Download Circular Motion - Manchester HEP

Physics 1 - Circular Motion
Circular Motion
Aims
 To understand angular velocity and angular acceleration.
 To become familiar with Torque and Moment of Inertia.
 To demonstrate conservation of angular momentum
In this experimental tutorial you will first undertake a tutorial question to analyse the
rotation of a disk when constant torque is applied and then perform an experiment to
measure angular acceleration. The final part of the experiment demonstrates
conservation of angular momentum.
Part 1: Tutorial Question (15 mins)
A freely rotating disk has a thread coiled around it at a distance r from the centre of
the disk. The free end of the thread is passed over a pulley and connected to a hanging
weight of mass m. This setup is illustrated in Figure 1.
r
Rotating disk
disk
pulley
T
T
Table
mg
thread
Plane view
Side view
Figure 1
1.1
Show that the moment of inertia is;
I  mr (r 
g

),
where m is the hanging mass, g is gravitational acceleration,  is the angular
acceleration of the disk, and r is the point at which the torque is applied to the
disk.
Hints:
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Physics 1 - Circular Motion
(a) Consider Newton’s 2nd law for the hanging mass
(b) Consider the rotational analogue of Newton’s 2nd law for the rotating disk
dL

where L  I .  is the applied Torque, L is the angular momentum
dt
and  the angular velocity.
1.2
Recall from the lecture notes, what the Moment of Inertia, I, is of an annulus
(i.e. a disk with the centre removed) of inner radius R1, outer radius R2 and
Mass M?
Part 2: Angular Velocity & Acceleration (20 mins)
The experiment analysed in the tutorial question is now setup and performed. A light
gate is used to determine the angular velocity of the disk. The light gate is triggered
on and off by a plastic wedge of 10o size that is attached to the disk.
Please be careful with this experiment as the plastic pieces attached to the disk
are fragile.
 The thread should be wound around the central cog by first wrapping it over the
metal hook and gradually spinning the turn-table.
 A weight of mass 50g should be suspended over the pulley on the other end of the
thread.
 Open the EasySense software package on the computer and ensure that the light
gate is connected via the data hub.
 Click on the Timing option from the Experiments menu.
 Select Raw Times then click finish. If the computer has not recognised the light
sensor then seek help from a demonstrator.
 Hold the wheel still and press Start from the toolbar. At the same time release the
wheel to lower the suspended mass.
 Let the wheel spin freely and once the 50g mass has hit the floor, allow the wheel
to spin for a few seconds, then click Stop.
 From File select Transfer data to Excel to open up a spreadsheet.
 Delete column A as this is unwanted. You will observe the light gate readings
record 1 when covered and 0 for open.
2.1
Create a column to calculate the time interval for which the 10o flag cuts the
light at the light gate each time the disk rotates. From this information you
should be able to create another column to calculate the angular velocity of the
wheel.
2.2
Plot the angular velocity of the disk against time and explain the main features
of the graph. Using the rotational analogue of the equation v=u+at, find from
your graph the angular acceleration of the disk during the period that the
torque is applied.
Hints:
360 o = 2 rad,
 = 2 f,
2
f= 1/T
Physics 1 - Circular Motion
When fitting a trend line you may need to change the range of data considered
in your spreadsheet.
2.3
Using your calculation for angular acceleration and the equation from 1.1 find
the moment of inertia of the setup.
Part 3: Adding another annulus to verify theoretical prediction (15 mins)
 Now select one of the other annulus’s and measure the radius’ R1, R2 and its mass
M.
3.1
Determine the moment of inertia of the annulus using your result from 1.2 and
work out the combined moment of inertia once added to the original setup
 Remove the cog annulus from the turntable and slot the other on top, then replace
the cog annulus.
 Wind the thread around the central cog as before, then click on the EasySense
window and select New. It will ask you to save the previous data but choose No
then check Raw Times is selected and click Finish.
3.2
Repeat the experiment as before with the new combined system to find the
new angular acceleration and hence find the new moment of inertia.
3.3
Compare your prediction with your experimental results and discuss any
deviation.
Part 4: Conservation of Angular Momentum (Optional)
 Remove the metal disks from the perspex wheel.
 Add an additional perspex disk, with the same moment of inertia as the original
disk.
 Both disks have magnets attached to them to allow elastic collisions.
 Leave disk 1 stationary
 Rotate disk 2 with a very low angular velocity.
4.1
Describe and explain the resulting motion. What is the linear motion analogue
to this experiment?
4.2
Describe and explain what happens if you add an additional metal disk to just
one of the Perspex disks?
Further work
The following questions are related to the topic covered by this experimental tutorial.
 Exercise book questions D23-25.
 Mastering Physics Dynamics 6: Circular Motion and Angular Momentum
assignment. Particularly try, ‘The End of the Song’, ‘Ladybugs on a rotating disk’
and ‘Record and Turntable’.
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Physics 1 - Circular Motion
Demonstrators' Answers, Hints, Marking Scheme and Equipment List.
Marking Scheme
Section
1.1
1.2
2.1
2.2
2.3
3.1
3.2
3.3
Discretionary mark
TOTAL
Answers
1.1
Newton 2 for hanging mass
T  mg  ma (1)
Newton 2 for rotating disk
  I
Torque from tension applied at distance r
  Tr
Hence
T  I / r (2)
Angular and linear acceleration
v  r hence differentiating
a  r (3)
Sub (2),(3) into (1):
I
 mg  mr
r
I  mr 2  mgr
g

I  mr r  


2
2
1
M (R  R )
2
1
2
1.2
I
2.1
see pages following
2.2
4
Mark
1
1
1
1
1
1
1
1
2
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Physics 1 - Circular Motion
Angular Velocity (rad/s)
Angular Velocity v Time
20
18
16
14
12
10
8
6
4
2
0
Series1
0
2
4
6
8
10
12
Time (s)
Graph shows uniform angular acceleration while torque applied. The graph
then plateaus at constant angular velocity, in practice friction means that it
uniformly decelerates slowly.
  0  t
Typical value of  can be seen in appendix.
2.3
Typical moment of inertia of original setup can bee seen in appendix –
original setup.
3.1
Typical value of combined I for one annulus can be found in the appendixadded annulus, theoretical.
3.2
Typical value for the angular acceleration and inertia of the combined system
can be found in appendix – added annulus, experimental results.
3.3
Typical differences can be found in appendix – combined.
4.1
Conservation of angular momentum
I11  I 22
The two disks have the same moment of inertia. After the collision the 2nd disk
has the same angular velocity as the 1st disk had before the collision and the 1st
disk is stationary. On the following collision the angular momentum is again
exchanged between the two disks.
The motion is analogous to that of collisions between ball bearings of equal
mass, as arranged in the Newton’s cradle toy.
4.2
This is now analogous to collisions between balls of different masses. Angular
momentum is still conserved, but the moment of inertia of the two masses is
no longer equal. When the lighter (lower I) object hits the heavier (higher I)
one it bounces off reversing the direction of its angular velocity and imparting
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Physics 1 - Circular Motion
angular momentum to the heavier one. When the heavier (higher I) object hits
the lighter (lower I) one it continues in the same direction but at lower angular
velocity and imparts its additional angular momentum to the lighter one which
moves off faster.
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Physics 1 - Circular Motion
Results
Original Setup:
Event Time
s
0
0.088160001
1.57731998
1.603510022
2.425179958
2.444210052
3.086950064
3.102679968
3.65019989
3.663939953
4.149909973
4.162270069
4.604320049
4.615660191
5.024260044
5.034800053
5.416769981
5.426670074
5.786819935
5.796179771
6.140999794
6.15019989
6.494800091
6.504020214
6.849979877
6.859250069
7.206540108
7.215849876
7.564509869
7.573860168
7.9238801
7.933259964
8.284660339
8.294059753
8.646849632
8.656299591
9.0104599
9.019940376
9.375519753
9.38504982
9.74203968
State at
A
State
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Time to pass
s
w
rad/s
0.088160001
1.979729
0.026190042
6.664095
0.019030094
9.171417
0.015729904
11.09561
0.013740063
12.70248
0.012360096
14.12068
0.011340141
15.39072
0.010540009
16.55909
0.009900093
17.62942
0.009359837
18.64701
0.009200096
18.97077
0.009220123
18.92957
0.009270191
18.82733
0.009309769
18.74729
0.0093503
18.66602
0.009379864
18.60719
0.009399414
18.56849
0.009449959
18.46917
0.009480476
18.40972
0.009530067
18.31392
a
rad/s/s
2.9075
7
I
kgm2
0.002201571
Physics 1 - Circular Motion
Added annulus:
Theoretical
Mass of
additional
annulus'
kg
Experimental
Results
0
0.12306
2.03106
2.0637
3.08568
3.10926
3.90501
3.92443
4.60064
4.61758
5.21674
5.23195
5.77608
5.79002
6.2918
6.30471
6.77226
6.78438
7.22479
7.23625
7.65398
7.66487
8.06461
8.07522
8.4728
8.48345
8.88238
8.89306
9.2933
9.30402
9.70557
9.71632
10.11918
10.12997
10.53416
10.54498
10.95052
10.96138
11.36827
11.37917
Radius
R2
m
0.56839
Event
Time
s
Radius
R1
kgm2
m
0.03
State at
A
State
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Moment
of inertia
0.082
0.002167
Time to pass
s
w
rad/s
0.123060003
1.418275
0.03263998
5.347213
0.023580074
7.401712
0.019419909
8.98732
0.016940117
10.30294
0.015209675
11.47513
0.013939857
12.52042
0.012909889
13.51932
0.01211977
14.40068
0.011459827
15.22998
0.010890007
16.02689
0.01061058
16.44895
0.010649681
16.38856
0.010680199
16.34173
0.010720253
16.28067
0.010749817
16.2359
0.010789871
16.17563
0.010820389
16.13
0.010860443
16.07052
0.010900497
16.01146
a
rad/s/s
1.9297
8
I
kgm2
0.003293
Physics 1 - Circular Motion
Combined:
Theoretical
prediction
Experimental
results
Momenty of
Inertia 1
Momenty
of Inertia 2
Combined
Moment
of Inertia
kgm2
kgm2
kgm2
0.00220157
0.0010824
Angular
Acceleration
Moment of
Inertia
Rad/s2
kgm2
1.9297
Experimental
Value for
Moment of
Inertia
Theoretical
Value for
moment of
Inertia
0.001091401
0.0010824
0.003293
Deviation
0.00829986
9
0.003284
Physics 1 - Circular Motion
Equipment List
Dynamics Kit
Circular motion kit
50g mass hanger
Computer with Easysense software installed
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