3B SCIENTIFIC PHYSICS U10362 User manual

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Instruction sheet
3B SCIENTIFIC3B SCIENTIFIC
3B SCIENTIFIC3B SCIENTIFIC
3B SCIENTIFIC® PHYSICSPHYSICS
PHYSICSPHYSICS
PHYSICS
U10362 Ballistic pendulum
®
9/04 MH
1Projectile launcher (U10360)
2Back plate
3Guide for swing pointer
4Bearing screw
5Counter bearing
6Swing pointer
7Angle scale
8Pendulum
9Projectile catcher
bl Base plate
bm Table clamp
bn Knurled screw
bo Ramrod (for U10360)
bp Extra weights, 2 pcs.
1234 5
6
7
8
9
bl
bm
bn
Fig. 1: Components
bp bo
1. Safety instructions
•This instruction sheet is concerned mainly with the
ballistic pendulum. You should also read the in-
structions for the projectile launcher U10360.
•To check whether a projectile is located in the pro-
jectile launcher and the spring is cocked, only use
the observation holes at the sides. Do not look into
the barrel from the front. Risk of injury!
•Never aim at people!
•Protective goggles should be worn during the ex-
periments.
•The projectile launcher should always be stored
with the spring loose and with no projectile in the
barrel.
2. Description
•The ballistic pendulum is for experiment-based
determination of the launch velocity of a projec-
tile when it leaves the projectile launcher. It is also
possible to determine trajectories when the pro-
jectile is launched horizontally or at an angle.
Launch heights of 5, 10, 15, 20 or 30 cm can be
selected easily with the aid of the drilled holes.
•Thanks to the extreme lightness of the pendulum,
the experiment can be performed using compara-
tively safe plastic projectiles instead of steel balls.
Experiments involving inelastic collisions (quanti-
tatively) and elastic collisions (qualitatively) can be
evaluated. The velocity of the projectiles deter-
mined from trajectory and pendulum experiments
typically agree to within about 3%.
•Extra weights allow various pendulum travels to
be investigated for constant speeds.
3. Operation and maintenance
•First the ballistic pendulum is screwed to a stable
bench by means of its clamp. The projectile
launcher is then screwed to the back plate 2rom
behind either in a horizontal position in front of
the pendulum as in Fig. 1 or as shown in Fig 3.
Tip: if the workbench is not stable enough, it may
be that when the pendulum swings to its maximum
extent and then swings back, it may jog the appa-
ratus when striking the projectile launcher, caus-
ing the swing pointer to be shifted out of line. If
this happens, the pendulum should rather be
stopped by hand.
•Projectiles should always be loaded when the sp-
ring is not under tension by placing the sphere in
loosely through the front of the plastic cylinder

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within the device. The sphere is then pushed down
inside the barrel using the ramrod until the de-
sired spring tension has been reached. The ram-
rod should not be removed too quickly, otherwise
the suction its removal produces may pull the
sphere out with it. The position of the sphere may
only be checked using the observation holes. Never
look into the barrel!
•Before launching, ensure that no one is in the way
of the trajectory. To launch, the cord of the launch-
ing lever is briefly pulled perpendicularly to the
lever.
•The pendulum 8can be removed by undoing the
bearing screw 4and turned by 180° so that it is
installed with the rear of the projectile catcher
9pointing towards the launcher (experiments on
elastic collision). The counter bearing 5is designed
so that the pendulum hangs at a slight angle if the
bearing screw is only light tightened. This means
that the projectile catcher is not precisely in front
of the launch aperture of the launcher. For this rea-
son, the bearing screw should be tightened until
the catcher and the launch aperture are in line.
•After turning the pendulum round, or if necessary,
the guide 3for the swing pointer 6should be
adjusted so that the pointer just touches it when
the pendulum is suspended at rest. The screw on
the guide should only be finger-tightened to avoid
the appearance of pressure on the pendulum rod.
•Maintenance: the ballistic pendulum principally
requires no maintenance. If necessary some non-
acidic grease (Vaseline) can be applied to the bear-
ing screw 4and the knurled screw bn. Other than
in the vicinity of the scale, the apparatus may be
cleaned using acetone, ethanol (white spirit) or
petroleum ether as required. Avoid submerging the
equipment in water.
4. Experiment procedure and evaluation
4.1 Ballistic pendulum
4.1.1 Experiment setup
•The experiment setup corresponds to Fig. 1 for
experiments on inelastic collision. For experi-
ments on elastic collisions, the pendulum should
be turned round by 180° (cf. Section 3 “Opera-
tion”).
4.1.2 Experiment procedure
•It is practical for these experiments to enter the
experiment number, the spring tension (1, 2 or
3), the type of collision (inelastic “i” or elastic “e”),
the number of extra weights used and the mea-
sured angle ϕ. In order to obtain the most accu-
rate experiment results, after one shot, a second
should be performed with the swing pointer not
having been reset to 0° in between. This mini-
mizes the unavoidable frictional losses of the
swing pointer.
•Example experiment sequence:
No Spring Type of Extra Angle ϕ
tension collision weights
11 i0 17.5
22 i0 25.0
33 i0 36.0
41 i2 9.5
52 i2 13.5
63 i2 19.0
71 e0 29.5
82 e0 42.0
93 e0 60.0
4.1.3 Experiment evaluation
4.1.3.1 Inelastic collision
•The following equation is valid for the swinging
pendulum due to conservation of energy
Epot = Ekin (1)
where the potential energy is
Epot = mtot g∆h(2)
Here is mtot the total mass of the pendulum in-
cluding the projectile and any extra weights, gis
the acceleration due to gravity and ∆his the dif-
ference in height of the center of gravity of the
pendulum at rest and at the maximum extent of
its swing.
•From the measured angle ϕ and the measured
length Is to the center of gravity according to Fig.2
the following is derived:
∆h= Is(1 – cos ϕ) (3)
Fig. 2: Determining the required lengths. Distance between center of gravity
and axis of rotation (Is) should be measured including the projectile and any
additional weights when the collision is inelastic. To perform the measure-
ment, the pendulum may, for example, be balanced on a ruler mounted on
its side. The distance between the center of the projectile and the axis of
rotation is IK= 280 mm.
•The kinetic energy can be calculated from the
moment of inertia Itot relative to the axis of rota-
tion and the maximum angular speed ωaccord-
ing to the equation
EI
kin tot
=1
2
2
ω
(4)
•If Equations 2 and 4 are inserted into Equation 1
and ∆h eliminated using Equation 3 then the
equation can be rearranged to:

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ωϕ =2m gl ( )
tot s
tot
cos1−
I
(5)
•However, we are not seeking ω, but the initial ve-
locity of the projectile v0. The relationship be-
tween the two values is given be the equation for
the conservation of angular momentum directly
before and after the collision:
LK= Ltot (6)
with the angular momentum of the projectile
LK= mKIKv0(7)
before the collision and the total angular momen-
tum
Ltot = Itot ω(8)
after the collision. Inserting Eqs. 7 and 8 into Eq.
6 gives:
mKIKv0= Itot ω(9)
•Resolving this for ωand equating with Eq. 5 leads
to the following relationship
vml Imgl
s0
KK
tot tot
121cos=−
()
ϕ
(10)
•The moment of inertia is in principle determined
from the integral
Ildm
tot 2
m
=∫(11)
where lis the distance of each mass element from
the axis of rotation. Since in this case it is not the
moment of inertia that we seek to derive Itot can
also be calculated from the period Tof the pen-
dulum (with projectile and any extra weights). For
a physical pendulum the following is valid for
small deflections1:
Imgl
T
stot tot
=




2
2
π
(12)
•This means that all the variables are now known
or calculable. For the above example, the follow-
ing table emerges:
No mK / kg mtot / kg Is/ m T/ s v0in m/s
1 0.00695 0.06295 0.218 1.01 3.39
2 0.00695 0.06295 0.218 1.01 4.82
3 0.00695 0.06295 0.218 1.01 6.88
4 0.00695 0.09795 0.252 1.07 3.51
5 0.00695 0.09795 0.252 1.07 4.98
6 0.00695 0.09795 0.252 1.07 6.99
•The numeric values should be determined sepa-
rately for every pendulum, since material and
manufacturing tolerances mean that values may
differ from one to another.
4.1.3.2 Elastic collision
•For a swinging pendulum Eq. 5 is still valid for
the motion after a collision, the only difference
being that the moment of inertia IP is determined
without the projectile but with any extra weights
(pendulum mass mP):
ωϕ =−
()
21cos
Ps
P
mgl
I
(13)
•To determine the relationship between ωand the
initial velocity v0both the conservation of angu-
lar momentum and the conservation of energy
before and after the collision must now be used.
The additional equation is required since the sys-
tem has an additional degree of freedom in the
projectile velocity v2 after the collision. As for Eq.
9, the following is true for the angular momen-
tum:
mKIKv0= mKIKv2+ IPω
⇔
=−vv I
mI
20
P
KK
ω
(14)
•If this velocity v2is inserted into the equation for
the conservation of energy
1
2
1
2
1
2
0
22 22
mv mv I
KKP
=+ω
(15)
by rearranging in various steps the following ex-
pression is obtained for v0
vl
I
mI s
02
1
21=+







ωKP
KK
(16)
•If Eq. 13 is plugged in here and IPdetermined as
in Eq. 12, then v0can be calculated for an ideal
inelastic collision:
No mK / kg mP / kg Is/ m T/ s v0in m/s
7 0.00695 0.0560 0.211 1.008 2.88
8 0.00695 0.0560 0.211 1.008 4.05
9 0.00695 0.0560 0.211 1.008 5.65
•These values for v0are about 18% smaller than
those obtained for inelastic collisions. This can
be explained by the fact that the elastic collisions
are not entirely ideal.
1Recknagel, A.: Physik Mechanik, 3te Auflage, VEB Verlag Technik Berlin, 1958.

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4.2 Determination of trajectories
4.2.1 Experiment setup
•One possible experiment setup is shown schemati-
cally in Fig. 3 (not to scale). The drill holes in the
back plate of the pendulum are placed so that
when a projectile is fired to land directly on the
workbench, the launch heights are 50, 100, 150,
200 and 300 mm.
Fig. 3: Experiment setup, key: 1Projectile launcher, 2Launch position of
projectile, 3Paper, 4Carbon paper, 5Easel with whiteboard (for ex-
ample)
•When launching against a vertical wall (e.g.
whiteboard U10030 mounted on easel U10381)
the radius of the projectile (1.25 cm) should be
subtracted from the distance between the point
of launch and the wall to obtain the distance
measurement xM. The height measurement yMrela-
tive to the launch height is given by the height of
the impact on the wall minus 62.5 mm, 112.5 mm,
162.5 mm, 212.5 mm or 312.5 depending on the
hole used.
4.2.2 Experiment procedure
•It is practical for these experiments to note the
experiment number, the spring tension (1, 2 or
3), the launch angle and the values xMand yM.
Example:
No Spring Launch- Projectile Target
tension angle distance height
ϕ/ ° xM/ cm yM/ cm
1 1 0 171.3 –30
2 2 0 125.4 –30
3 3 0 86.9 –30
4 1 0 62.3 –15
5 2 0 90.5 –15
6 3 0 120.7 –15
4.1.3 Experiment evaluation
•It is practical to take as the origin of the coordi-
nate system the mid-point of the projectile at the
moment of launch. Then the following applies:
vX= v0cos ϕ(17)
vY= v0sin ϕ(18)
yvt gt=−
Y
1
2
2
(19)
x= vXt(20)
From Eq. 20 t= x/ vX, whereby the time can be
eliminated from Eq.19.
•If vXand vyare then eliminated from the resulting
equation using Eqs. 17 and 18, the following is
obtained
yx x g
v
=−tan cos ϕϕ
2
0
22
2
(21).
This is the equation for the trajectory.
•In this equation only the launch velocity v0is un-
known since the distances xand ywere measured
during the course of the experiments. If v0is cal-
culated for the various experiments, the follow-
ing results are obtained:
Spring tension v0in m/s
1 3.53
2 5.10
3 6.85
•The numbers are based on a total of 25 experi-
ments, of which only 6 are explicitly listed in the
above table. The trajectory can now be obtained
from these using Eq. 21 and compared to the
measured values. The result is shown in Fig. 4.
Fig. 4: Comparison of measurements and calculated curve, x= horizontal
projectile distance, y= vertical height, symbols = measured values
(circles = spring tension 1, squares = spring tension 2, rhombuses = spring
tension 3), lines = calculated trajectories
234
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