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Lab 7 Ballistic Pendulum!
!
Introduction
We will use the ballistic pendulum to examine how a complex set of physical processes can be
broken down into simpler single processes for analysis.
Equipment
Ballistic pendulum.
Description
The ballistic pendulum is a tool for measuring the launch speed, or muzzle velocity, of a usually
small and fast projectile. The projectile is fired at the bob of a pendulum that catches the
projectile. Subsequently, the pendulum swings up to a certain height or angle to indirectly indicate
the speed of the projectile.
Below are the before and after photos of a trial. The gun is a horizontal spring launcher. Notice,
in this case, that only the angle indicator has moved.
The discussion below explains the processes involved in the functioning of the ballistic pendulum.
Different physical principles apply to each process. Primarily, we are interested in whether we can
apply momentum and/or energy conservation to each process.
1. Firing
The process starts with the launch. A projectile of mass, m, is launched. At the muzzle of the
launcher, the speed of the projectile is called vlaunch.
pendulum
v=0
vlaunch
launcher
During the launch, the projectile picks up momentum and kinetic energy.
momentum and the kinetic energy of the projectile are not conserved.
h
Therefore, both the
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2. Traversing the Gap
The projectile flies through a small gap between the muzzle and the bob of the pendulum.
pendulum
vlaunch
v
h
launcher
Let’s look at momentum first. The bearing does not interact with anything other than the Earth so
let’s make the bearing our system. The question to ask is whether there is an external net force
acting on the system. There is. It is due to Earth’s gravity. Therefore, the momentum is not
conserved. However, since the launch speed is horizontal and gravity acts vertically, gravity does
not alter the horizontal component of the momentum. Therefore, the momentum we care about
does not change and is conserved, so the velocity is unchanged as well.
pi,x = p f ,x
⇒ mvlaunch = mv
⇒ v = vlaunch
As for the energy, the question to ask is whether there are non-conservative forces acting on the
system. Assuming air resistance is negligible, there is not. Therefore, the energy of the system is
conserved.
3. Next, the projectile is caught by the bob of the pendulum. This results in the combined object of
both the projectile and the pendulum bob traveling at a reduced speed vpendulum.
pendulum
vlaunch
vpendulum
launcher
h
Since the bearing interacts with the pendulum, let’s make the system be both objects. For the
momentum, there are external forces on the system. They are gravity and tension. It happens
that they act in opposite directions during this process so that they cancel. The external net force
is zero. Momentum is conserved.
Momentum conservation for the system in the horizontal direction gives the following equation.
pi,x = p f ,x
⇒ mvlaunch = (M + m)v pendulum
As for the energy, this is a completely inelastic collision so it is not conserved. You can hear the
energy coming out of the system. You can feel it in the vibration of the apparatus too.
4. The last process is when the pendulum swings up, in a circular arc, to its final, resting position.
pendulum
launcher
h
The bearing and the pendulum are moving together so let’s keep them as the system. For the
momentum, the velocity of the system changes direction, so the momentum is not conserved.
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For the energy, is there any no non-conservative work done to the system. Gravity does work, but
it is conservative. Tension is a non-conservative force. The work that it does is the following.




W = F ⋅ Δx = T ⋅ Δs
The tension points toward the center around which the pendulum rotates while the pendulum
travels along the edge of the circle.
This means the tension is always perpendicular to
displacement. Hence, the work cone is zero.
Even though there is an non-conservative force on the pendulum, the non-conservative work is
still zero. The energy is still conserved.
Ei = E f
⇒ Ki +U i = K f +U f
1
(M + m)v 2pendulum + 0 = 0 + (M + m)gh
2
Combining the two equations results in an equation for the launch speed based on the height
increase.
vlaunch =
M +m
2gh
m
Recall from the pictures that we are actually measuring the angle rotated by the pendulum rather
than the height traveled by the pendulum. We need two things. One, how is the height related to
the angle. The following diagram shows how the height is calculated.
R
Rcosθ
R
R–Rcosθ
h = R − R cos θ = R(1 − cos θ)
Two, where do measure the position of the pendulum? The mass of the entire pendulum is
distributed unevenly along the pendulum; it is what is called a physical pendulum. unsurprisingly,
the “location” of the mass is the center of mass, RCM.
Together, we have this for the launch speed.
vlaunch =
M +m
2gRCM (1 − cos θ)
m
Experiment: Ballistic Pendulum
Measure the mass of the bearing. Let’s estimate its uncertainty to be 0.5 gram. Measure the
mass of the entire pendulum assembly including the stem. Let’s estimate its uncertainty to be 0.5
gram as well.
Next, you are going to find the distance of the center of mass from the pivot point, RCM. Place the
bearing inside the pendulum. Balance the pendulum on the edge of a ruler. Measure the distance
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from the pivot to this location.
millimeters, so be careful.
Let’s estimate the uncertainty in this measurement to be 2
Now, we are ready to measure the angle. Measure the angle indicated when the pendulum is at
its resting position in case it is not zero so that you can calculate the actual angle traversed.
Park the pendulum to the out-of-the-way position.
Load the spring launcher to three clicks. Release the pendulum to its normal, resting position.
Rotate the angle indicator to about 30°. Launch the bearing. If the resulting angle is more than 5°
above 30°, start with a larger angle for the indicator. Measure the final angle 10 times. Find the
average and the standard deviation.
Use the equation above to calculate the launch speed. Include the uncertainty.
Experiment: Zero Launch Angle
Notice that we don’t know the expected value for this experiment.
We will try to verify the above results by doing a completely different experiment. We will do a
zero launch angle experiments.
Park the pendulum out of the way. Place the apparatus at a designated location pointing at an
empty area on the floor. Load and fire the bearing once to find the location where it will land.
Tape a piece of paper centered on where the projectile landed. Fire the projectile at the paper a
total of 10 times.
Measure all distances from the bottom of the “launch position” of the bearing. This is marked on
the launcher. Since this is all about free fall, the launcher must not have influence. Also, since we
are measuring where the bearing lands, we need the location of the bottom of the bearing in the
beginning.
Measure the height of the bottom of the launch position from the floor. Measure the ranges of 10
trials. Calculate the mean and standard deviation of the ranges.
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Let’s also assume there is some uncertainty in the launch angle. The experiment might not be
perfectly level for a number of reasons. Let’s estimate a maximum of 0.3° uncertainty. This
affects the range by a certain amount. You can calculate this amount by calculating the range with
a launch angle of ±0.3°. Add this uncertainty to the range uncertainty above.
Calculate the initial speed from this data. Include the uncertainty.
Analysis
Here is a case where two measurements of the same value are done two different ways without
an expected value. How do we know if the two experiments agree?
We now have two pdf’s, one for each experiment. One question we want to answer first is “does
experiment 1 agree with experiment 2?” The value to calculate is “what is the probability that a
result from experiment 1 will fall within some tolerance (usually within one or two standard
deviations) from the mean of the results from experiment 2?”
For example, let’s say our two sets of results have the following distributions.
Exp. 1
Exp. 2
results
limits
When we compare experiment 1 to experiment 2, the result of experiment 2 provides the limits
from which to calculate the probability and experiment 1 provides the distribution. The probability
that a result from experiment 1 will fall within one standard deviation from the mean of the results
from experiment 2 is the shaded area. In the above diagram, the probability is about 16%.
The way to calculate this is again using the “normalcdf” function.
normalcdf (mean of exp. 2 − std. dev. of exp. 2,
mean of exp. 2 + std. dev. of exp. 2,
mean of exp. 1,
std. dev. of exp. 1)
So what is good agreement and what is poor agreement? This is just a matter of degree. With a
tolerance of one standard deviation, let me say that 0% to 33% is poor, 33% to 67% is good,
and 67% to 100% is excellent.
Note also that you can reverse the role of the two results and calculate a different value for the
probability. Experiment 1 can agree with experiment 2 but not the other way around. Calculate
both probabilities, “the probability that results from experiment 1 will fall within one standard
deviation from the mean of experiment 2” and “the probability that results from experiment 2 will
fall within one standard deviation from the mean of experiment 1”.
Conclusion
Present your results from the two experiments and address whether the results from the two
independent experiments agree with each other.
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