Mechanics Moment of inertia and angular acceleration with

29PHYWE Systeme GmbH & Co. KG · D-37070 Göttingen Laboratory Experiments Physics

Dynamics Mechanics

Moment of inertia and angular acceleration with Cobra3 1.3.13-11/15

Principle:If a constant torque is applied to abody that rotates without frictionaround a fixed axis, the changingangle of rotation increases propor-tionally to the square of the time andthe angular velocity proportional tothe time.

Potential energy and additionally the rotational energy.

Tasks:1. Measurement of the laws of angle

and angular velocity according totime for a uniform rotation move-ment.

2. Measurement of the laws of angleand angular velocity according totime for a uniformly acceleratedrotational movement.

3. Rotation angle � is proportional tothe time t required for the rota -tion.

Experiment P2131315 with precision pivot bearingExperiment P2131311 with air bearingCobra3 Basic-Unit, USB 12150.50 1 1Tripod base -PASS- 02002.55 1 1Precision pivot bearing 02419.00 1Inertia rod 02417.03 1Power supply 12V/2A 12151.99 1 1Software Cobra3, Translation/ Rotation 14512.61 1 1Light barrier, compact 11207.20 1 1Blower 230V/50Hz 13770.97 1Pressure tube, l = 1.5 m 11205.01 1Air bearing 02417.01 1Turntable with angular scale 02417.02 1 1Holding device with cable release 02417.04 1 1Aperture plate for turntable 02417.05 1 1Slotted weights, 1 g, polished 03916.00 9 20Slotted weights, 10 g, coated black 02205.01 3 10Slotted weight, 50 g, coated black 02206.01 2 2Silk thread on spool, l = 200 mm 02412.00 1 1Weight holder for slotted weights 02204.00 1 1Bench clamp -PASS- 02010.00 2 2Stand tube 02060.00 1Support rod, stainless steel 18/8, l = 250 mm, d = 10 mm 02031.00 1 1Measuring tape, l = 2 m 09936.00 1 1Circular level with mounting, d = 35 mm 02122.00 1 1Right angle clamp -PASS- 02040.55 1 1Connecting cable, 4 mm plug, 32 A, red, l = 100 cm 07363.01 1 1Connecting cable, 4 mm plug, 32 A, blue, l = 100 cm 07363.04 1 1Connecting cable, 4 mm plug, 32 A, yellow, l = 100 cm 07363.02 1 1Weight holder, 1 g 02407-00 1PC, Windows® XP or higher

What you need:

Complete Equipment Set, Manual on CD-ROM includedMoment of inertia and angular accelerationwith Cobra3 P2131311/15

What you can learn about …

� Angular velocity� Rotation� Moment� Torque� Moment of inertia� Rotational energy

Set-up of experiment P2131311 with air bearing

Related topicsAngular velocity, rotation, moment, torque, moment of inertia,rotational energy

PrincipleIf a constant torque is applied to a body that rotates withoutfriction around a fixed axis, the changing angle of rotationincreases proportionally to the square of the time and theangular velocity proportional to the time.

Task1. 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 rotationalmovement.

3. Rotation angle w is proportional to the time t required for therotation.

EquipmentCobra3 Basic Unit 12150.00 1Power supply, 12 V- 12151.99 1RS232 cable 14602.00 1Translation/Rotation Software 14512.61 1Light barrier, compact 11207.20 1Blower 13770.97 1Pressure tube, l = 1.5 m 11205.01 1

Air bearing 02417.01 1Turntable with angle scale 02417.02 2Holding device with cable release 02417.04 1Aperture plate for turntable 02417.05 1Slotted weight, 1 g, polished 03916.00 9Slotted weight, 10 g, black 02205.01 3Slotted weight, 50 g, silver bronze 02206.02 2Silk thread, l = 200 m 02412.00 1Weight holder, 10 g 02204.00 1Bench clamp -PASS- 02010.00 2Tripod -PASS- 02002.55 1Stand tube 02060.00 1Support rod, l = 250 mm 02031.00 1Measuring tape, l = 2 m 09936.00 1Circular level 02122.00 1Boss head 02043.00 1Connecting cord, l = 100 cm, red 07363.01 1Connecting cord, l = 100 cm, blue 07363.04 1Connecting cord, l = 100 cm, yellow 07363.02 1

PC, WINDOWS® 95 or higher

Alternative experimental set-ups are to be found at the end ofthis experimental description.

Set-up and procedureIn accordance with Fig. 1.

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2131311 1

LEP1.3.13

-11Moment of inertia and angular acceleration with Cobra3

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

Perform the electrical connection of the compact light barrierto the Cobra3 Basic Unit according to Fig. 2. Ensure that thethread that connects the axis of rotation with the wheel of thelight barrier is horizontal. Wind the thread approximately 15times 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 lift-ed by the air pressure and rotates without vibration on its cush-ion of air.Set the measuring perameters according to Fig. 3.Lay the silk thread across the wheel on the light barrier andadjust the experimental set-up in such a manner that the 10-gweight holder hangs freely. The cord groove on the wheel mustbe in alignment with the silk thread.

Place the stop plate (aperture plate for turntable) in the startingposition and fix it in position with the holding device. Enter thediameter of the turntable’s axle (30 mm), around which the silkthread will be rolled up, in the ”Axle diameter” dialog box sothat the differing rotational velocities of the compact light bar-rier and the axle of the turn table can be synchronised. The end of the silk thread is loaded with the 10-g weight hold-er and further additional weights. Switch on the blower, actuate the cable release. The turntablemust not begin to vibrate. As soon as the turntable has startedto rotate, click on the ”Start measurement” icon. Just beforethe weight holder reaches the floor, click on the ”Stop mea-surement” icon. The mass must not oscillate during measure-ment recording.Remarks:If the turntable does not rotate uniformly, check to see whetherallowing it to rotate in the opposite direction improves the sit-uation. If necessary, change the air supply at the blower.

Theory and evaluationThe relationship between the angular momentum of a rigidbody in the stationary coordinate system with its origin at thecentre of gravity, and the moment acting on it (see Fig. 4),is

(1)

The angular momentum is expressed by the angular velocityand the inertia tensor from

In the present case, has the direction of a principal inertiaaxis (Z-axis), so that has only one component:

LZ = IZ · v

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

TZ � IZ dvdt

.

L Sv S

L S

� I ˆ · v S

.

I ˆv S

T S

�d

dt L .

S

T S

L S

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P21313112

LEP1.3.13


Fig. 2. Connection of the compact light barrier to the Cobra3Basic Unit

Fig. 3. Measuring parameters

redyellow blue

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

The moment of the force (see Fig. 2)

gives for

TZ = r · m · g ,

so that the equation of motion reads

From this, one obtains

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 andthe angular velocity-time-law of the uniformly acceleratedrotary motion verifies the explained theory.

For the evaluation of the measured data do as follows: After clicking on the ”Autoscale” icon, all measured data aredisplayed in full-screen mode (cf. Fig. 5). In addition to theinteresting measured points themselves (the rising branch ofthe velocity-time curve), some points also may have beenmeasured which can be attributed to the termination of move-ment phase (possible contact of the accelerating mass with thefloor or something similar). These measured points can bedeleted before proceeding with the further evaluation.

<omega (t)> shows the angular velocity-time curve, a straightline which conforms to the relationship v = a · t (Fig. 5). Theproportionality factor a represents the angular acceleration

. If the Regression icon is clicked upon, a regression lineis drawn through the measured points; the slope m indicatesthe angular acceleration a. In the example in Fig. 5, for exam-ple a = 0.463 rad/s2. (The very noise onset of the measurementis due to the low resolution of the spoked wheel at low veloci-ties!)Fig. 6 shows the time course of the angular acceleration. Here,too, a linear regression line has been drawn. The segment ofthe y axis b = 0.443 rad/s2 supplies the initial value of the angu-lar acceleration a. For a uniformly accelerated rotary move-ment, the angular acceleration as a function of time is con-stant.Fig. 7 shows the curve of the path-time law, which exhibits aparabolic course, in which the measured points have beenstrongly emphasised. The parabolic course of the path-time law can be verified asfollows (Fig. 8):The time axis is squared to obtain a linearized curve course.Using the Measurement / Channel Manager, the time is placedon the x and the y axes. The is necessary as only the y axescan be mathematically reworked.Using Analysis / Channel modification, the operation x := x * xis performed on the y axis. This new channel is exported intothe original measurement (Export Measurement / MeasuringChannel). Finally, using Measurement / Channel Manager, thenew squared time is assigned to the x axis and the angle j, tothe y axis. The regression line in Fig. 8 proves that the curvecourse is now linear and thus also the original quadraticdependence of the path on the time.

v#

� a

IZ � mgr

a .

mgr � IZ dvdt

� IZ · a .

r S � F S

:

TS

� r S � F S

F S


LEP1.3.13


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

Fig. 8. Angle-time2 diagram

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

Fig. 6. Regression line in the at diagram

The turntable is accelerated by the vertically moving mass. Theeffective torque M is calculated according to

M = r · m · g

where:r Radius of the axle bolt or of the driving

wheel used m Accelerated massg = 9.81 m/s2 Acceleration of gravity

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

M · t = J · v.

Thus, for the moment of inertia J the following is true:

In an v(t) graph (Fig 5) the v(t) relationship is exactly the slopeof the regression line. To calculate J, the accelerating mass mand the radius r (1.5 cm) of the rotational axis around which thethread is wound must be taken into consideration.

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

= 0.0075 kg m2/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 = J · a .therefore

= 0.0169 kg m2.

From Fig. 6 one obtains a from the y axis segment of theregression line.The rotational energy (Fig. 9): Erot(t) = 0.5 J v2 , in this case J= 0.0165 kg/m2. Conversion by: Analysis / Channel modifica-tion / 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 / Operationx := 0.051 * 9.81 * (0.77 - x * 0.015), where x = w(t).The law of conservation of energy states that the sum of thekinetic and potential energy in this closed system must be con-stant. This statement can be easily checked by the addition ofpotential and kinetic energy (Fig. 11).Remark: The accelerated mass m becomes increasingly morerapid in the course of the experiment and thus receives anincreasing kinetic energy. However, this energy is extremelysmall compared to the two other energy forms present and canthus be neglected in the calculation.

J � Ma

� 0.0075 kg m2>s2

a

J � 0.0075 kg m2 >s2

0.463 rad > s2 .

J � M

v >t �

r · m · g

v > t .


LEP1.3.13


Fig. 9. Rotational energy of accelerated rotary movement

Fig. 10. Potential energy and additionally the rotational energy Fig. 11. Energy balance, Etotal = Erot + Epot

RemarksAt extremely slow angular velocities, signal transients or defor-mations can occur. These can be reduced if the sampling rateis changed.Angular velocities that are too small cannot be measured bythe wheel on the light barrier and are plotted as a referenceline.Instead of the compact light barrier (11207.20), the movementsensor (12004.10) can also be used (see Fig. 12: The thread ishorizontal and is placed in the larger of the two cord grooveson the movement sensor.) In this case the following additionalequipment is required:

EquipmentMovement sensor with cable 12004.10 1Adapter, BNC-socket/4mm plug pair 07542.27 1Adapter, socket-plug, 4 mm 07542.20 1


LEP1.3.13


Fig. 12. Connection of the movement sensor to the Cobra3Basic Unit

redblackyellowBNC1BNC2


LEP1.3.13


LEP1.3.13



Related TopicsRotation, angular velocity, torque, angular acceleration, angu-lar momentum, moment of inertia, rotational energy.

PrincipleA known torque is applied to a body that can rotate about afixed axis with minimal friction. Angle and angular velocity aremeasured over the time and the moment of inertia is deter-mined. The torque is exerted by a string on a wheel of knownradius with the force on the string resulting from the knownforce of a mass in the earth's gravitational field. The knownenergy gain of the lowering mass is converted to rotationalenergy of the body under observation.

Tasks1. Measure the angular velocity and angle of rotation vs. time

for a disc with constant torque applied to it for different val-ues of torque generated with various forces on three differ-ent radii. Calculate the moment of inertia of the disc.

2. Measure the angular velocity and angle of rotation vs. timeand thus the moment of inertia for two discs and for a barwith masses mounted to it at different distances from theaxis of rotation.

3. Calculate the rotational energy and the angular momentumof the disc over the time. Calculate the energy loss of theweight from the height loss over the time and compare.

EquipmentTripod -PASS- 02002.55 1Precision bearing 02419.00 1Inertia rod 02417.03 1Turntable with angle scale 02417.02 2Aperture plate for turntable 02417.05 1Cobra3 Basic Unit, USB 12150.50 1Cobra3 power supply unit 12151.99 1Translation/Rotation Software 14512.61 1Light barrier, compact 11207.20 1Connecting cord, l =150 cm, red 07364.01 1Connecting cord, l =150 cm, blue 07364.04 1Connecting cord, l =150 cm, yellow 07364.02 1Boss head 02043.00 1Support rod, l = 250 mm 02031.00 1Bench clamp -PASS- 02010.00 2Silk thread, l = 200 m 02412.00 1Circular level 02122.00 1Weight holder, 10 g 02204.00 1Weight holder, 1 g 02407.00 1Slotted weight, 1 g, natur. colour 03916.00 20Slotted weight, 10 g, black 02205.01 10Slotted weight, 50 g, black 02206.01 2Holding device w.cable release 02417.04 1Measuring tape, l = 2 m 09936.00 1PC, WINDOWS® 95 or higher

Fig.1: Experimental set up with turntable

LEP1.3.13


P2131315 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

Set-up and procedure1. Set the experiment up as seen in Fig.1. Connect the com-pact light barrier to the Cobra3 unit according to Fig.2.Connect the COBRA3 Basic Unit to a USB port of the com-puter. Adjust the turntable to be horizontal – it must not startto move with an imbalance without other torque applied. Fixthe silk thread (with the weight holder on one end) with thescrew of the precision bearing or a piece of adhesive tape tothe wheels with the grooves on the axis of rotation and wind itseveral times around one of the wheels – enough turns, thatthe weight may reach the floor. Be sure the thread and thewheel of the compact light barrier and the groove of theselected wheel are well aligned. Place the holding device withcable release in a way that it just holds the turntable on the"Aperture plate" and does not disturb the movement afterrelease.

Start the "measure" program and select "Gauge" > "Cobra3 –Translation / Rotation", select the tab "Rotation" and set theparameters as seen in Fig.3. In the field "Axle diameter" youhave to enter the diameter of the chosen wheel with groove onthe axis of rotation. It may be best to start with the one with15 mm radius and 30 mm diameter (the smallest), becausewith this one the highest number of turns of the disc is observ-able before the weight reaches the floor. Start the measurement with the "Continue" button and release

the turntable. Data recording should start automatically. Stopdata recording just before the weight reaches the floor by keypress (in order not to have irrelevant data at the end of yourrecorded data).

1. Record measurements with different accelerating weightsma 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 tohigher values. Record measurements with the thread running in differentgrooves – adjust the position of the light barrier to align threadand groove and wheel of light barrier (thread has to be hori-zontal) and enter the correct axle diameter in the "Rotation"chart – and choose especially the weight on the end of thethread ma such that the torque is constant e.g.ma = 60 g for ra = 15 mm and ma = 30 g for ra = 30 mm andma = 20 g for ra = 45 mm, each time the torque being T =magra= 8.83 mNm with the earth's gravitational accelerationg = 9.81 N/kg = 9.81 m/s2. Also choose a weight with whichyou take a measurement for each groove radius.You may also take a measurement with the weight ma so lightthat it nearly does not affect the movement (like theweightholder of 1 g) but is enough to drive the light barrier'swheel. Start the turntable by a shove with the hand. Nearlyunaccelerated movement should be observable.

2. Record measurements with two turntables mounted on theprecision bearing with weight values used on the singleturntable and with double the weight used on the singleturntable for comparison.Remove the turntables and mount the inertia rod to the preci-sion bearing and the two weight holders symmetrically to therod with both the same distance to the axis ri.Take measurements with varied masses mi at constant ri andalso with constant masses mi at varied ri (both masses ofcourse still mounted symmetrically) – accelerated with thesame weight ma (or with the same series of weights for highprecision).

T S

� ra S

� F S

Fig.3: Cobra3 – Translation / Rotation settings

Fig. 2. Connection of the compact light barrier to the Cobra3Basic Unit

redyellow blue

Theory and evaluation

The angular momentum of a single particle at position with velocity , mass m and momentum isdefined as

and the torque from the force is defined as

,

with torque and angular momentum depending on the originof the reference frame. The change of in time is

and with

and Newtons's law the equation of movement

becomes

(1).

For a system of N particles with center of mass and

total linear momentum the angular momentum

is

Now the movement of the center of mass is neglected, the ori-

gin set to the center of mass and a rigid body assumed with

fixed. The velocity of particle i may be written as

with vector of rotation

(2)

constant throughout the body. Then

With is

with

and

The inertial coefficients or moments of inertia are defined as

(3)

and with the matrix it is

(4)

and for the rotational acceleration is then

The rotational energy is

Sum convention: sum up over same indices using

The coordinate axes can allways be set to the "principal axesof inertia" so that none but the diagonal elements of the matrixIk,k ≠ 0.In this experiment only a rotation about the z-axis can occurand with the unit vector . The energy isthen

(5)

The torque T = ma (g-a)ra is nearly constant in time since theacceleration a = a · ra of the mass ma used for accelerating therotation is small compared to the gravitational accelerationg = 9.81 m/s2 and the thread is always tangential to the wheelwith ra. So with (1), (2) and (3)

(6)

(7)

(8).

The potential energy of the accelerating weight is

E = magh(t) = -magw(t) · ra

thus verifying (4).

�182 1

2 ·

ma2g2ra

2

Iz,z · t2 �172 1

2 Iz,z v

2 ,

w1t 2 � w 1t � 0 2 �12

· ma gra

Iz,z · t2

v1t 2 � v 1t � 0 2 � ma gra

Iz,z · t

T � Iz,z d2w

dt2 � ma gra

E �12

Iz,zv2

ezv S � ez vz � ez v

1 aS � bS2 1 cS � d

S2 � 1 aS· cS 2 1 bS· d

S2 � 1 aS · d

S2 1 b

S · cS 2 .

E �12ami ni

2 �12ami 1v S � rSi 2

2 �12

Ik,l vk vl

T S

�dJ

S

dt� I ˆ ·

dv S

dt� I ˆ · a S

a S �dv S

dt

JS

� I ˆ· vS

I ˆ � 5Ik,l6

Ix,z � � ami xi zi

Ix,y � � ami xi yi

Ix,x � ami 1ri2 � xi

2 2

Jz � vzami 1ri2 � zi

2 2 � vzamizixi � vyamiziyi .

rSi · vS � xi vx � yi vy � zi vz

J S

� ami 1 vS · ri

2 � rSi 1 rSi ·vS 2 2

aS � 1 bS

� cS 2 � bS1 aS· cS 2 � cS1 aS· b

S2

JS

� ami rSi � nSi � ami rSi � 1 vS � rSi 2 .

vS �d wS

dt

nSi � vS � rSi

rSi � rSj

� JS

c.m. � RS

c.m. � P S

.

J S

� aN

i�1mi 1 r

Si � R

Sc.m. 2 � nSi � a

N

i�1mi R

Sc.m. � nSi �

PS

� ami nS

i

R S

c.m.

T S

�dJ

S

dt

FS

�dp S

dt

d r S

dt� p S � n S � m n S � 0

d J S

dt�

ddt

1 r S � p S 2 � d r S

dt� p S � r S �

d p S

dt

J S

TS

� r S � F S

F S

T S

JS

� r S � p S

p S � m n Sn Sr SJ

S

LEP1.3.13



LEP1.3.13



If a weight mi is mounted to a rod that can rotate about thefixed axis z perpendicular to it in a distance ri, maybe the rodlying along the y-axis, then are the coordinates of the weight(0,ri,0)and according to (3) is the moment of inertia about thez-axis , about the x-axis it is andabout 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 timet0 (the begin of movement without digital distortion) may befound with the "Regression" tool of measure used on the"omega" values or on the data table as the first value wherethe angle exceeds zero. Select time as source channel andenter the operation "(t-t0)^2" with your actual value t0 in yourchannel modification window. Then use "Measurement" >"Channel manager…" to select the square of time as x-axisand "angle" as y-axis. The slope of the obtained curve is halfthe angular acceleration and may be determined with the"Regression" tool. Fig. 4 and Fig. 5 show obtained data ofseveral measurements which were put into one diagram with"Measurement" > "Assume channel…" and scaled to thesame value with the "Scale curves" tool and the option "set tovalues". 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. 6were plotted vs. the used torque T = gmara with"Measurement" > "Enter data manually…". The inverse slopeis then the moment of inertia and reads I = 13.3 mNms2 =133 kg cm2 for one disc and I = 26.8 mNms2 = 268 kgcm2 fortwo discs.

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 angularacceleration data were evaluated from the measurements withthe "Regression" function from the "omega" values and enteredwith "Measurement" > "Enter data manually…" into a "Manuallycreated measurement". The moment of inertia values were cal-culated by "Analysis" > "Channel modification".

Ix,x � mi ri2Iz,z � mi ri

2

Fig. 4: Angle vs. square of time for one turntable

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

Fig. 6 : Angular acceleration vs. accelerating torque.

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

LEP1.3.13



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 – butthen it's crucial to correct the data for the right zero point oftime and angle using "Analysis" > "Channel modification…"subtracting t0 and adding/subtracting a w0 and making a plotof the changed channels with "Measurement" > "Channelmanager…", erasing the values lower than zero in the datatable and using the "Display options" tool to set the scaling ofboth axes to "logarithmic". Fig. 7 shows an example for thebar with 2 · 80 g mounted 21 cm from the axis and accelerat-ed with a weight of 21 g at 15 mm i.e. a torque of 3.09 mNm.

Since , reads the plot

,

on the other hand T = 3.09 mNm compared to theoretical

= 143 kgcm2.

I ˆz � 2 · 80 g · 121 cm 22 � I ˆrod,z � 70.6 kgcm2 � 72 kgcm2

a · I ˆ 1 I ˆz � 150 kgcm2

12 a � 0.103 ·

1s2 1 a � 0.203

1s2

w � 12 a · t

2

Fig. 8: Moment of inertia vs. square of radius with m = 2 ·100 gand constant torque of 3.09 mNm

Fig. 9 Bilogarithmic plot of angle vs. time

LEP1.3.13



Documents

Mechanics Moment of inertia and angular acceleration with