot (the others are
not of so much interest, so it may not be necessary to reset them also). Select the last
axis values (horizontal and vertical) and reset them. But, Autoscale destroys your
limits reset.
c) For inspection, select region of interest and magnify.
> You may redefine the definitions of KE and PE (masses, angles) after a run
changes the plots - it is not necessary to re-run if the wrong parameters were used
initially during the run. The data won't change, just the calculated columns.
> An energy graph may show positive initial kinetic energy. There is nothing wrong
with this. It could arise from a little shove on release, or from the sonic ranger turning
on after cart the cart has started moving.
Even when the unwanted portions are removed from the graphs, the acceleration
graphs probably will look jagged. This is due to the second differentiation of the basic
distance data. But the average value is rather constant in the valid region (as
expected).
A start from rest involves zero initial total energy. In this case, small numerical differences
between the calculated KE and PE can eventually become significant. Also, frictional
effects can have cumulative effect on KE, as can neglected pulley KE. Try to interpret
deviations from conservation which appear in your graphs. How should the appearance of
the total energy curve be affected if there is friction (energy loss)? Will either of the other
two energy curves be affected? Which?
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Figure 3
The cart position curve represents data. The other kinematic curves and the energy
curves are generated in calculated columns. If a mass or angle is changed,
definitions in these calculated columns must be changed for the plots to be valid.
You can change after the run; before is best to avoid confusion.
Look at the Logger Pro sample screen capture above. Interpretation is key. The 9 gm
bucket contained a 20 gm additional weight. 0.029 (kg) was entered in both the Total KE
and the Total KE column definitions. The cart mass was 0.484 kg, and angle was 0.
Inspect your KE and PE definitions initially, and every time you change masses or track
angle (enter in degrees, + or -). Sin(θ) is obtained by measurement: sin(θ) = h/x, where x
is track length (slant distance along the track, the hypotenuse of the triangle), and
h is the difference in height between track ends (not height off the table), but you
can measure up from the table and take the difference.
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The upper left (position) and lower left (velocity) curves show that the cart did not start at
rest – it was launched toward the motion sensor (d decreases, velocity is negative and
becoming less so as time increases). Position is minimum (sensor may not measure
correctly if cart gets too close) when velocity changes sign (goes positive).
Inspect the upper right acceleration graph. It is choppy, due to the numerical
differentiation, and slopes down a little. (Possible explanations of slope?) But it is
roughly constant, and clearly does not change sign – the acceleration is always away from
the motion sensor (in the direction of the net horizontal force on it (tension T of string)
After the mass hits the floor, the string goes slack and the definition of KE and PE
in Logger Pro will no longer hold. Cart and bucket velocities and accelerations are
identical before impact.
A Modern View of Energy Conservation
Emmy Noether: Symmetries and Conservation Laws
“ Emmy Amalie Noether
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Born: 23 March 1882 in Erlangen, Bavaria, Germany
Died: 14 April 1935 in Bryn Mawr, Pennsylvania, USA
The key to the relation of symmetry laws to conservation laws is Emmy
Noether's celebrated Theorem. ... Before Noether's Theorem the principle
of conservation of energy was shrouded in mystery, leading to the
obscure physical systems of Mach and Ostwald. Noether's simple and
profound mathematical formulation did much to demystify physics. --Feza Gursey
Energy conservation is a direct consequence of the fact that the laws of nature
do not change with time. It is therefore not something special to Newton's laws
of motion, but is true for much more general physical theories.
Lavoisier's careful accounting for mass changes in chemical reactions gave rise to the idea
that total mass is conserved. Newton gave attention to the concept of conservation of
energies in collisions, concluding that the kinetic energy was conserved in “hard” collisions
(we would now call them “elastic”collisions). Liebniz, a contemporary, sharpened
Newton's views, calling kinetic energy “vis viva” (“living force”). Now, we would say that
“mechanical” (organized) energy is manifestly conserved in elastic collisions, but not in
others, termed “inelastic”. But, taking into account random internal motions (“heat”), we
can consider total energy to be conserved in all cases.
In the 20th century, Einstein's special relativity modified these separate insights into a
combined conservation law of inter-convertible mass-energy. And the existence of this
energy conservation law was identified with time symmetry of physical law (symmetry,
invariance, sameness -- “no change”, conservation) – the experimental observation that an
experiment gives the same result tomorrow as last week or next week.
Other conservation laws (linear momentum, angular momentum, etc.) have also been
identified with symmetry principles of our universe. The mathematical contributions of
Emmy Noether formalized and clarified this connection between a symmetry of nature and
the necessary existence of a corresponding conservation law.
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Emmy Noether's father Max Noether was a distinguished mathematician and a professor
at Erlangen. Her mother was Ida Kaufmann, from a wealthy Cologne family. Both Emmy's
parents were of Jewish origin and Emmy was the eldest of their four children, the three
younger children being boys.
Emmy Noether attended the Höhere Töchter Schule in Erlangen from 1889 until 1897.
She studied German, English, French, arithmetic and was given piano lessons. She loved
dancing and looked forward to parties with children of her father's university colleagues.
At this stage her aim was to become a language teacher and after further study of English
and French she took the examinations of the State of Bavaria and, in 1900, became a
certificated teacher of English and French in Bavarian girls schools.
However Noether never became a language teacher. Instead she decided to take the
difficult route for a woman of that time and study mathematics at university. Women were
allowed to study at German universities unofficially and each professor had to give
permission for his course. Noether obtained permission to sit in on courses at the
University of Erlangen during 1900 to 1902. Then, having taken and passed the
matriculation examination in Nürnberg in 1903, she went to the University of Göttingen.
During 1903-04 she attended lectures by Blumenthal, Hilbert, Klein and Minkowski.
In 1904 Noether was permitted to matriculate at Erlangen and in 1907 was granted a
doctorate after working under Paul Gordan. Hilbert's basis theorem of 1888 had given an
existence result for finiteness of invariants in n variables. Gordan, however, took a
constructive approach and looked at constructive methods to arrive at the same results.
Noether's doctoral thesis followed this constructive approach of Gordan and listed systems
of 331 covariant forms.
Having completed her doctorate the normal progression to an academic post would have
been the “habilitation”. However this route was not open to women so Noether remained
at Erlangen, helping her father who, particularly because of his own disabilities, was
grateful for his daughter's help. Noether also worked on her own research, in particular
she was influenced by Fischer who had succeeded Gordan 1911. This influence took
Noether towards Hilbert's abstract approach to the subject and away from the constructive
approach of Gordan.
Noether's reputation grew quickly as her publications appeared. In 1908 she was elected
to the Circolo Matematico di Palermo, then in 1909 she was invited to become a member
of the Deutsche Mathematiker Vereinigung and in the same year she was invited to
address the annual meeting of the Society in Salzburg. In 1913 she lectured in Vienna.
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In 1915 Hilbert and Klein invited Noether to return to Göttingen. They persuaded her to
remain at Göttingen while they fought a battle to have her officially on the Faculty. In a
long battle with the university authorities to allow Noether to obtain her habilitation there
were many setbacks and it was not until 1919 that permission was granted. During this
time Hilbert had allowed Noether to lecture by advertising her courses under his own
name. For example a course given in the winter semester of 1916-17 appears in the
catalogue as:
Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr
E Noether, Mondays from 4-6, no tuition.
Emmy Noether's first piece of work when she arrived in Göttingen in 1915 is a result
in theoretical physics sometimes referred to as Noether's Theorem, which proves a
relationship between symmetries in physics and conservation principles. This basic
result in the general theory of relativity was praised by Einstein in a letter to Hilbert
when he referred to Noether's penetrating mathematical thinking. “
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html
She was never elected to the Königl. Gesellschaft der Wissenschaften zu Göttingen .
[h1970cr] Her great 1918 paper on symmetries and conservation laws was
communicated to the Gesellschaft by Felix Klein.
Auguste Dick raises interesting questions regarding the fact that Noether was never
appointed to a paid position in the faculty of the University of Göttingen:
"How was it then that in her academic career she did not go beyond the
[unpaid] level of nicht-beamteter ausserordentlicher Professor? ... Was it
because she was Jewish? There were several Jewish Ordinarii in Göttingen.
Was it because she was a member of the social- democratic party? ... Or was it
her firm stance as a pacifist that was frowned upon? ..." -- [en1981ad]
As a Jewish woman, in 1933 Emmy Noether was fired from her position as a privat
docent in Göttingen. By decree no Jew was allowed to teach after Hitler came to power.
(In 1934 women were dismissed from University posts.)
Hermann Weyl wrote about her in this period
" A stormy time of struggle like this one we spent in Göttingen in the summer of
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1933 draws people closely together; thus I have a vivid recollection of these
months. Emmy Noether - her courage, her frankness, her unconcern about her own
fate, her conciliatory spirit - was in the midst of all the hatred and meanness,
despair and sorrow surrounding us, a moral solace." [sm1935hw]