Download Physics Laboratory 1 Last modified : 2007.4.2 Experiment 3. Worry

Physics Laboratory 1
Last modified : 2007.4.2
Experiment 3. Worry of Sisyphus
purpose of experiment
The energy in nature is a conserved quantity. That is, the energy of an isolated
system does not change as time varies. (It has time invariance) However, as
the types of energy may vary, when we consider one type of energy it seems to
disappear when the energy changes its type.
The energy relevant to motion of a body under the earth’s gravitation is the
gravitational potential energy and the kinetic energy, and the sum of these two
is called mechanical energy and we call a system where the mechanical energy
is conserved, a conservative system. When friction is involved in the motion of
a body then the mechanical energy decreases as time passes because it can
change its type into sound or heat due to friction. In this experiment, we
investigate the transformation between the kinetic energy and the potential
energy of a body in motion under the gravity of the earth and conservation of
mechanical energy and the effect of friction.
Experiment Outline
 Investigate the change of the kinetic energy as per motion of a body and that
of potential energy as per the displacement under gravity. Measure each type of
the mechanical energy and confirm that the sum is constant. (In this experiment,
we investigate the ball rolling down a linear track with a PC camera. Plan on
how to do the experiment in advance.)
 Investigate the difference when it rolls with and without slipping.
 The kinetic energy of a rolling body has both translational and rotational
components. What is the expression for each component?
 What about the slipping case?
 What can you infer from this experiment? Draw a conclusion.
 Is the mechanical energy of the system you investigated conserved?
 If not, what is the reason? Can you quantify it?
Experiment method
The following devices are prepared for the experiment. (in parentheses are the
number of the devices)
A linear track (1) A circular track (shared by group 1 and 2)
A ball (1)
A ruler (1)
A plastic dish to catch a ball (1)
A linear(circular) track stand (2)
A CCD (1)
A balance (2, shared)
If you need anything else, check with your teaching assistant or the experiment
preparation room(19-111, 25-418) or prepare it yourself. You should study the
balance and the micrometer in advance and make up a plan in advance.
The following is the recommended experiment method.
(movie)
1) Measure the time △t, as the function of height h, it takes the ball to pass the two
points Z1 and Z2 and compare with the case in which the mechanical energy is
conserved.
① set up the apparatuses as in the movie. Set up the camera so that the motion is
observed from the side. Put the reference ruler on the same plane (where motion
occurs) of the track. (*think about why we put the camera and the ruler there and
discuss it in the result analysis.
Turn on the computer and run “I-CA”. click [filecamera configuration] on the menu and make sure the
CCD screen is turned on. (movie)
* Tip. Camera configuration
1. After running I-CA program, select the camera
configuration in the file menu.
2. Push the menu of the remote control till the setup
menu pops up.
3. Select the ALS/AES on the third row.
4. Select the LEVEL -FIX- OFF on the last row.
5. Select 1/250
6. Push BACK on the remote control and return to the
previous.
7. Select AGS/SENS
8. Select LIGHT and NORMAL
9. Select SENS
10. Select X32(32times)
11. Push BACK twice to exit
② put the bead at the height of h and start the experiment. Movie. When saving data is
finished, select [picture-screen analysis] and analyze the saved data. Specify the datasaving path, determine the first and the last frame to analyze and the reference color of
the object. Set up the reference point and ratio reference line and start the analysis.
Movie movie think about what you should take for the reference line. (you can see the
searching for the object in the designated analysis area from the movie)
movie
③ after the analysis, save the data and you can see the image file used for analysis and
the location data file of the object on the screen. (x and y coordinates as time flows)
④ you can obtain the coordinates of two points and the time △t it took for the bead to
pass these two points from the data. Make a measurement more than five times for each
height h and obtain the average and the stddev of each △t, and investigate for the height
h of more than five different cases.
⑤ you can draw a graph obtained using Excel. (if you would like open data using Origin,
open the data using Excel and select the file type as [text(delimited by tabs)] and save it
and then open it with Origin) you can obtain the angle and the time △t of passing two
intervals from the data.
⑥ Obtain the relation between the increase of the kinetic energy of the ball and the
decrease of the potential energy from the measurement. (using two data (time) in a row
from the data obtained by measurement you can obtain the average speed in the
perpendicular direction at the certain interval, and using this average height of two points
you can obtain the increase and decrease change of the kinetic and the potential energy.
⑦ What can you infer about the mechanical energy this system from this? Make
sure you obtained the correct kinetic energy of the bead rolling on the track.
2) repeat the above experiment using beads of different masses. What difference can
you see when the masses change? What is the reason?
3) find hmin the minimum starting height where the bead starts to fall without
reaching the maximum of the circular track and compare with the theoretically
expected value. Can you explain the difference, if any? If the bead is to overcome
the maximum of the circular track without falling, what should be the speed of the bead
at the maximum? Compare the result from the data and the theory. If there is a
difference, think about why.
4)investigate the effect of friction from the rail track.
Taking an appropriate height where the bead begins to roll, measure the height (from
data) to which the ball returns and see the effect of friction that reduces the kinetic
energy.
background theory
How do the bodies of two at a distance affect (gravitation) each other?
Consider a gravitational field generated in the neighborhood of a body and
suppose that this field applies a force on another body, by way of
understanding the force between bodies at a distance. Since a body under a
force moves with acceleration, its speed and kinetic energy increase. If an
external force exists then the change of the mechanical energy may be
understood as the work done by the external force, however, taking account of
the energy of the external force source, the mechanical energy is still
conserved. (i.e. total energy considering the energy of external force source is
still conserved.) Acceleration of a body by the gravitational field is regarded as
by the potential energy of the body under the gravitational field. That is, a
potential energy means a power that can be transformed into a kinetic energy
or that can do work. Quantitatively, the potential energy is defined as follows.
To move a body of mass m with constant velocity under the gravity
Fg = - mg (g is the gravitational acceleration)
the external force
(1)
f = - Fg
(2)
that cancels this gravitational force should be applied. The work done by the
external force to move this body from the displacement xo to x with constant
velocity is
(3)
since the kinetic energy of a body with constant velocity does not change, this
work done by the external force has to increase other type of mechanical
energy. That is, it increases the potential energy and the change of the
potential energy is
(4)
since the potential energy is defined by change as in the equation (4), its
absolute value is undetermined and has no meaning. Therefore take an
appropriate position of reference(eg. The surface of the earth) and set the
potential energy at that position as some convenient value(eg. Uearth’s surface=0)
then the gravitational potential energy at the height h of this body is
U = mgh
In this process of obtaining the potential energy, the relation,
(5)
E = T + U = 1/2 mv2 + mgh
(6)
that the sum of kinetic and potential energy, i.e., the fact that the body’s
mechanical energy is constant is already used, therefore the conservation of
mechanical energy holds. The motion of the steel ball cannot be explained
sufficiently by the motion of the center. This is obvious because the ball can
have a rotating motion without changing the location of its center. Generally,
the kinetic energy of a rigid body like a steel ball is the sum of the translational
kinetic energy of center of mass and the rotational kinetic energy about the axis
through its center of mass. The kinetic energy of a ball of mass m, the radius r,
the speed vcm of motion of center of mass and the angular velocity ω (rotating
angle per second) about the axis through the center is
(7)
where I is the moment of inertia about the axis through the center of the ball.
The moment of inertia could be thought of as the inertia of rotational motion,
and for a homogeneous sphere,
(8)
When a ball rolls without slipping,
(9)
Therefore, the mechanical energy in the equation (6) in the case of a ball rolling
without slipping,
E = T + U = (7/10) mv2cm + mgx
(10)
Beware that the velocity v is the velocity of the ball on the slope and x is the
height(perpendicular) of the ball. Let us investigate the time △t it took the ball
at rest at the height h to start rolling on the slope and to pass h1 and h2.
Since the mechanical energy of the ball at rest at the height h is
E = mgh
(11)
the center of mass velocity (of the center parallel to the slope) of the ball when
passing the height x is, if conservation of mechanical energy is assumed,
(7/10)mv2cm + mgx = mgh
therefore
(12)
(13)
Let θ the angle of the slope then the velocity vcm is
(14)
(15)
(16)
and integrating
(17)
(18)
i.e.
(19)
is obtained. Since this equation is obtained by the assumption of conservation
of mechanical energy, by comparing the equation (19) with measurement of
time t it took the ball at rest at the height h to start rolling on the slope and to
pass h1 and h2, it can be shown whether the mechanical energy is conserved.
In the case of the ball rolling not on a slope like this experiment but on a
slanted track, the effective radius of rotation changes as
(20)
in the picture. d is the distance between the points where the ball touches the
track.
The equation (9) can be expressed using this effective radius
(21)
How, then, does the expression of (1) change? What about the case where the
steel ball rolls with slipping? The minimum height for revolution of the circular track
of radius Rcir can be calculated as follows. At the maximum of the circular track the
total mechanical energy is
(22)
if the bead falls from the maximum of the circular track, even when there is no
normal force of the circular track, the force of gravity on the bead at the maximum
height when falling is the same as the centripetal force.
(23)
With this, you can obtain the minimum starting height hmin from where the bead can
reach the maximum without falling
※ Questions
The roller coaster must have a certain speed or more to move on the circular track at
the maximum point. (this is obtained from the experiment) Also, at the end of the
linear track (generally at the minimum height) it gives the maximum speed to double
the fun. How high, then, the height should rise to double the speed at the minimum
height? (ignore the attenuation from friction) when the speed is doubled, how much
do the momentum and the kinetic energy increase?
The law of energy conservation and translational symmetry in time of physical
phenomena
The law of energy conservation is one of the several conservation laws as the
properties of nature that are important in physics. As the law of momentum
conservation and the law of angular momentum conservation, the law of energy
conservation states that the total amount of energy of an isolated system does
not change as time flows, this means the energy itself cannot vanish or created
even though the inner parts constituting an isolated system can exchange
energy among them or that the type of energy of the isolated system can
change. What does it mean that the energy of a system does not change as
time flows? It means that the energy of the system and natural phenomena
related to the energy stays the same whenever measurement is made. This
property is called symmetry with respect to translations in time. The laws of
conservation are attributed to the symmetrical properties the system or nature
has, the law of momentum conservation is from the fact that the properties of a
system do not change when observer’s coordinate is translated in space, and
the law of angular momentum conservation is from the fact that the properties
do not change when the direction of the coordinate is rotated. These laws of
conservation coincide with the statement that there is no absolute time axis or
spatial axis in nature.
- Fundamentals of physics by Halliday & Resnick
Without friction between a bob (a) hooked to a wire and a body (b) connected
to a spring, mechanical energy is conserved and their motion continues forever.
In this case, regardless of the time when observation of motion starts, the law
of nature (law of motion) can be found.
※ friction and the law of energy conservation
When friction acts on a moving body, the mechanical energy of the system is
not conserved. That is, the mechanical energy of the system changes as time
varies. Since friction is an interaction between the body and other systems(a
desk or air, etc.), the body is not an isolated system so the law of (mechanical)
energy conservation for an isolated system does not hold. However, as friction
converts part of the mechanical energy to thermal energy, the total energy
including the heat of the total system (body + desk, body + air, etc.) is
conserved. This is the law of energy conservation in a broad sense. The
exchange between the kinetic energy and the potential energy occurs freely but
the transformation between the thermal and mechanical energy has a direction.
That is, the mechanical energy is transformed by friction into the thermal energy,
but the thermal energy is not transformed into the mechanical energy
spontaneously. Why is that? This directional phenomenon in nature is called the
second law of thermodynamics. Without this direction in transformation
between the mechanical energy and the thermal energy, a huge cruise ship
could sail by using the thermal energy for cooling the seawater without fuel.
This property of nature called the second law of thermodynamics provides us
with food for thought such as the impossibility of perpetual engine and time’s
arrow flowing in one direction, etc.
References

A method of processing the measurement data

analysis by graphs
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