Download Conservation of Momentum and Energy

Conservation of Momentum and Energy
Purpose
To experimentally verify the laws of conservation of momentum and energy.
In particular you will do:
Part A. The Linear Track (set-up in room 266)
1. Newton’s First Law
2. Elastic Collisions
3. Inelastic Collisions
4. (optional). Test of Newton’s First Law using the Kinesthetic Cart.
Part B. The Ballistic Pendulum (set-up in room 267)
Introduction
The laws of conservation of energy and momentum are among the most
fundamental and useful laws of physics. They aid in the solution of many mechanics
problems and come up frequently in many fields of science. What these laws say is
that if there are no net forces on a system, then that system will have the same
momentum, p = mv, at all times. In addition, if there are no external or internal forces
acting in or on a system, then the energy of that system will remain constant.
Newton’s First Law is hidden in these conservation laws. Newton’s First Law states
that bodies at rest will remain at rest as long as no forces act upon them, and bodies in
motion will remain in motion as long as no forces act upon them. As one can see,
Newton’s First Law is a statement about conservation of momentum and energy.
Things stay the same, as long as they are left alone.
Despite their fundamental nature, the conservation laws are often difficult to
observe in ordinary experiences, primarily because of the presence of friction. Friction
between moving bodies and their surroundings means there are external forces acting
on the system, therefore, the conservation laws do not apply. So, to observe the
conservation laws, friction must be eliminated as much as possible.
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This lab will deal primarily with the conservation laws as they apply to collisions
between material objects. These collisions can be divided into two different classes;
elastic collisions and inelastic collisions. If the kinetic energy of a particle is the same
before and after the collision, then the collision is said to be elastic. Notice the
reference to particles. Solid bodies are not particles, but have structure. If the
collision, however, leaves there structure unchanged, they can be treated as particles.
For the other type of collision, energy will flow between the two objects, and the kinetic
energy will not be conserved. In this case, the collision is said to be inelastic. Note that
in the absence of friction, the momentum will be conserved in both types of collisions.
In one dimension, the conditions before and after an elastic collision between two
bodies of masses m1 and m2, initial velocities v1i and v2i, and final velocities v1f and v2f,
are given by
1
2
2 m1 v 1i
1
1
1
+ 2 m 2v22 i = 2 m 1v 21f + 2 m2 v22f
(Kinetic Energy)
m1 v1i  m2 v2i  m1 v1f  m2 v2 f
(Momentum)
For an inelastic collision only the momentum equation is valid.
Prelab Homework
The prelab homework must be done at home and handed to the lab TA before you
start the lab.
1) A small ball of mass m1 and velocity v1i has an elastic collision with a large, stationary
object of mass m2. Show that the velocity v1f of the ball and v2f of the large object
after the collision in terms of the two masses and v1i are
m1  m2
v 1f =  m1  m2  v 1 i
v 2f =  m12m 1m2  v 1i
2) For the previous problem, show that the velocity of the ball is reversed after the
collision if the stationary object is extremely large, m2 going to infinity. Indicate the
limits of the velocities.
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Part A: The Track:
For these experiments you will be using a track with carts that have very low friction
wheels. The track setup with a cart used in the experiments is shown in figure 1. You
will be able to measure the time intervals of a cart using the photo gate timers (see
handout on photo gate timers), from which you can calculate the velocity of the cart.
This is an electronic timer controlled by the interruption of an invisible infrared light
beam when the cart passes the gate. If an object of length L interrupts the beam for a
time interval t while passing through the gate, then the average velocity of the object
during that time is given by
v
L
t
(3.1)
This is the basic measurement you will make in this experiment. By measuring the
velocity of the carts before and after various collisions, as well as weighing the carts,
you will be able to calculate the energy and momentum and test the conservation laws.
Operation of the Timers:
For a complete description of the timers, see the handout.
Fig. 1
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Operation and Setup of the Track:
There are several things that must be done in preparation for the experiments.
Most importantly is to check if the track is level. Levels are provided for this purpose.
Take one, and place it over the track’s feet at one end of the track. First, place it along
the length of the track and observe the level’s “bubble”. You want this bubble to be
exactly in the middle. If it is not, adjust the feet of the track by turning the screws at
the base of the feet. When the bubble is in the middle of the level, repeat this
procedure at the opposite end of the track with the level again length wise. Once both
ends have been adjusted, repeat this again at both ends with the level along the width
of the track. Make sure that one of the end stops is on the right end of the track and the
launcher is on the left side of the track (see Fig. 2). For a description of the launchers
see the handout. To move the end stop and the launcher, loosen the screws on the side
and slide them into position. Lastly, align the two photo gates equidistant from each
other, making sure that the display is facing you. The track is equipped with a
measuring tape so that distances can be measured. Place the first photo gate at 50cm
from the launcher. Place the second photo gate at about 100cm from the launcher so
that the two photo gates are about 50cm apart.
Let us turn our attention to the carts. One end of the cart has a magnet inside and
the other end has a plunger and Velcro strips labeled in figure as Hook-and-pile Pads
(see fig. 3). You will use the plunger for elastic collisions and the Velcro for inelastic
collisions. It might be the case that the Velcro will not stick together well. To test this,
push the plunger in all the way on both carts. Put the two carts together by their
Velcro ends. If they are not sticking well, you can roughen the Velcro by repeatedly
putting the carts together and
pulling them apart. Check that the
carts have no irregularities by
testing them on the track. Place
each cart on the track, making sure
that the wheels are in the grooves.
Give each cart a small push and see
how well it moves on the track.
There should be minimal loss in
velocity along the track. If there is
noticeable loss in velocity, check
the level of the track again. If that
Fig. 2
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Fig. 3
is fine, check the wheels of the cart and ask the TA for assistance.
Overview
The track will be used to create various collisions between the carts. The collisions
will be arranged so that an initially moving cart will pass through a photo gate before
hitting another object, so as to measure the initial velocity. After the collision, the cart
will again pass through a photo gate so as to measure the final velocity. With these
measured velocities, as well as the measured masses, the momentum and kinetic energy
of the carts before and after the collisions can be calculated and compared.
Steps will be taken to deal with systematic errors in this experiment. The
approximate effects of friction will be measured. Launchers are used so the initial
conditions of consecutive trials can be reproduced.
Procedure for experiment A.1: Newton’s First Law
The object of this experiment is to test Newton’s First Law, which states that an
object will remain in its state of motion so long as no forces act upon the object. In this
case, a cart moving with no friction would not change its velocity as it moves along the
track.
1) Place two of the rectangular masses on top of one of the carts. Place a cardboard
strip between the two masses length wise so that it fits firmly.
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2) You must measure the length of the cardboard and weigh the cart with the mass and
the cardboard. Record these measurements.
3) Fix the height of the photo gates so that the beam is interrupted by the cardboard.
Do this by bringing the cart up to each of the photo gates and adjusting the height of
bracket so that it is the cardboard that sets the timer off. You can adjust the height
of the photo gate by turning the screw on the post and moving the bracket up or
down. Make sure you adjust both of the photo gates.
4) Adjust the cart launcher so the scale reads 3.2cm when it is cocked. You can adjust
the compression by loosening the screw on the latching clamp and sliding it into
position. Take one of the carts and bring it up to the launcher. Make sure the
launcher is aimed at the center of the cart. Make sure that the cart’s wheels are in
the grooves of the track. Launch the cart by pulling the string on the launcher.
Once it has passed through both of the photo gates, stop the cart and record the
times displayed on the photo gates.
5) Repeat step 4 two more times for a total of three trials.
6) Repeat steps 4 and 5, this time with a friction block instead of the masses. Place a
piece of cardboard into the groove on the friction block, (you might need two to fit
tightly). Make sure you record the new mass of the cart.
Procedure for Experiment A.2: Elastic Collisions
For this experiment, you will create elastic collisions with the two carts. You will
explore all the possible scenarios; two carts of the same mass colliding, a cart of lower
mass colliding with a cart of a higher mass, a cart of a higher mass colliding with one of
a lower mass, and a cart colliding with an infinite mass (the end stops). It is important
that you set the launchers to the settings described. These settings demonstrate each
scenario best.
1) Place one of the carts with two of the rectangular masses on it and the cardboard
placed as before, between the two photo gates, about three quarters of the way to the
second photo gate. Set the launcher to 3.5cm. Place the other cart with the friction
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block and the cardboard in position to be launched. Make sure that the carts’
plungers are facing each other and that the plungers are all the way out.
2) Launch the cart toward the stationary cart that is in the middle. The cart will pass
through the photo gate and repel off of the stationary cart. The two carts will then
pass through their respective photo gates. Record the initial time that the cart took
to pass through the photo gate and the time the carts took to pass through the gates
after the collision. Remember that the time displayed in memory is the total time.
You must subtract to get the actual second time.
3) Repeat this procedure two more times.
4) Repeat steps 1-3, but this time both carts should have the friction blocks on them so
their masses are equal. Set the launcher to 2cm.
5) Repeat steps 1-3, but this time the cart that you launch should have the two masses
and the cart in the middle should have the friction block. Adjust the launcher to
3cm.
6) Repeat steps 1-3 once again, but this time, use only one cart colliding with an end
stop. Move the photo gates to the opposite end of the track spaced the same way.
Slide the launcher to about 100cm from it’s current position. Set the launcher to
3.5cm. Move the cart into position and launch it. Record the time displayed on
both the photo gates before and after the collision.
Procedure for Experiment A.3: Inelastic Collisions
This final collision experiment will deal with inelastic collisions.
Repeat the
procedure for part B, but this time the Velcro end of the carts should be facing front. In
each collision, the carts should collide and stick together. When the carts pass through
the photo gate together, the beam will be broken only when the cardboard passes
through. This means there will be a space where the beam is not severed, so make sure
you measure the length of the two cardboard pieces and not the length of the two carts.
Procedure for Experiment A.4: optional
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You and your partner with a third person will repeat the experiment demonstrated
by you TA. The instructions on how to do this experiment is in a hand-out which your
TA will give to you.
Data Analysis for Part A.1
The goal in all the trials is to compare the initial momentum of the carts with the
final momentum, and likewise the initial kinetic energy to the final. This will be done
by dividing the final value of the momentum by the initial, giving the fraction of the
momentum that is conserved. If the conservation laws hold, this should be equal to
one. Results less than one indicate that momentum was lost. In the same way, the
fraction of kinetic energy that is conserved will be found.
Newton’s First Law states that if there is no net force acting on an object, its
momentum will not change. Use the data you have recorded to test this by calculating
the fraction of the momentum conserved in each trial. Because there is only one cart, its
mass and length factor out of this ratio, which will then be
pf
pi
(3.2)
where ti and tf are the earlier and later time intervals, respectively. Kinetic energy
should also be conserved. calculate the fraction that is conserved,
kf
ki
(3.3)
The presence of any external forces, such as friction or gravity, will create a systematic
error in all further measurements. To get a rough estimate of the effect this will have,
calculate the average fraction of the initial energy and momentum lost for your data.
To do this, first average all the momentum ratios and all the kinetic energy ratios
calculated above, (to get the average fractions conserved), then subtract these results
from one (to get the average fraction lost),
p f 
p
 1   
p
 pi 
(3.4)
K f 
K
 1   
K
 Ki 
(3.5)
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These results will be used in the error analysis.
Data Analysis for Part A.2
Calculate the fraction of the momentum and kinetic energy that was conserved in all
the collisions; that is calculate
pf
pi

Kf
Ki

m1v 1 f  m2v 2 f
m1v 1 i
(3.5)
m1 v 21f  m2v 22 f
m1 v 21
i
(3.6)
Velocity and momentum have directions, so give them signs in your calculation (i.e.
right is positive). You did three trials each, average these results.
Data Analysis for Part A.3
Again, find the average fraction of momentum and energy conserved. In this case,
v1f and v2f are the same (because the carts are joined together).
Error Analysis for Part A
If the conservation laws are correct, it is still unreasonable to expect the fractions
calculated to have values of exactly one, because of experimental errors. Errors in
length and time measurements will have only a small effect on the results. The
systematic error caused by friction Is more important. Once the losses are found, we
will need to know how much can be attributed to friction before concluding the
conservation laws were not followed. The losses due to friction were calculated for the
trials in part A (with equation 3.4 and 3.5). Although this will not give the exact error
due to friction, it should not be more than a few times larger. So, as long as the results
differ from one by at most three or four times the losses found for part A, we can
assume those differences are likely due to friction. Decide which conservation laws
were verified in each part.
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Part B: Velocity of a Projectile
In this experiment, the velocity of a projectile as it leaves a spring gun will be
measured using two methods. First, the conservation laws will be used in a somewhat
subtle way to determine the velocity. For comparison, it will also be measured using
projectile motion.
Part B1 Measuring velocity with a Ballistic Pendulum
A ball is shot by a spring gun into a catcher arranged to swing as a pendulum
(figure 3.3). When the ball is caught, the combination of the ball and catcher become
the bob of the pendulum. Although the collision between ball and catcher is inelastic
and energy is not conserved, momentum is. With the catcher at rest, the initial
momentum of the system is provided by the ball, shot with velocity vb. Just after the
ball is caught, the momentum that is due to motion of the center of mass of the
pendulum assembled from the catcher and ball, having velocity vp . Conservation of
momentum requires these be the same,
mvb  M  mvp
pendulum
(3.7)
cone bearing
screw
ball
catcher
bob
ball
latch
spring
C. M.
indicator
spring
gun
ball
h
pawl
h
1
h
2
trigger
Figure 3.3
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where m and M are the masses of the ball and pendulum respectively.
After the ball is caught, the energy of the combined ball and catcher system is
conserved. Initially the center of mass is at a height h1, with velocity vp. As it moves
upwards against the force of gravity, the kinetic energy is converted into potential
energy. A pawl mechanism latches it at its highest point, h2, where all the initial
kinetic energy of the pendulum has been used to produce a change in the potential
energy,
1
2
M  mgh  M  mv p2
(3.8)
which can be determined by measuring the increase in height h = h1 - h2 of the center
of mass of the assembly. The value of the initial velocity of the ball is,
 M
vb  1  2gh
 m 
(3.9)
Procedure for Part B1
Be sure your apparatus is firmly fastened down so it will not move during the
experiment. Check to make sure the base is level and adjust if necessary. The ball
mounts on the gun's push rod via a hole through its diameter. Make sure this fits
smoothly on your apparatus. Make sure that the ball catcher is aligned with the spring
gun.
1) Measure and record the weight of your ball. To remove the ball from the catcher, lift
the latch spring with your finger while pushing out from the rear. The ball should
come out easily. Do not bend the spring backwards, or it may break.
2) Weigh the pendulum. To do this, carefully unscrew the cone bearing screw until
the pendulum is released. Replace
lift lat ch spr ing
push out
the pendulum and tighten the cone
t o r elease ball
fr om r ear
bearing gently but firmly back in
place.
3) To cock the gun, first rest the
pendulum on the rack to get it out
cat cher
ball
Figure 3.4
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of the way. Then, with the ball in place on the rod, push on the ball until the trigger
latches.
4) With the pendulum hanging freely measure the height of the Center of Mass (C.M.)
indicator from the base.
5) Starting with the pendulum at rest, fire the ball into the catcher nine times (be
careful not to get your hand caught in the gun when you fire it) and measure the
height of the C.M. indicator from the base each time. It will not always be the same.
Part B2: Measuring velocity with projectile motion
With the catcher removed from its path, the ball will fall freely after it leaves the
gun. It will follow a parabolic arc until it hits the table top, traveling a horizontal
distance L while falling a vertical distance h. The equations of projectile motion can be
used to analyze the motion in these directions, as shown in figure 3.5. Combining the
two equations to eliminate time, the initial velocity of the ball is then
vb  L
g
2h
Procedure for Part B2
scr een
h=
1
2
gt
2
L = vb t
Figure 3.5
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(3.10)
Make sure the apparatus is securely fastened and leveled on a block on the table.
Latch the pendulum bob out of the way and practice firing the ball from the gun out
over the table. Note where the ball lands. Be careful of wild shots.
1) Tape a piece of white paper over the area of the table top where the ball lands.
Cover it with carbon paper, carbon side down, so that the impact point of the ball
will produce a mark.
2) With the ball in place and the gun released, find the position on the table top directly
below the center of the ball and mark it with a piece of tape.
3) Measure the height from the bottom of the ball to the table top.
4) Fire five shots, marking and numbering each of the impacts.
5) Measure the range L of each of these shots from your tape mark.
Data analysis for Part B1
Calculate the average increase in height of the center of mass of the pendulum and
its standard error (from the standard deviation of the measurements). Use this data to
calculate the initial velocity of the ball and its error (see Appendix on Error Analysis).
You may assume there is no error in the masses.
Data Analysis for Part B2
Calculate the average range L and its error ∆L from the standard deviation. Make a
reasonable estimate of ∆h, the possible error in your height measurement. Calculate
the initial velocity of the ball from the measurements of h and L. Calculate the error in
the initial velocity ∆vb obtained with this method. The result from propagation of
errors (see Appendix on Error Analysis) is
2
2
 L  h 
vb  vb

 L  2 h 
13
(3.11)
Compare the results you obtained for the initial velocity of the ball by the two different
methods. Are they consistent within your errors? Can you account for discrepancies?
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For the Lab Tech and TA
Equipment
Part A (Room 266)
 Track
 2 photo gate timers
 2 carts with plungers
 2 rectangular masses
 2 friction blocks
 scale
 4 cardboard strips
Part B (Room 267)
 spring gun & ballistic pendulum apparatus
 carbon paper
For the Person giving the Lab Lecture
1. Description of lab 3, explain vectors, formula.
2. Discuss prelab for Lab 3
3. If there is time, mention origin of conservation of energy and momentum and
relation to invariance principles with respect to position and time.
For the Lab TA
1. Include, as part of your lecture, a demonstration of the Kinesthetic Cart.
2. Divide the class into two groups. The first group will begin with experiment A.
The other group will start experiment B. The groups will switch experiments half way
through the lab session.
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To: The student doing this experiment
From: David Douglass
B&L 156
e-mail: [email protected]
I am involved in the improvement of the undergraduate physics laboratories. This
experiment is one of fifteen total. Your input is important and essential for future
upgrades.
I would appreciate any comments that you wish to make. Simply write them
below, remove, fold, and drop into any university mailbox. I will respond individually
to all signed comments
David Douglass
B&L 156
revised 8/20/96 eb/dhd
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