Moment of Inertia and Angular Acceleration With Cobra3

Mechanics

Dynamics

Moment Moment of inertia inertia and and angular angular accelerati acceleration on with with Cobra3 1.3.13-1 1.3.13-11 1 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.

What you need: Cobra3 Basic Unit Power supply, 12 VRS232 cable Transl Translati ation/ on/Rot Rotati ation on Softwa Software re Light barrier, compact Blower Pressure tube, l = 1,5 m 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 PC, Windows® 95 or higher

12150.00 12151.99 14602.00 14512.6 14512.61 1 11207.20 13770.97 11205.01 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 1 1 1 1 1 1 1 2 1 1 9 3 2 1 1 2 1 1 1 1 1 1 1 1 1

Potential energy and additionally the rotational energy.

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

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

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

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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.

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:

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

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 (1) T  L . dt

The angular momentum is expressed by the angular velocity S v and the inertia tensor Iˆ from S

In the present case,

Iˆ ·

S

.

L



S

has the direction of a principal inertia

v

v

S

axis (Z-axis), so that 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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S

The moment of the force F (see Fig. 2) S

T

gives for

S

r



S

r





S

F

S

F : TZ = r · m · g ,

so that the equation of motion reads mgr  IZ

dv dt



IZ · a .

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

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

m g = 9.81 m/s2

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



M v

>t



r · m · g 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



M a



0.0075 kg m2> s2 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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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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21313-11

Moment of Inertia and Angular Acceleration With Cobra3

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