Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
TEP
Related topics
Rotation, angular velocity, torque, angular acceleration, angular momentum, moment of inertia, rotational
energy.
Principle
A known torque is applied to a body that can rotate about a fixed axis with minimal friction. Angle and
angular velocity are measured over the time and the moment of inertia is determined. The torque is exerted by a string on a wheel of known radius with the force on the string resulting from the known force of
a mass in the earth's gravitational field. The known energy gain of the lowering mass is converted to rotational energy of the body under observation.
Equipment
1
1
1
1
1
1
1
1
1
1
1
1
20
10
1
1
2
1
1
Cobra4 Wireless Manager
Cobra4 Wireless-Link
Movement sensor with cable
Cobra4 Sensor-Unit Timer/Counter
Software measure Cobra4 multi-user licence
Tripod -PHYWEPrecision bearing
Turntable with angle scale
Inertia rod
Aperture plate for turntable
Right angle clamp -PHYWESilk thread, l = 200 m
Slotted weight, 1 g
Slotted weight, 10 g, black
Weight holder, 1 g
Weight Holder, 10 g
Barrel base PHYWE
Holding device w. cable release
Support rod PHYWE,square,l = 250 mm
12600-00
12601-00
12004-10
12651-00
14550-61
02002-55
02419-00
02417-02
02417-03
02417-05
02040-55
02412-00
03916-00
02205-01
02407-00
02204-00
02006-55
02417-04
02025-55
Additionally required
PC with USB interface, Windows XP or higher
Fig. 1:
Experimental setup.
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TEP
Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
Tasks
1. Measure the angular velocity and angle of rotation vs. time for a disc with constant torque applied to
it for different values of torque generated with various forces on three different radii. Calculate the
moment of inertia of the disc.
2. Measure the angular velocity and angle of rotation vs. time and thus the moment of inertia for two
discs and for a bar with masses mounted to it at different distances from the axis of rotation.
3. Calculate the rotational energy and the angular momentum of the disc over the time. Calculate the
energy loss of the weight from the height loss over the time and compare.
Set-up and procedure
Set the experiment up as seen in Fig.1. Connect the movement sensor to the Cobra4 Sensor-Unit Timer/Counter and the Cobra4 Wireless Manager a USB port of the computer. Adjust the turntable to be horizontal – it must not start to move with an imbalance without other torque applied. Fix the silk thread (with
the weight holder on one end) with the screw of the precision bearing or a piece of adhesive tape to the
wheels with the grooves on the axis of rotation and wind it several times around one of the wheels –
enough turns, that the weight may reach the floor. Be sure the thread and the wheel of the movement
sensor and the groove of the selected wheel are well aligned. Place the holding device with cable release in a way that it just holds the turntable on the "Aperture plate" and does not disturb the movement
after release.
Start the "measure" program and load the experiment. Start the measurement with the "Continue" button
and release the turntable. Data recording should start automatically. Stop data recording just before the
weight reaches the floor by key press (in order not to have irrelevant data at the end of your recorded data).
1. Record measurements with different accelerating weights 𝑚𝑎 up to 100 𝑔. For heavy weights you
may set the "Get value" time to 200 𝑚𝑠 and for low weights you may set this time to higher values. Record measurements with the thread running in different grooves – adjust the position of
the light barrier to align thread and groove and wheel of light barrier (thread has to be horizontal)
and enter the correct axle diameter in the "Rotation" chart – and choose especially the weight on
⃗ = 𝑟𝑎 × 𝐹 is constant e.g. 𝑚𝑎 = 60 𝑔 for 𝑟𝑎 =
the end of the thread 𝑚𝑎 such that the torque 𝑇
15 𝑚𝑚 and 𝑚𝑎 = 30 𝑚𝑚 for 𝑟𝑎 = 30 𝑚𝑚 and 𝑚𝑎 = 20 𝑔 for 𝑟𝑎 = 45 𝑚𝑚, each time the torque being 𝑇 = 𝑚𝑎 𝑔𝑟𝑎 = 8.83 𝑚𝑁𝑚 with the earth's gravitational acceleration 𝑔 =
9.81 𝑁
𝑘𝑔
𝑚
= 9.81 𝑠2 . Also
choose a weight with which you take a measurement for each groove radius. You may also take
a measurement with the weight 𝑚𝑎 so light that it nearly does not affect the movement (like the
weightholder of 1 𝑔) but is enough to drive the light barrier's wheel. Start the turntable by a shove
with the hand. Nearly unaccelerated movement should be observable.
2. Record measurements with two turntables mounted on the precision bearing with weight values
used on the single turntable and with double the weight used on the single turntable for comparison.
Remove the turntables and mount the inertia rod to the precision bearing and the two weight
holders symmetrically to the rod with both the same distance to the axis 𝑟𝑖 . Take measurements
with varied masses 𝑚𝑖 at constant 𝑟𝑖 and also with constant masses 𝑚𝑖 at varied 𝑟𝑖 (both masses
of course still mounted symmetrically) – accelerated with the same weight 𝑚𝑎 (or with the same
series of weights for high precision).
Theory and evaluation
The angular momentum 𝐽 of a single particle at position 𝑟 with velocity 𝑣 , mass 𝑚 and momentum 𝑝 =
𝑚 𝑣 is defined as
2
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P2131363
Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
TEP
𝐽= 𝑟 × 𝑝
⃗ from the force 𝐹 is defined as
And the torque 𝑇
⃗ = 𝑟 × 𝐹
𝑇
with torque and angular momentum depending on the origin of the reference frame. The change of 𝐽 in
time is
𝑑𝐽
𝑑
𝑑𝑟
𝑑𝑝
(𝑟 × 𝑝) =
=
× 𝑝+ 𝑟 ×
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
and with
𝑑𝑟
× 𝑝 = 𝑣 × 𝑚𝑣 = 0
𝑑𝑡
and Newton`s law 𝐹 =
𝑑𝑝
𝑑𝑡
the equation of movement becomes
⃗ =
𝑇
𝑑𝐽
𝑑𝑡
(1)
For a system of N particles with center of mass 𝑅⃗𝑐.𝑚. and total linear momentum 𝑃⃗ = ∑ 𝑚𝑖 𝑣𝑖 the angular
momentum is
𝑁
𝑁
𝐽 = ∑ 𝑚𝑖 (𝑟𝑖 − 𝑅⃗𝑐.𝑚. ) × 𝑣𝑖 + ∑ 𝑚𝑖 𝑅⃗𝑐.𝑚. × 𝑣𝑖 = 𝐽𝑐.𝑚. + 𝑅⃗𝑐.𝑚. × 𝑃⃗
𝑖=1
𝑖=1
Now the movement of the center of mass is neglected, the origin set to the center of mass and a rigid
body assumes with 𝑟𝑖 − 𝑟𝑗 fixed. The velocity of particle 𝑖 may be written as 𝑣𝑖 = 𝜔
⃗ × 𝑟𝑖 with vector of
rotation
𝜔
⃗ =
𝑑𝜑
⃗
𝑑𝑡
(2)
Constant throughout the body. Then
𝐽 = ∑ 𝑚𝑖 𝑟𝑖 × 𝑣𝑖 = ∑ 𝑚𝑖 𝑟𝑖 × (𝜔
⃗ × 𝑟𝑖 )
With
𝑎 × (𝑏⃗ × 𝑐) = 𝑏⃗ (𝑎 ∙ 𝑐) − 𝑐 (𝑎 ∙ 𝑏⃗)
Is
𝐽 = ∑ 𝑚𝑖 (𝜔
⃗ ∙ 𝑟𝑖2 − 𝑟𝑖 (𝑟𝑖 ∙ 𝜔
⃗ ))
with
𝑟𝑖 ∙ 𝜔
⃗ = 𝑥𝑖 𝜔𝑥 + 𝑦𝑖 𝜔𝑦 + 𝑧𝑖 𝜔𝑧
and
𝐽𝑧 = 𝜔𝑧 ∑ 𝑚𝑖 (𝑟𝑖2 − 𝑧𝑖2 ) − 𝜔𝑧 ∑ 𝑚𝑖 𝑧𝑖 𝑥𝑖 − 𝜔𝑦 ∑ 𝑚𝑖 𝑧𝑖 𝑦𝑖
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TEP
Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
The inertial coefficients or moments of inertia are defined as
𝐼𝑥,𝑥 = ∑ 𝑚𝑖 (𝑟𝑖2 − 𝑥𝑖2 )
(3)
𝐼𝑥,𝑦 = − ∑ 𝑚𝑖 𝑥𝑖 𝑦𝑖
𝐼𝑥,𝑧 = − ∑ 𝑚𝑖 𝑥𝑖 𝑧𝑖
And with the matrix 𝐼̂ = {𝐼𝑘,𝑙 } it is
𝐽 = 𝐼̂ ∙ 𝜔
⃗
And for the rotational acceleration 𝛼 =
⃗⃗⃗
𝑑𝜔
𝑑𝑡
(4)
is then
⃗ =
𝑇
𝑑𝐽
𝑑𝜔
⃗
= 𝐼̂ ∙
= 𝐼̂ ∙ 𝛼
𝑑𝑡
𝑑𝑡
The rotational energy is
𝐸=
1
1
1
∑ 𝑚𝑖 𝑣𝑖2 = ∑ 𝑚𝑖 (𝜔
⃗ × 𝑟𝑖 )2 = 𝐼𝑘,𝑙 𝜔𝑘 𝜔𝑙
2
2
2
Sum convention: sum up over same indices using
(𝑎 × 𝑏⃗)(𝑐 × 𝑑) = (𝑎 ∙ 𝑐)(𝑏⃗ ∙ 𝑑 ) − (𝑎 ∙ 𝑑)(𝑏⃗ ∙ 𝑐)
The coordinate axes can always be set to the "principal axes of inertia" so that none but the diagonal elements of the matrix 𝐼𝑘,𝑘 ≠ 0. In this experiment only a rotation about the z-axis can occur and 𝜔
⃗ ≡
𝑒̂𝑧 𝜔𝑧 ≡ 𝑒̂𝑧 𝜔 with the unit vector 𝑒̂𝑧 . The energy is then
𝐸=
1
𝐼 𝜔2
2 𝑧,𝑧
(5)
The torque 𝑇 = 𝑚𝑎 (𝑔 − 𝑎)𝑟𝑎 is nearly constant in time since the acceleration 𝑎 = 𝛼 ∙ 𝑟𝑎 of the mass 𝑚𝑎
𝑚
used for accelerating the rotation is small compared to the gravitational acceleration 𝑔 = 9.81 𝑠2 and the
thread is always tangential to the wheel with 𝑟𝑎 . So with (1), (2) and (3)
𝑇 = 𝐼𝑧,𝑧
𝑑2 𝜑
= 𝑚𝑎 𝑔𝑟𝑎
𝑑𝑡 2
(6)
𝜔(𝑡) = 𝜔 (𝑡 = 0) +
𝑚𝑎 𝑔𝑟𝑎
∙𝑡
𝐼𝑧,𝑧
(7)
𝜑(𝑡) = 𝜑 (𝑡 = 0) +
1 𝑚𝑎 𝑔𝑟𝑎 2
∙
∙𝑡
2
𝐼𝑧,𝑧
(8)
The potential energy of the accelerating weight is
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Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
𝐸 = 𝑚𝑎 𝑔ℎ(𝑡) = −𝑚𝑎 𝑔𝜑(𝑡) ∙ 𝑟𝑎 =
TEP
1 𝑚𝑎2 𝑔2 𝑟𝑎2 2 1
∙
∙ 𝑡 = 𝐼𝑧,𝑧 𝜔2
2
𝐼𝑧,𝑧
2
Thus verifying (4).
If a weight 𝑚𝑖 is mounted to a rod that can rotate about the fixed axis 𝑧 perpendicular to it in a distance
𝑟𝑖 , maybe the rod lying along the y-axis, then are the coordinates of the weight (0, 𝑟𝑖 , 0)and according to
(3) is the moment of inertia about the z-axis 𝐼𝑧,𝑧 = 𝑚𝑖 𝑟𝑖2 , about the x-axis it is 𝐼𝑥,𝑥 = 𝑚𝑖 𝑟𝑖2 and about the yaxis it is 𝐼𝑦,𝑦 = 0.
Plot the angle of rotation vs. the square of time using "Analysis" > "Channel modification". The correct
zero of time 𝑡0 (the begin of movement without digital distortion) may be found with the "Regression" tool
of measure used on the "omega" values or on the data table as the first value where the angle exceeds
zero. Select time as source channel and enter the operation "(𝑡 − 𝑡0 )2 " with your actual value t0 in your
channel modification window. Then use "Measurement" > "Channel manager…" to select the square of
time as x-axis and "angle" as y-axis. The slope of the obtained curve is half the angular acceleration and
may be determined with the "Regression" tool. Fig. 2 and Fig. 3 show obtained data of several measurements which were put into one diagram with "Measurement" > "Assume channel…" and scaled to the
same value with the "Scale curves" tool and the option "set to values". The linearity vs. the square of
time can be seen well. Compare these slope values with the slope of the "omega" values of the raw data:
In Fig. 4 the angular acceleration values of Fig. 2 and Fig. 3 were plotted vs. the used torque 𝑇 = 𝑔𝑚𝑎 𝑟𝑎
with "Measurement" > "Enter data manually…". The inverse slope is then the moment of inertia and
reads 𝐼 = 13.3 𝑚𝑁𝑚𝑠 2 = 133 𝑘𝑔 𝑐𝑚2 for one disc and 𝐼 = 26.8 𝑚𝑁𝑚𝑠 2 = 268 𝑘𝑔 𝑐𝑚2 for two discs.
Fig. 2:
Angle vs. square of time for one turntable.
Fig. 4:
Fig. 3:
Angle vs. square of time for two turntables.
Angular acceleration vs. accelerating torque.
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Moment of inertia and angular acceleration with
Cobra4 and a precision pivot bearing
TEP
𝑇
Fig. 5 shows a plot of moment of inertia 𝐼𝑧,𝑧 = 𝛼 = 𝑚𝑖 𝑟𝑖2 + 𝐼𝑅𝑜𝑑;𝑧,𝑧 vs. the weight 𝑚𝑖 . The regression val-
ues yield 𝑟𝑖2 = 0.047 𝑚2 ⇒ 𝑟𝑖 = 21.7 𝑐𝑚 and 𝐼𝑅𝑜𝑑;𝑧,𝑧 = 83 𝑘𝑔 𝑐𝑚2 . The angular acceleration data were
evaluated from the measurements with the "Regression" function from the "omega" values and entered
with "Measurement" > "Enter data manually…" into a "Manually created measurement". The moment of
inertia values were calculated by "Analysis" > "Channel modification".
Fig. 5:
Moment of inertia vs. weight at r = 200 mm with constant
torque of 3.09 mNm.
Else you may evaluate the data using a bilogarithmic plot – but then it's crucial to correct the data for the
right zero point of time and angle using "Analysis" > "Channel modification…" subtracting 𝑡0 and adding/subtracting a 𝜑0 and making a plot of the changed channels with "Measurement" > "Channel manager…", erasing the values lower than zero in the data table and using the "Display options" tool to set the
scaling of both axes to "logarithmic". Fig. 6 shows an example for the bar with 2 ∙ 80 𝑔 mounted 21 𝑐𝑚
from the axis and accelerated with a weight of 21 𝑔 at 15 𝑚𝑚 i.e. a torque of 3.09 𝑚𝑁𝑚.
1
Since 𝜑 = 2 𝛼 ∙ 𝑡 2 , reads the plot
1
1
1
𝛼 = 0.103 ∙ 2 ⇒ 𝛼 = 0.203 2
2
𝑠
𝑠
On the other hand 𝑇 0 3.09 𝑚𝑁𝑚 𝛼 ∙ 𝐼̂ ⇒ 𝐼̂𝑧 = 150 𝑘𝑔 𝑐𝑚2 compared to theoretical
2
𝐼̂𝑧 = 2 ∙ 80 𝑔 ∙ (21 𝑐𝑚) + 𝐼̂𝑟𝑜𝑑,𝑧 = 70.6 𝑘𝑔 𝑐𝑚2 = 143 𝑘𝑔 𝑐𝑚2
Fig. 6:
6
Bilogarithmic plot of angle vs. time.
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