Physics 31210 Spring 2006
Experiment 5
Energy and Momentum Conservation
The Ballistic Pendulum
I. Introduction.
In this experiment we will test the principles of conservation of energy and conservation of
momentum. A ball is shot into a cup which is itself the bob of a pendulum. The pendulum
swings up and is caught at its maximum height by a toothed rack. In the first step, when the
ball collides with the cup, one expects linear momentum to be conserved. In the second,
when the pendulum recoils and swings up, one expects mechanical energy to be conserved
as the initial kinetic energy of the pendulum is transformed into potential energy. (Note that
mechanical energy is NOT conserved in the inelastic collision when the ball is trapped in
the cup.) For both of these processes you will use the appropriate conservation principle to
calculate, from independent measurements of other quantities, the initial speed V of the
pendulum bob (including the ball) just after the ball strikes it, but before it has recoiled
significantly. The consistency of the two determinations of the speed V provides a
demonstration of the conservation laws.
NOTE: Both determinations of V involve possible sources of systematic error, which are
not accounted for in the calculations. You are expected to look for and identify possible
sources of systematic error and to list them and estimate their importance in your Laboratory
Report.
II. Required Equipment.
Beck ballistic pendulum assembly including spring gun, ball, swinging pendulum,
meter sticks, rulers, computer. (See Figure 1.) Make sure that the pendulum is securely
fastened to the support arm assembly at the top before beginning the experiment. Make
sure the ball is in the same position each time the spring gun is fired.
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Experiment 5
FIGURE 1
III. Procedure.
A.
V from Conservation of Mechanical Energy.
1.
After the ball hits the pendulum cup, the kinetic energy of the pendulum plus ball is
converted into potential energy as they swing up and come to rest at a some height
ymax. From conservation of mechanical energy applied to the ball-pendulum system:
● E = K + U = (1/2) (M+m)v2 + (M+m)gy.
where M is the mass of the pendulum;
m is the mass of the ball;
v is the magnitude of the instantaneous velocity (the speed) of the system;
and y is the elevation of the ball in some convenient coordinate system,
for example y = 0 could be chosen for the elevation of base of the ballistic
pendulum.
● When the ball is projected into the cup, y = yo, and the system recoils with
velocity v = V. At this instant, E is given as:
E = (1/2) (M+m)V2 + (M+m)gyo.
● When the pendulum reaches its maximum elevation (y = ymax), its
instantaneous velocity is zero, and the kinetic energy term vanishes. Hence E
may now be written:
E = (M+m)gymax.
● If total mechanical energy is conserved, then E is a constant and thus:
(M+m)gymax = (1/2) (M+m) V2 + (M+m)gyo
● Hence we can solve for V:
V = sqrt [2g(ymax-yo)].
● Figure 2 shows the geometry of the system.
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Experiment 5
y max
V
yo
FIGURE 2
2.
3.
To measure y properly, we need to locate the center of mass of the ball/pendulum
system.
● To do so, place the ball in the pendulum cup. Remove the pendulum from
the support arm - but keep all bolts and nuts attached. Carefully place the
pendulum on a “knife edge”, a thin metal strip that you used in the earlier
inertial mass experiment, and locate the position of the pendulum on the knife
edge where it is in balance. Mark the pendulum with a “dot” using a felt-tip
pen to locate this position which is the center-of-mass of the ball/pendulum
system. You will use this dot as a reference for all y measurements. [The
location of this “dot” may or may not correspond to other dots or marks on
the pendulum. Your measurement is the one that counts!]
● Next, measure and record the mass of the pendulum (M) and ball (m)
separately for later use. We will ignore possible errors in these mass values
for the purposes of this experiment.
● Reattach the pendulum securely to the support arm.
● Measure and record the height (yo) of your dot (the center of mass of the ballpendulum system) above the base. Also estimate and record the error (∆yo).
You are now ready to begin data taking.
● To load the spring-gun, put the ball in place and pull the ball back until the
spring is latched. Lower the pendulum and make sure that it can swing freely
and that the opening in the bob is lined up with the gun. Each time you fire
the spring gun, make sure that the pendulum is securely fastened to the
support arm assembly at the top.
● Starting with the pendulum at rest, shoot the ball into the cup. The pendulum
(including the ball) will swing up and lock into some particular tooth in the
rack. Record the number of this tooth in the data table below (Table 1).
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●
●
●
Experiment 5
Repeat the process for a total of ten times, so you have enough data to
determine a mean value and standard deviation. NOTE: If the pendulum
position for a trial is much different from the others, something may have
happened during the firing and you should discard this measurement and
repeat it. For each measurement (i= 1,...,10) and observed tooth position,
measure the elevation (yi) of the center of mass above the pendulum base.
Calculate the mean value of these elevations (ym) from your measurements.
Measure and record the difference ∆ytooth in elevation of the center of mass
caused by a shift in position of one tooth. This tooth “size” contributes to the
measurement of elevation and also contributes an error: δytooth = (∆ytooth / 2).
Record this quantity for future use.
You should now be able to fill all elements of the data Table 1 below. The
calculated quantities in several rows and columns will be important in later
analysis and discussion. Section 3 of Measurement and Error describes the
elements of the table and the procedure we are following here.
Measurement (i=1,...,n=10)
Tooth
number
yi
∆yi = (yi - ym )
(∆yi)2
1
2
3
4
5
6
7
8
9
n=10
Σ (yi)
ym = Σ (yi) / n
Σ (∆yi)2
σ2= Σ (∆yi)2 / (n-1)
σ2m = Σ (∆yi)2 / [n(n-1)]
4. You can also check the above values, by using your lab bench computer. To do so,
open Graphical Analysis. When the data table appears, insert “y” for the header, and
type in your column of ten yi values, one below the other. Then select the column
and from the Data Menu, select “statistics”. The mean value and sigma that appear in
the dialog box should correspond to ym and σ in your data table above. You can then
get the σm from the expression: σm = σ/√[n]. Record these values in your physics
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laboratory report. The purpose of using both methods - the table filled out by hand
and the computer tabulation - is to make sure that you see how these quantities are
determined.
B. V from Momentum Conservation.
1.
If momentum is conserved in the collision of the ball and pendulum, the speed V of
the ball and pendulum just after the collision can be calculated from the masses of
ball and pendulum and the speed of the ball just before the collision.
● Pi = Pf is the statement of momentum conservation. This is a vector relation.
However, if we consider the collision to occur along the x-axis, we can write
this as an equation for x components: Pix = Pfx.
●
Pix = mu, where u is the initial speed of the ball prior to collision with the
pendulum cup. Again m is the mass of the ball measured above.
●
Pfx = (M+m)V, where V is the recoil velocity of the ball and pendulum
system, and M is the mass of the pendulum.
●
Thus: mu = (M+m)V, or V = mu / (M+m).
●
We know everything but u.
2. A useful method to determine the initial speed u of the ball is projectile motion. (See
Figure 3 for choice of coordinate system.) For example if the ball, initially at (xi,yi)
= (0,Y), is projected horizontally with velocity components (vix , viy) = (u,0), we
may write:
● y = yi + viyt - (1/2) gt2 = Y - (1/2) gt2
● x = xi + vixt = ut
3. If the ball travels a horizontal distance X before hitting the floor, then (xf,yf) = (X,0).
By eliminating t in the above equations, we can solve for u:
● u = X * sqrt [g/(2Y)].
u
Y
(0,0)
X
FIGURE 3
4.
To make the required measurement, remove the pendulum from the support arm and
set it aside on your lab bench. Move your apparatus to a designated "target range"
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●
Experiment 5
where the floor is covered with paper. Place the apparatus on a table provided and
make sure it remains in a fixed position while you take all your measurements. Fire
the ball horizontally and note where it lands on the paper.
● Place a piece of carbon paper face down at this spot and fire the ball 10 times
so it lands on the carbon paper. Measure Y, the ball's initial elevation above
the carbon paper, and X, the horizontal distance the ball travels after leaving
the spring gun until it hits the carbon paper. You must also determine the
errors in the quantities, δX and δY.
●For example, if your measurement of X requires several separate
measurements, X = x1 + x2 + x3, then the error in X will be given as:
δX = sqrt [ (δx1)2 + (δx2)2 + (δx3)2 ].
A similar expression may be assumed for δY.
One or more of the error terms may dominate the others. For example the hits
on the carbon paper will form a distribution. You will need to determine the
mean x value for these “hits” and estimate the error in this mean value. To
do this, set up a data table and find xm and σxm . You may then identify the
error in xm as δxm = σxm. Table 2 is set up below to assist you. Save your
paper with the “hits” on it and include it in either yours or one of your group
members’ lab writeups.
Point i= 1,...,n=10
1
2
3
4
5
6
7
8
9
n=10
Σ (xi)
xm = Σ (xi) / n
Σ (∆xi )2
σx2 = Σ (∆xi )2 / (n-1)
σ2xm = Σ (∆xi )2 / [n(n-1)]
∆xi = (xi - xm)
xi
(∆xi )2
5. Again, you may use the Graphical Analysis program on your lab bench
computer to type in a table column of xi values, and to verify xm and σx using
the Data Menu and Statistics dialog box. You can then obtain σxm from the
relation: σxm = σx /√[n]. Record these computer-determined values in your
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physics laboratory report form.
6. Lastly, remount the pendulum properly and securely on the support arm of the
apparatus so it swings freely. Make sure the ball is also returned to the lab
bench. You have now completed all required work in the laboratory.
7. Before you leave the laboratory you must have completed the data tables,
have measured y0 and estimated δy0, measured m and M, and have sufficient
information to determine X, Y and δX and δY. You can make final
calculations of values, errors, and comparisons outside the laboratory.
IV. Analysis
A. V from Energy Conservation.
•
1. V, the initial recoil speed of the pendulum and ball after the collision, may
be found using the relation:
• V = sqrt [2gY ]
where Y = (ym+(∆ytooth /2) - yo )
● You should explain in your own words why each term is there, and make a
sketch.
2. The error in V is then given by the expression:
● δV = (V/2) δY / Y
● where δY = sqrt [ (δym)2 + (δytooth)2 +(δyo)2]
● and use: (δym)2 = (σm)2.
B. V from Momentum Conservation.
1. From momentum conservation, the speed V of the ball and pendulum just after they
collide is given by:
● V = mu/(M+m)
● where u = X * sqrt (g/2Y).
2. The error in V is then given by the expression (ignoring errors in masses):
● δV = m δu / (M+m)
● where δu = u * sqrt [ (δX/X)2 + (1/4) * (δY/Y)2 ].
V. Comparison of Results and Discussion.
1. In this experiment there is no "known" or "standard" value of V with which to
compare your results. Both of your determinations involve possible sources of
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systematic error which may be significant.
● Try to identify and list sources of systematic error for both the case of the
energy conservation and momentum conservation experiments.
2. Calculate the difference between the values for V and the error in this difference
based on your calculated values for δV.
● Do your values agree within the calculated errors?
● Do these results suggest that there are additional, significant systematic
errors?
3. Based on your estimates of random and systematic errors, do you conclude that your
results are consistent with the principles of conservation of momentum and energy?
● Explain your conclusions.
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