Moments of inertia of different bodies / Steiner’s theorem with Cobra3
Related topics
Rigid body, moment of inertia, centre of gravity, axis of rotation,
torsional vibration, spring constant, angular restoring force.
Principle
The moment of inertia of a solid body depends on its mass distribution and the axis of rotation. Steiner’s theorem elucidates
this relationship.
Task
The moments of inertia of different bodies are determined by
oscillation measurements. Steiner’s theorem is verified.
Equipment
Cobra3 Basic Unit
Power supply, 12 VRS232 cable
Translation/Rotation Software
Light barrier, compact
Angular oscillation apparatus
Portable balance, CS2000
Block battery, 9 V
Silk thread, l = 200 m
Weight holder, 1 g
Slotted weight, 1 g
Bench clamp -PASSTripod base -PASSStand tube
Measuring tape, l = 2 m
Connecting cord, l = 100 cm, red
Connecting cord, l = 100 cm, blue
Connecting cord, l = 100 cm, yellow
PC, WINDOWS® 95 or higher
12150.00
12151.99
14602.00
14512.61
11207.20
02415.88
48892.00
07496.10
02412.00
02407.00
03916.00
02010.00
02002.55
02060.00
09936.00
07363.01
07363.04
07363.02
1
1
1
1
1
1
1
1
1
1
3
1
1
1
1
1
1
1
Alternative experimental set-ups are to be found at the end of
this experimental description.
LEP
1.3.28
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Set-up and procedure
The experimental set-up is shown in Fig. 1.
Perform the electrical connection of the compact light barrier
to the Cobra3 Basic Unit according to Fig. 2. Ensure that the
thread that connects the axis of rotation with the wheel of the
light barrier is horizontal. Wind the thread approximately 7
times around the rotation axis of the angular oscillation apparatus below the fixing screw.
Set the measuring perameters according to Fig. 3. Enter
20 mm for the ”Axle diameter” to synchronise the rotations of
the wheel on the compact light barrier and of the angular oscillation apparatus, which rotate at different speeds.
Lay the silk thread across the wheel of the light barrier and
adjust the set-up in such a manner that the 1-g weight holder
hangs freely and is located approximately in the middle of the
silk thread. In any case, the thread must be long enough to not
block the oscillation.
Deflect the body that is fitted on the spring by approximately
180°, release it and start recording the measured values by
clicking on the ”Start measurement” icon.
After approximately 10 to 15 s, click on the ”Stop measurement” icon.
If the values (50 ms) in the ”Get values every (50) ms” dialog
box are too high or too low, noisy or non-uniform measurements can occur. In such cases adjust the sampling rate of the
measurement accordingly.
The 1-g weight holder tautens the connecting thread between
the rotational axis of the angular oscillation apparatus and the
compact light barrier. If the thread bulges too easily during the
movement of the body on the light barrier, load additional 1-g
mass pieces onto the weight holder.
Progressively mount the following bodies on the rotary axis:
disc without holes, hollow cylinder, solid cylinder, sphere, rod,
disc with holes. Fix the masses symmetrically and successively at varying distances from the rotational axis on the rod.
Beginning in the centre, rotate the disc with holes around all
bores which lie on a radius.
Fig. 1. Experimental set-up with the compact light barrier
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
21328-11
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LEP
1.3.28
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Moments of inertia of different bodies / Steiner’s theorem with Cobra3
After the evaluation, an image similar to Figure 4 appears on
the screen. Determine the period T with the aid of the freely
movable cursor lines.
The angular directive force D* is given through the restoring
spring (the spiral spring in our case). The duration of oscillation
T 2p
Theory and evaluation
For small oscillation amplitudes, the angle of rotation and the
corresponding returning momentum are proportional; one
obtains harmonic circular oscillations with the frequency
v
D*
.
B J
of the circular oscillation is determined through the measurement. The unknown momentum of inertia can be calculated
from the values for the angular directive force D* and the period of oscillation T by means of the following equation;
J
With known J this formula can be used to calculate D*.
Fig. 2. Connection of the compact light barrier to the Cobra3
Basic Unit
J
.
B D*
T2 *
D .
4p2
Theoretical considerations yield the following relations for the
moments of inertia J of the used test bodies:
Circular disk:
The moment of inertia J of a circular disk depends on its mass
m and its radius r according to
J
1
m · r2 .
2
Massive and hollow cylinder:
The moment of inertia Jv of a massive cylinder depends on its
mass mv and its radius r. The following relation is valid:
Jv 1
m · r2 .
2 v
For a hollow cylinder of comparable mass and exterior radius,
the moment of inertia must be larger. It depends both on the
mass mH of the hollow cylinder and on the radii ri and ra:
JH 1
m 1r2 r2i 2 (hollow cylinder)
2 H a
For a density r and a height h, the mass of the hollow cylinder
can be expressed through
yellow
blue
Fig. 3. Measuring parameters (Light barier)
red
mH r · h · p 1r2a r2i 2 .
For the moment of inertia JH one obtains the following equation:
1
r · h · p 1r2a r2i 2 1r2a r2i 2
JH 2
Fig. 4. Typical measuring result
2
21328-11
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Moments of inertia of different bodies / Steiner’s theorem with Cobra3
which yields:
JH 1
1
r · h · p r4a r · h · p r4i .
2
2
This equation expresses that the moment of inertia of the hollow cylinder can be expressed as moment of inertia of two
massive cylinders of the same density. The mass of the larger
massive cylinder of radius ra is:
ma r · h · p · r2a
and the mass of the smaller massive cylinder of radius r1 is:
LEP
1.3.28
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Rod with movable masses:
The moment of inertia of a rod with masses which can be shifted depends on the distance between the masses and the
rotating axis. The measured moments of inertia of the rod without masses and with the masses set at different points are listed in table 2. The table also gives the calculated moments of
inertia of the rod without masses:
1
J
M L2
12
1
· 0.1715 · 602 kg cm2 51.45 kg cm2 ,
12
Fig. 6. Connection of the movement sensor to the Cobra3
Basic Unit
mi r · h · p · r2i .
One thus obtains
JH 1
1
m · r2 m r2i .
2 a a
2 i
Sphere:
The moment of inertia JK of a homogeneous spherical body
related to an axis passing through its central point is:
JK 2
m · r2 .
5
red
black
yellow
BNC1
BNC2
Thus, the moment of inertia of a cylinder with the same radius r
and the same mass m is larger than that of a comparable
sphere.
Next to the dimensions and masses of the test bodies, table 1
also contains the measured and theoretically calculated
moments of inertia.
Fig. 5. Experimental set-up with the angular oscillation apparatus. The thread runs horizontally and is wrapped once around the
larger of the two cord grooves of the movement sensor
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
21328-11
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LEP
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Moments of inertia of different bodies / Steiner’s theorem with Cobra3
the moments of inertia of the masses without the rod:
2 · m · l2
and finally, the sum of the moments of inertia of the rod and of
the masses:
1
M L2 2 · m · l2 J0 2 · m · l2 .
12
The moment of inertia measured every time corresponds to the
sum of the single moments of inertia.
The following parameters are valid for the measurement examples described here:
total length of the rod L
60 cm
mass M of the rod
0.1715 kg.
Mass of a single weight m
0.2115 kg.
Steiner’s theorem:
According to Steiner’s theorem, the total moment of inertia J of
a body rotated about an arbitrary axis is composed of two
parts according to
J J0 m · r2 .
Remarks
At extremely slow angular velocities, signal transients or deformations can occur. These can be reduced if the sampling rate
is changed. In any case, error-free recorded intervals can be
selected from the measuring signal after completion of the
measurements.
Since the movement recording is not performed without contact, slight damping of the measured oscillations do occur.
Angular velocities that are too small cannot be measured by
the wheel on the compact light barrier and are plotted as the
reference line.
When using the compact light barrier, the program can only be
used to record strictly harmonic oscillations
Instead of the compact light barrier (11207.20), the movement
sensor (12004.10) can also be used (cf. Fig. 6 and Fig. 7: The
thread is horizontal and is placed in the larger of the two cord
grooves on the movement sensor.) In this case the following
additional equipment is required:
Equipment
Movement sensor with cable
Adapter, BNC-socket/4mm plug pair
Adapter, socket-plug, 4 mm
12004.10
07542.27
07542.20
1
1
1
Fig. 7: Measuring parameters (movement sensor)
J0 is the moment of inertia related to an axis running parallel to
the rotation axis and passing through the centre of gravity (axis
of gravity). The expression mr2 is the moment of inertia of the
mass of the body concentrated at the centre of gravity, related
to the rotation axis. Table 3 compares the measured moment
of inertia Jmeasured with the sum of the single moments of inertia J0 + mr2 and shows the validity of Steiner’s theorem.
Table 1
Body
m/kg
r /cm
J/kgcm2
JT/kgcm2
Disc
0.292
11
17.42
17.66
Solid cylinder
0.370
4.78
4.44
Hollow cylinder
0.390
8.15
7.49
Sphere
0.798
16.14
15.64
4.9
ra/ri : 5/4.6
7
Table 2
l / cm
Jmeasured / kgcm2
–
5
10
15
20
25
42.14
53.81
86.28
144.52
223.77
322.75
2 · l2 m / kgcm2
–
10.58
42.3
95.18
169.2
264.38
2 · l2 m + J0 / kgcm2
51.45
62.03
93.75
146.63
220.65
315.83
Table 3 (disc mass m = 0.454 kg)
r / cm
0
3
6
9
12
4
21328-11
Jmeasured / kgcm2
50.44
55.25
68.91
106.47
119.00
m r2 / kgcm2
–
4.09
16.34
36.77
65.38
J0 + m r2 / kgcm2
–
54.53
66.78
87.21
115.82
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen