lab5 lep1313 01 angular momentum

1.3.13-01/05 Moment of inertia and angular acceleration

Principle:A moment acts on a body which canbe rotated about a bear ing withoutfriction. The moment of inertia is determined from the angular accel-eration.

Moment of inertia of a mass point as a function of the square of its distancefrom the axis of rotation.

Tasks:From the angular acceleration, themoment of inertia is determined as a function of the mass and of thedistance from the axis of rotation.

1. of a disc,

2. of a bar,

3. of a mass point.

What you can learn about …

� Angular velocity� Rotary motion� Moment� Moment of inertia of a disc� Moment of inertia of a bar� Moment of inertia of a mass

point

Experiment P2131305 with precision pivot bearingExperiment P2131301 with air bearingTripod base -PASS- 02002.55 2 1Precision pivot bearing 02419.00 1Inertia rod 02417.03 1 1Turntable with angular scale 02417.02 1 2Aperture plate for turntable 02417.05 1 1Air bearing 02417.01 1Holding device with cable release 02417.04 1 1Precision pulley 11201.02 1 1Blower 230 V/50Hz 13770.97 1Pressure tube, l = 1.5 m 11205.01 1Light barrier with counter 11207.30 1 1Power supply 5 V DC/2.4 A with 4 mm plugs 11076.99 1 1Supporting blocks, 10, 20, 30, 40 mm 02070.00 1Slotted weights, 1 g, polished 03916.00 20 20Slotted weights, 10 g, coated black 02205.01 10 10Slotted weight, 50 g, coated black 02206.01 2 2Weight holder, 1g, silver bronzing 02407.00 1 1Silk thread on spool, l = 200 mm 02412.00 1 1Support rod -PASS-, square, l = 1000 mm 02028.55 1Support rod -PASS-, square, l = 400 mm 02026.55 1 1Right angle clamp -PASS- 02040.55 3 2Bench clamp -PASS- 02010.00 2 2Connecting cord, 32 A, 1000 mm, red 07363.01 1Connecting cord, 32 A, 1000 mm, blue 07363.04 1Adapter, BNC-plug/socket 4 mm. 07542.26 1Capacitor 100 nF/250V, G1 39105.18 1Circular level 02122.00 1Weight holder for slotted weights 02204.00 1Measuring tape, l = 2 m 09936.00 1

What you need:

Complete Equipment Set, Manual on CD-ROM includedMoment of inertia and angular acceleration P2131301/05

28 PHYWE Systeme GmbH & Co. KG · D-37070 GöttingenLaboratory Experiments Physics

Mechanics Dynamics

Set-up of experiment P2131305 with pivot bearing

LEP1.3.13

-01Moment of inertia and angular acceleration

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2131301 1

Related topicsAngular velocity, rotary motion, moment, moment of inertia ofa disc, moment of inertia of a bar, moment of inertia of a masspoint.

PrincipleA moment acts on a body which can be rotated about a bear-ing without friction. The moment of inertia is determined fromthe angular acceleration.

EquipmentTurntable with angle scale 02417.02 1Aperture plate for turntable 02417.05 1Air bearing 02417.01 1Inertia rod 02417.03 1Holding device w. cable release 02417.04 1Precision pulley 11201.02 1Blower 13770.97 1Pressure tube, l = 1.5 m 11205.01 1Light barrier with Counter 11207.30 1Power supply 5 V DC/2.4 A 11076.99 1Supporting blocks, set of 4 02070.00 1Slotted weight, 1 g, natur. colour 03916.00 20Slotted weight, 10 g, black 02205.01 10Slotted weight, 50 g, black 02206.01 2

Weight holder 1 g 02407.00 1Silk thread, 200 m 02412.00 1Tripod base -PASS- 02002.55 2Support rod -PASS-, square, l = 1000 mm 02028.55 1Support rod -PASS-, square, l = 400 mm 02026.55 1Right angle clamp -PASS- 02040.55 3Bench clamp, -PASS- 02010.00 2

TasksFrom the angular acceleration, the moment of inertia aredetermined as a function of the mass and of the distance fromthe axis of rotation.1. of a disc,2. of a bar,3. of a mass point.

Set-up and procedureThe experimental set-up is arranged as shown in Fig. 1. Therotary bearing, with the blower switched on, is alignet hori-zontally with the adjusting feet on the tripod. The release tripmust be so adjusted that it is in contact with the inserted sec-tor mark in the set condition.

The precision pully is clamped so that the thread floats hori-zontally above the rotating plane and is aligned with the pul-ley.

Fig. 1: Experimental set-up for investigating the moment of inertia of bodies.

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P2131301 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

While using the bar, the fork type light barrier is positioned insuch a way that, the end of the bar, lying opposite to the angu-lar screen standing just in front of the light path, is held by theholder.While using the turntable, before the beginning of the experi-ment, the pin of the holder through the bore hole near theedge fixes the turntable. The fork type light barrier is position-ed such that, the screen connected to the turntable, is in frontof the light ray.Whenever the holder is released, the light barrier must beinterrupted at that moment.

The measurement is done in the following manner:

1. Measurement of the angular velocity v:

– Set the light barrier selection key at “ ” and press the“Reset” button

– Release the holder to start the movement flow

The light barrier measures at first, the initial darkening timewhich is of no great importance.

– During the flow movement, press the “Reset” button afterthe screen has attained end velocity but before the screenpasses the light barrier. The time measured now, ∆t is usedfor the measurement of angular velocity (∆w is the angle ofthe used rotary disc shutter)v = ∆w /∆t

2. Measurement of angular acceleration a:

– The experiment is repeated under the same conditions,required for the measurement of angular velocity. However,the light barrier key must be set at “ ” and the “Reset”button is pressed.

– The time ‘t’ indicated is the time for the acceleration.According to a = v/tthe acceleration is obtained.

NoteIt is to be noted, that the supporting block stops the weightholder used for acceleration at that moment, when the screenenters the path of light of the fork type light barrier. More ac-celeration should not be effected during the measurement of∆t.

Theory and evaluationThe relationship between the angular momentum T

�of a rigid

body in the stationary coordinate system with its origin at thecentre of gravity, and the moment T

�acting on it, is

(1)

The angular momentum is expressed by the angular velocityv� and the inertia tensor I from

L�

= I · v� ,

that is, the reduction of the tensor with the vector.

In the present case,v� has the direction of a principal inertiaaxis (Z-axis), so that L

�has only one component:

LZ = IZ · v

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

The moment of the force F�

(see Fig. 2)

T�

= r�

� F�

gives for :

TZ = r · m · g,

so that the equation of motion reads

mgr = IZ � IZ · a.

From this, one obtains

IZ = .

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

a) For a flat disc of radius r and mass m, one obtains

IZ = m r2.

From the data of the disc

2r = 0.350 mm = 0.829 kg

one obtains

IZ = 12.69 · 10–3 kgm2.

The mean value of the measured moment of inertia is

IZ = 12.71 · 10–3 kgm2.

12

mgr

a

dv

dt

rS � FS

TZ � IZ dv

dt .

TS

�d

dt LS

Fig. 2: Moment of a weight force on the rotary plate.

b) For a long rod of mass m and length l, one obtains

IZ = ml2.

From the data for the rod

m = 0.158 kgl = 0.730 m

one obtains

IZ = 7.017 · 10–3 kgm2.

The mean value of the measured moment of inertia is

IZ = 6.988 · 10–3 kgm2.

c) For a mass point of mass m at a distance r from the axis ofrotation, one obtains

IZ = m r2.

For the measurements, a distance

r = 0.15 m

was selected.

From the regression line to the measured values of Fig. 3, withthe exponential statement

Y = A · XB + I0

(I0 is the moment of inertia of the rod), the exponent

B = 1.00 ± 0.02 (see (2))

is obtained.

The measurement was carried out with m = 0.2 kg.

From the regression line to the measured values of Fig. 4,

B = 1.93 ± 0.03 (see (2))

is obtained.NoteThe pivot pin is not taken into account for the theoretical cal-culation of the moment of inertia, since with a mass of 48 g, ithas a moment of inertia of only 4.3 · 10–6 kgm2.

The “support face” and bar retaining ring are balanced by thesector mask and plug, so that a uniform mass distribution canbe assumed for the bar over its whole length.

112

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Fig. 3: Moment of inertia of a mass point as a function of themass.

Fig. 4: Moment of inertia of a mass point as a function of thesquare of its distance from the axis of rotation.

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Related TopicsRotation, angular velocity, torque, angular acceleration, angu-lar moment, 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 angular velocity and angle of rotation vs. time for

a disc with constant torque applied to it for different valuesof torque generated with various forces on three differentradii. Calculate the moment of inertia of the disc.

2. Measure angular velocity and angle of rotation vs. time andthus the moment of inertia for two discs and for a bar withmasses mounted to it at different distances from the axis ofrotation.

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 comparethese two energies.

EquipmentTripod -PASS- 02002.55 1Precision bearing 02419.00 1Inertia rod 02417.03 1Turntable with angle scale 02417.02 2Aperture plate for turntable 02417.05 1Connecting cord, l = 150 cm, red 07364.01 1Connecting cord, l = 150 cm, blue 07364.04 1Light barrier with Counter 11207.30 1Adapter, BNC plug/4 mm socket 07542.26 1Foil capacitor in casings G1, 0.1 µF 39105.18 1Power supply 5 V DC/2.4 A 11076.99 1Precision pulley 11201.02 1Right angle clamp –PASS- 02040.55 2Support rods –PASS- l = 400 mm 02026.55 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, 03916.00 20Slotted weight, 10 g, black 02205.01 10Slotted weight, 50 g, black 02206.01 2Holding device with cable release 02417.04 1Measuring tape, l = 2 m 09936.00 1

Fig.1: Experimental set up with turntable

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Set-up and procedure1. Set the experiment up as seen in Fig.1. Connect the lightbarrier with counter according to Fig.2 to the cable release.Adjust the turntable to be horizontal – it must not start to movewith an imbalance without other torque applied. Fix the silkthread (with the weight holder on one end) with the screw ofthe precision bearing or a piece of adhesive tape to the wheelswith the grooves on the axis of rotation.Wind it several timesaround one of the wheels – enough turns, that the weight mayreach the floor. Be sure the thread and the wheel of the preci-sion pulley and the groove of the selected wheel are wellaligned. Place the holding device with cable release in a waythat it just holds the turntable on the sector mask and doesnot disturb the movement after release.Note down the angle w from the point of release to the pointwhere the aperture plate enters the beam of the light barrierand the radius ra of the selected wheel with groove. Measure the time t1 necessary for the turntable to rotate aboutthe angle w by setting the mode selection switch of the lightbarrier to the symbol " ". Press the reset button on thelight barrier. Release the holder – the clock should start count-ing. To enable the counter to stop on the interruption of thelight beam, the cable release has to be pushed back into it'sholding position. After pushing the cable release back thecounter should stop, when the mask enters the light beam.Measure the time necessary for the angle w + 2p by pushingthe release back only after the mask has completed a wholeturn and has passed again and so on for two and more turns.(Catch the mask by hand before it hits the holder.)Measure the angular velocity v after a rotation about theangle w by measuring the time t2 that the mask needs to passthe light barrier: Set the mode selection switch to the symbol" " and after releasing the holder the counter starts count-ing. Press the reset button before the mask enters the lightbarrier. Then the counter starts when the mask enters the lightbeam and counts as long as the light beam is interrupted. Ifthe measured time was t2 and the sector mask covers theangle 15°, then the radian of this angle is w = 2 · p · 15/360and the frequency or angular velocity v = w/t2. You may pressthe reset button after one or more passes of the mask mea-suring the angular velocity after more than one turn of therotation.

1. Take measurements with different accelerating weights maup to 100 g. Note down measurement values with the thread running in dif-ferent grooves, i.e. different radii ra for the accelerating torque– adjust the position of the precision pulley to align thread andgroove and wheel of the pulley (thread has to be horizontal).Choose especially the weight on the end of the thread masuch that the torque 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 =magr = 8.83 mNm with earth's gravitational acceleration g=9.81 N/kg = 9.81 m/s2. Also choose a weight with which youtake a measurement for each groove radius.You may also take a measurement with no weight ma. Start theturntable by a shove with the hand. Nearly unacceleratedmovement should be observable. Also measure the angularvelocity after converting a defined amount of potential energyto rotational energy by allowing the weight ma to pass only agiven height before touching the ground.

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 various 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).

Theory and evaluation

The angular momentum of a single particle at place 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

is the equation of motion becomes

(1).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

TS

� rSa � FS

Fig. 2: Connection of the light barrier (Lb).

For a system of N particles with center of mass andtotal linear 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).If a weight mi is mounted to a rod in a distance ri to the fixedaxis z, which is the axis of rotation perpendicular to the rod,with the rod lying along the y-axis, then are the coordinates ofthe weight (0, ri,0). According to (3) the moment of inertiaabout the z-axis is , about the x-axis and about the y-axis it is Iy,y, = 0.For evaluation both the speed vs. angle (v calculated by t2)and the time vs. angle (using t1) values may be used to deter-mine Iz,z. The time t1 values are more precise in case themovement is still accelerated while the mask passes the lightbarrier.

Fig. 3 shows t1 in dependance on torque T for fixed Iz,z, i.e.the moment of inertia of the turntable, and fixed w(t1) =215° = 3.75 rad. The slope of the curve in the bilogarithmicplot is – 0.519 compared to theoretical – 0.5, since it followsfrom (8) that

.t1 � B2w1t1 2 Iz,z

T

Ix,x � mi ri2Iz,z � mi ri

2

�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 � w1t � 0 2 �

12

· T

Iz,z · t2

v1t 2 � v 1t � 0 2 � ma gra

Iz,z · t � v1t � 0 2 �

T

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.

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Fig. 4 shows a bilogarithmic plot of angle vs. time with con-stant Iz,z (of the turntable): With both values of torque theslope of the curve is nearly 2 as predicted by (8).

Fig. 5 shows a bilogarithmic plot of angular velocity vs. timewith constant Iz,z (of the turntable): With all three values oftorque the slope of the curve is nearly 1 as predicted by (7).

Fig. 6 shows the Iz,z values calculated from the measured t1values that the bar needed to reach 185 ° = 3.23 rad with atorque applied of 3.09 mNm (ra = 15 mm, ma = 21 g, g =9.81 m/s2), according to (8):

- compared to the theoretical values, according to (3): Iz,z =Irod + mi · ri

2, Irod = 72 kg cm2. Two weights of 100 g each weremounted at ri from the axis of rotation – the plot shows a goodlinear dependence on ri

2.

Fig. 7 shows measured Iz,z values in dependence on theweight mi mounted to the bar at ri = 20 cm in comparison to

theoretical Iz,z = Irod + mi · ri2. The linear dependence on mi

can be seen.

Iz,z �T · t1

2

2 w

Fig. 7: Measured and theoretical values of Iz,z in dependenceon mi

Fig. 3: Bilogarithmic plot of t1 vs. T torque

Fig. 4: Bilogarithmic plot of w vs. time t1 for different torques

Fig. 5: Bilogarithmic plot of v vs. time t1

Fig. 6: Measured and theoretical values of Iz,z in dependenceon ri

2

lab5 lep1313 01 angular momentum

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