Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
TEP
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.
Equipment
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1
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3
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Cobra4 Wireless Manager
Cobra4 Wireless-Link
Movement sensor with cable
Cobra4 Sensor-Unit Timer/Counter
Angular oscillation apparatus
Support rod, stainless steel, d = 10 mm, l = 400 mm
Right angle clamp PHYWE
Weight holder, 1 g
Slotted weight, 1 g
Barrel base -PASSTripod base -PASSPortable Balance, OHAUS CS2000 - AC adapter included
Flat cell battery, 9 V
Measuring tape, l = 2 m
Software measure Cobra4 multi-user licence
12600-00
12601-00
12004-10
12651-00
02415-88
02039-00
02040-55
02407-00
03915-00
02006-55
02002-55
48917-93
07496-10
09936-00
14550-61
Additionally required
1 PC with USB interface, Windows XP or higher
Fig. 1:
Experimental setup.
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TEP
Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
Task
The moments of inertia of different bodies are determined by oscillation measurements. Steiner’s theorem is verified.
Setup and procedure
The experimental set-up is shown in Fig. 1.
Ensure that the thread that connects the axis of rotation with the wheel of the movement sensor is horizontal. Wind the thread approximately 7 times around the rotation axis of the angular oscillation apparatus below the fixing screw.
Set the measuring configuration “movement of inertia of different bodies/ Steiner´s Theorem with Cobra4”. Lay the silk thread across the wheel of the movement sensor and adjust the setup 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 360°, 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.
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.
After the evaluation, an image similar to Figure 2 appears on the screen. Determine the period T with the
aid of the freely movable cursor lines.
Fig 2: Typical measuring result.
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Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
TEP
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
With known this formula can be used to calculate
The angular directive force
duration of oscillation
.
is given through the restoring spring (the spiral spring in our case). The
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
and the period of oscillation by
means of the following equation;
Theoretical considerations yield the following relations for the moments of inertia J of the used test bodies:
Circular disk:
The moment of inertia of a circular disk depends on its mass
and its radius
according to
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:
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:
For a density r and a height h, the mass of the hollow cylinder can be expressed through
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Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
For the moment of inertia JH one obtains the following equation:
Which yields:
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:
and the mass of the smaller massive cylinder of radius r1 is:
One thus obtains
Sphere:
The moment of inertia JK of a homogeneous spherical body related to an axis passing through its central
point is:
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.
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:
The moment of inertia measured every time corresponds to the sum of the single moments of inertia.
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PHYWE Systeme GmbH & Co. KG © All rights reserved
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Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
TEP
The following parameters are valid for the measurement examples described here:
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
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
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 J measured with the sum of the single moments of inertia
and shows the validity of Steiner’s theorem.
Body
Mass
Radius
0.760
7
0.500
15
0.300
11
0.380
5
0.380
10
0.21
30
Formula
Sphere
Thin circular disc
Thick circular disc
Massive cylinder
Hollow cylinder
Dumbbell
Moment of inertia
14.9
56.3
18.2
4.8
9.4
9.6
31.5
Table 1
5
10
15
20
25
42.14
53.18
86.28
144.52
223.77
322.75
10.58
42.3
95.18
169.2
264.38
51.45
62.03
93.75
146.63
220.65
315.83
0
3
6
9
12
50.44
55.25
68.91
106.47
119.00
4.09
16.34
36.77
65.38
54.53
66.78
87.21
115.82
Table 2
Table 3
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Moments of inertia of different bodies/
Steiner´s theorem with Cobra4
Place for notes:
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