Mechanics

Dynamics

Moment of inertia and angular acceleration with Cobra3

1.3.13-11/15

What you can learn about …  Angular velocity  Rotation  Moment  Torque  Moment of inertia  Rotational energy

Principle: If a constant torque is applied to a body that rotates without friction around a fixed axis, the changing angle of rotation increases proportionally to the square of the time and the angular velocity proportional to the time.

Set-up of experiment P2131311 with air bearing

What you need: Experiment P2131315 with precision pivot bearing Experiment P2131311 with air bearing Cobra3 Basic-Unit, USB 12150.50 Tripod base -PASS02002.55 Precision pivot bearing 02419.00 Inertia rod 02417.03 Power supply 12V/2A 12151.99 Software Cobra3, Translation/ Rotation 14512.61 Light barrier, compact 11207.20 Blower 230V/50Hz 13770.97 Pressure tube, l = 1.5 m 11205.01 Air bearing 02417.01 Turntable with angular scale 02417.02 Holding device with cable release 02417.04 Aperture plate for turntable 02417.05 Slotted weights, 1 g, polished 03916.00 Slotted weights, 10 g, coated black 02205.01 Slotted weight, 50 g, coated black 02206.01 Silk thread on spool, l = 200 mm 02412.00 Weight holder for slotted weights 02204.00 Bench clamp -PASS02010.00 Stand tube 02060.00 Support rod, stainless steel 18/8, l = 250 mm, d = 10 mm 02031.00 Measuring tape, l = 2 m 09936.00 Circular level with mounting, d = 35 mm 02122.00 Right angle clamp -PASS02040.55 Connecting cable, 4 mm plug, 32 A, red, l = 100 cm 07363.01 Connecting cable, 4 mm plug, 32 A, blue, l = 100 cm 07363.04 Connecting cable, 4 mm plug, 32 A, yellow, l = 100 cm 07363.02 Weight holder, 1 g 02407-00 PC, Windows® XP or higher

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1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 9 20 3 10 2 2 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Potential energy and additionally the rotational energy.

Tasks: 1. Measurement of the laws of angle and angular velocity according to time for a uniform rotation movement. 2. Measurement of the laws of angle and angular velocity according to time for a uniformly accelerated rotational movement. 3. Rotation angle ␸ is proportional to the time t required for the rotation.

Complete Equipment Set, Manual on CD-ROM included Moment of inertia and angular acceleration with Cobra3 P2131311/15 PHYWE Systeme GmbH & Co. KG · D - 37070 Göttingen

Laboratory Experiments Physics 29

LEP 1.3.13 -11

Moment of inertia and angular acceleration with Cobra3

Related topics Angular velocity, rotation, moment, torque, moment of inertia, rotational energy Principle If a constant torque is applied to a body that rotates without friction around a fixed axis, the changing angle of rotation increases proportionally to the square of the time and the angular velocity proportional to the time. Task 1. Measurement of the laws of angle and angular velocity according to time for a uniform rotation movement. 2. Measurement of the laws of angle and angular velocity according to time for a uniformly accelerated rotational movement. 3. Rotation angle w is proportional to the time t required for the rotation. Equipment Cobra3 Basic Unit Power supply, 12 VRS232 cable Translation/Rotation Software Light barrier, compact Blower Pressure tube, l = 1.5 m

12150.00 12151.99 14602.00 14512.61 11207.20 13770.97 11205.01

1 1 1 1 1 1 1

Air bearing Turntable with angle scale Holding device with cable release Aperture plate for turntable Slotted weight, 1 g, polished Slotted weight, 10 g, black Slotted weight, 50 g, silver bronze Silk thread, l = 200 m Weight holder, 10 g Bench clamp -PASSTripod -PASSStand tube Support rod, l = 250 mm Measuring tape, l = 2 m Circular level Boss head Connecting cord, l = 100 cm, red Connecting cord, l = 100 cm, blue Connecting cord, l = 100 cm, yellow

02417.01 02417.02 02417.04 02417.05 03916.00 02205.01 02206.02 02412.00 02204.00 02010.00 02002.55 02060.00 02031.00 09936.00 02122.00 02043.00 07363.01 07363.04 07363.02

1 2 1 1 9 3 2 1 1 2 1 1 1 1 1 1 1 1 1

PC, WINDOWS® 95 or higher Alternative experimental set-ups are to be found at the end of this experimental description. Set-up and procedure In accordance with Fig. 1.

Fig. 1. Experimental set-up with the compact light barrier

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P2131311

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Moment of inertia and angular acceleration with Cobra3

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 15 times around the air bearing’s rotation axis. Adjust the tripod’s feet such that the turntable is horizontal. Adjust the air supply in such a manner that the rotor is just lifted by the air pressure and rotates without vibration on its cushion of air. Set the measuring perameters according to Fig. 3. Lay the silk thread across the wheel on the light barrier and adjust the experimental set-up in such a manner that the 10-g weight holder hangs freely. The cord groove on the wheel must be in alignment with the silk thread. Fig. 2. Connection of the compact light barrier to the Cobra3 Basic Unit

Place the stop plate (aperture plate for turntable) in the starting position and fix it in position with the holding device. Enter the diameter of the turntable’s axle (30 mm), around which the silk thread will be rolled up, in the ”Axle diameter” dialog box so that the differing rotational velocities of the compact light barrier and the axle of the turn table can be synchronised. The end of the silk thread is loaded with the 10-g weight holder and further additional weights. Switch on the blower, actuate the cable release. The turntable must not begin to vibrate. As soon as the turntable has started to rotate, click on the ”Start measurement” icon. Just before the weight holder reaches the floor, click on the ”Stop measurement” icon. The mass must not oscillate during measurement recording. Remarks: If the turntable does not rotate uniformly, check to see whether allowing it to rotate in the opposite direction improves the situation. If necessary, change the air supply at the blower. Theory and evaluation S The relationship between the angular momentum L of a rigid body in the stationary coordinate system with its origin at the S centre of gravity, and the moment T acting on it (see Fig. 4), is d S S T
L. (1) dt The angular momentum is expressed by the angular velocity S v and the inertia tensor Iˆ from S S L
Iˆ · v .

In the present case, axis (Z-axis), so that

S v has the direction of a principal inertia S L has only one component: LZ = IZ · v

where IZ is the Z-compound of the principal inertia tensor of the body. For this case, equation (1) reads yellow

Fig. 3. Measuring parameters

2

blue

red

TZ
IZ

dv . dt

Fig. 4. Moment of a weight force on the rotary plate

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P2131311

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Moment of inertia and angular acceleration with Cobra3

S The moment of the force F (see Fig. 2) S S T
S r  F S gives for S r
F : TZ = r · m · g , so that the equation of motion reads mgr
IZ

dv
IZ · a . dt

From this, one obtains IZ


mgr . a

The moment of inertia IZ of a body of density r (x, y, z) is IZ = ∫∫∫ r (x, y, z) (x2 + y2) dx dy dz In this experiment the measurement of the angle-time-law and the angular velocity-time-law of the uniformly accelerated rotary motion verifies the explained theory. For the evaluation of the measured data do as follows: After clicking on the ”Autoscale” icon, all measured data are displayed in full-screen mode (cf. Fig. 5). In addition to the interesting measured points themselves (the rising branch of the velocity-time curve), some points also may have been measured which can be attributed to the termination of movement phase (possible contact of the accelerating mass with the floor or something similar). These measured points can be deleted before proceeding with the further evaluation.

shows the angular velocity-time curve, a straight line which conforms to the relationship v = a · t (Fig. 5). The proportionality factor a represents the angular acceleration # v
a . If the Regression icon is clicked upon, a regression line is drawn through the measured points; the slope m indicates the angular acceleration a. In the example in Fig. 5, for example a = 0.463 rad/s2. (The very noise onset of the measurement is due to the low resolution of the spoked wheel at low velocities!) Fig. 6 shows the time course of the angular acceleration. Here, too, a linear regression line has been drawn. The segment of the y axis b = 0.443 rad/s2 supplies the initial value of the angular acceleration a. For a uniformly accelerated rotary movement, the angular acceleration as a function of time is constant. Fig. 7 shows the curve of the path-time law, which exhibits a parabolic course, in which the measured points have been strongly emphasised. The parabolic course of the path-time law can be verified as follows (Fig. 8): The time axis is squared to obtain a linearized curve course. Using the Measurement / Channel Manager, the time is placed on the x and the y axes. The is necessary as only the y axes can be mathematically reworked. Using Analysis / Channel modification, the operation x := x * x is performed on the y axis. This new channel is exported into the original measurement (Export Measurement / Measuring Channel). Finally, using Measurement / Channel Manager, the new squared time is assigned to the x axis and the angle j, to the y axis. The regression line in Fig. 8 proves that the curve course is now linear and thus also the original quadratic dependence of the path on the time.

Fig. 5. Angular velocity-time laws of an accelerated rotational movement with regression line

Fig. 7. Angle-time diagram with individual measure points

Fig. 6. Regression line in the at diagram

Fig. 8. Angle-time2 diagram

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Moment of inertia and angular acceleration with Cobra3

The turntable is accelerated by the vertically moving mass. The effective torque M is calculated according to M=r·m·g where: r m g = 9.81 m/s2

In this exemplary measurement the following is valid: m (slope) = 0.463 rad/s2 = v/t M = r · m · g = 0.015 m · 0.051 kg · 9.81 m/s2 = 0.0075 kg m2/s2 J


Radius of the axle bolt or of the driving wheel used Accelerated mass Acceleration of gravity

The relationship between the torque impulse M t, the moment of inertia J and angular velocity v is the following:

0.0075 kg m2 >s2 0.463 rad > s2

.

= 0.0162 kg m2. The moment of inertia J is also obtained in another way: The dynamic action of torques is the angular acceleration. Torque and angular acceleration are proportional to each other:

M · t = J · v. Thus, for the moment of inertia J the following is true: J


r·m·g M
. v >t v>t

In an v(t) graph (Fig 5) the v(t) relationship is exactly the slope of the regression line. To calculate J, the accelerating mass m and the radius r (1.5 cm) of the rotational axis around which the thread is wound must be taken into consideration. Fig. 9. Rotational energy of accelerated rotary movement

Fig. 10. Potential energy and additionally the rotational energy

4

M=J·a. therefore J


0.0075 kg m2>s2 M
a a = 0.0169 kg m2.

From Fig. 6 one obtains a from the y axis segment of the regression line. The rotational energy (Fig. 9): Erot(t) = 0.5 J v2 , in this case J = 0.0165 kg/m2. Conversion by: Analysis / Channel modification / Operation x := 0.5 * 0.0165 * x * x, where x = v(t). Potential energy (Fig. 10): Epot(t) = m g (h - s (t )), where h = 0.77 m and s (t ) = w(t) r. Conversion using: Analysis / Channel modification / Operation x := 0.051 * 9.81 * (0.77 - x * 0.015), where x = w(t). The law of conservation of energy states that the sum of the kinetic and potential energy in this closed system must be constant. This statement can be easily checked by the addition of potential and kinetic energy (Fig. 11). Remark: The accelerated mass m becomes increasingly more rapid in the course of the experiment and thus receives an increasing kinetic energy. However, this energy is extremely small compared to the two other energy forms present and can thus be neglected in the calculation.

Fig. 11. Energy balance, Etotal = Erot + Epot

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P2131311

LEP 1.3.13 -11

Moment of inertia and angular acceleration with Cobra3

Remarks At extremely slow angular velocities, signal transients or deformations can occur. These can be reduced if the sampling rate is changed. Angular velocities that are too small cannot be measured by the wheel on the light barrier and are plotted as a reference line. Instead of the compact light barrier (11207.20), the movement sensor (12004.10) can also be used (see Fig. 12: 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

red black yellow BNC1 BNC2

1 1 1

Fig. 12. Connection of the movement sensor to the Cobra3 Basic Unit

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P2131311

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Moment of inertia and angular acceleration with Cobra3

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

P2131311

LEP 1.3.13 -15

Moment of inertia and angular acceleration with Cobra3

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. 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.

Equipment Tripod -PASSPrecision bearing Inertia rod Turntable with angle scale Aperture plate for turntable Cobra3 Basic Unit, USB Cobra3 power supply unit Translation/Rotation Software Light barrier, compact Connecting cord, l =150 cm, red Connecting cord, l =150 cm, blue Connecting cord, l =150 cm, yellow Boss head Support rod, l = 250 mm Bench clamp -PASSSilk thread, l = 200 m Circular level Weight holder, 10 g Weight holder, 1 g Slotted weight, 1 g, natur. colour Slotted weight, 10 g, black Slotted weight, 50 g, black Holding device w.cable release Measuring tape, l = 2 m PC, WINDOWS® 95 or higher

02002.55 1 02419.00 1 02417.03 1 02417.02 2 02417.05 1 12150.50 1 12151.99 1 14512.61 1 11207.20 1 07364.01 1 07364.04 1 07364.02 1 02043.00 1 02031.00 1 02010.00 2 02412.00 1 02122.00 1 02204.00 1 02407.00 1 03916.00 20 02205.01 10 02206.01 2 02417.04 1 09936.00 1

Fig.1: Experimental set up with turntable

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P2131315

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LEP 1.3.13 -15

Moment of inertia and angular acceleration with Cobra3

Set-up and procedure 1. Set the experiment up as seen in Fig.1. Connect the compact light barrier to the Cobra3 unit according to Fig. 2. Connect the COBRA3 Basic Unit to 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 compact light barrier 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 select "Gauge" > "Cobra3 – Translation / Rotation", select the tab "Rotation" and set the parameters as seen in Fig.3. In the field "Axle diameter" you have to enter the diameter of the chosen wheel with groove on the axis of rotation. It may be best to start with the one with 15 mm radius and 30 mm diameter (the smallest), because with this one the highest number of turns of the disc is observable before the weight reaches the floor.

Fig. 2. Connection of the compact light barrier to the Cobra3 Basic Unit

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blue

red

Fig.3: Cobra3 – Translation / Rotation settings

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 the end of the S S S thread ma such that the torque T
ra  F is constant e.g. ma = 60 g for ra = 15 mm and ma = 30 g for ra = 30 mm and ma = 20 g for ra = 45 mm, each time the torque being T = magra= 8.83 mNm with the earth's gravitational acceleration g = 9.81 N/kg = 9.81 m/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).

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LEP 1.3.13 -15

Moment of inertia and angular acceleration with Cobra3

The inertial coefficients or moments of inertia are defined as

Theory and evaluation S The angular momentum J of a single particle at position S r with velocity S n , mass m and momentum S p
mS n is defined as S S J
S r  p

Ix,x
a mi 1r2i  x2i 2

(3)

Ix,y
 a mi xi yi Ix,z
 a mi xi zi

S S and the torque T from the force F is defined as and with the matrix Iˆ
5Ik,l 6 it is

S S r  F , T
S with torque and angular momentum depending on the origin S of the reference frame. The change of J in time is S dS p dJ d S dS r
1 r  S p 2
 S p  S r  dt dt dt dt and with

S S dp and Newtons's law F
the equation of movement dt becomes

S dv and for the rotational acceleration S is then a
dt

The rotational energy is 1 1 1 SS r i 2 2
Ik,l vk vl E
a mi n2i
a mi 1v 2 2 2 Sum convention: sum up over same indices using

S S dJ T
dt

(1).

S For a system of N particles with center of mass R c.m. and S n i the angular momentum total linear momentum P
a mi S is N N S S S r i  R c.m. 2  S n i  a mi R c.m.  S ni
J
a mi 1 S i
1

S S S
J c.m.  R c.m.  P . Now the movement of the center of mass is neglected, the origin set to the center of mass and a rigid body assumed with S r S r fixed. The velocity of particle i may be written as i

(4)

S S dv S dJ T

Iˆ ·
Iˆ · S a dt dt

dS r  S p
S n mS n
0 dt

i
1

S J
Iˆ· S v

j

S v S r i with vector of rotation ni
S dS w S v
dt

(2)

S S S S S S 1S a  b 2 1S c  d 2
1S a·S c 2 1 b · d 2  1S a · d 21 b · S c2. The coordinate axes can allways 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 v
eˆ z vz
eˆ z v with the unit vector eˆ z . The energy is and S then 1 E
Iz,zv2 (5) 2 The torque T = ma (g-a)ra is nearly constant in time since the acceleration a = a · ra of the mass ma used for accelerating the rotation is small compared to the gravitational acceleration g = 9.81 m/s2 and the thread is always tangential to the wheel with ra. So with (1), (2) and (3) T
Iz,z

d2w dt2


ma gra

(6)

constant throughout the body. Then S J
a miS ri  S n i
a miS r i  1S v S r i2 . S S S a 1b S c 2
b 1S a·S c2 S c 1S a · b 2 is With S S J
a mi 1 S v · r2i  S r i 1S r i ·S v 2 2 with

v1t 2
v 1t
02  w1t 2
w 1t
02 

ma gra ·t Iz,z

(7)

1 ma gra 2 · ·t 2 Iz,z

(8).

The potential energy of the accelerating weight is

and

172 1 182 1 m2g2r2 a a E = magh(t) = -magw(t) · ra
· · t2
Iz,z v2 , 2 Iz,z 2

Jz
vz a mi 1r2i  z2i 2  vz a mizixi  vy a miziyi .

thus verifying (4).

S ri·S v
xi vx  yi vy  zi vz

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

P2131315

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Moment of inertia and angular acceleration with Cobra3

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 r2i , about the x-axis it is Ix,x
mi r2i and about the y-axis 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. 4 and Fig. 5 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. 6 the angular acceleration values of Fig. 5 and Fig. 6 were plotted vs. the used torque T = gmara with "Measurement" > "Enter data manually…". The inverse slope is then the moment of inertia and reads I = 13.3 mNms2 = 133 kg cm2 for one disc and I = 26.8 mNms2 = 268 kgcm2 for two discs.

Fig. 6 : Angular acceleration vs. accelerating torque.

Fig. 7 shows a plot of moment of inertia Iz,z = T/a = miri2 + IRod;z,z vs. the weight mi. The regression values yield ri2 = 0.047 m2 1 ri = 21.7cm and IRod;z,z = 83 kgcm2. 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. 4: Angle vs. square of time for one turntable

Fig. 5: Angle vs. square of time for two turntables

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Fig. 7: Moment of inertia vs. weight at r = 200 mm with constant torque of 3.09 mNm

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

LEP 1.3.13 -15

Moment of inertia and angular acceleration with Cobra3

Fig.8 shows a plot Iz,z = T/a = miri2 + IRod;z,z vs. square of radius ri2. The regression values yield a mass mi = 240 g and IRod;z,z.= 82 kg cm2

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 w0 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. 7 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. Since w
12 a · t2 , reads the plot 1 1 1 a
0.203 2 , s2 s on the other hand T = 3.09 mNm a · Iˆ 1 Iˆz
150 kgcm2 compared to theoretical 1 2

a
0.103 ·

Iˆz
2 · 80 g · 121 cm2 2  Iˆrod,z
70.6 kgcm2  72 kgcm2 = 143 kgcm2. Fig. 8: Moment of inertia vs. square of radius with m = 2 ·100 g and constant torque of 3.09 mNm

Fig. 9 Bilogarithmic plot of angle vs. time

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P2131315

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Moment of inertia and angular acceleration with Cobra3

P2131315

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen