L-14
T O R Q U E
(L-14)
There is a close parallel between the theoretical dynamics of linearly
translating objects and the dynamics of rotating objects. For instance, just as
Newton's Second Law predicts that a translational acceleration must come as a
consequence of a net applied force, the rotational analog maintains that a rotational acceleration must come as a consequence of a net applied torque. This lab is
designed to see if torques and angular acceleration are really related as the theory
predicts. It has the added delight of requiring you to use conservation of energy to
determine the moment of inertia of the system.
PROCEDURE--DATA
Part A: (general setup and procedure)
a.) Plug your Smart Pulley into the Lab Pro and open the Logger Pro
program. Find the velocity vs time graph and open it.
Note: In setting up the apparatus, be careful NOT to drag the device across
the table. In most cases, metal screws act as feet for the device--dragging will
damage the table.
b.) The apparatus is shown below. Use a 100 gram hanging mass. The
a r t pu lley
smart sm
pulley
st r in g
hub
r ot a t in g st a n d
(pa r a llel t o t a ble)
axle
ba se
su ppor t t a ble
pu lley cla m p
weigh t
WARNING: Do n ot dr a g a ppa r a t u s over t a ble!
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pulley should be positioned so that the string is parallel to the table (in that way,
the string will not wind off the hub while being rewound). The string should be
wound around the SECOND HUB of the apparatus (not the top hub).
c.) Measure the diameter of the hub around which the string is wound.
Record your measurement in MKS.
Part B: (data to determine rxF)
d.) In the first part of the Calculations, you will be asked to use rxFnet to
determine the net torque acting on the disk. To do so, you will need the radius
vector r (that shouldn't be too hard considering that you have already measured
the diameter of the hub in Procedure c) and the net force motivating the disk to
rotate. Assuming we neglect friction, that force will be the tension in the string.
To determine an expression for the tension T in the string, you will be asked to
use Newton's Second Law which, in turn, means you will need the linear acceleration a of the hanging mass.
With all this in mind, hit COLLECT, wait until that icon changes to STOP,
then allow the hanging mass to freefall. Hit STOP before the hanging mass hits
the floor (in fact, stop the hanging mass before it hits the floor, also). You should
end up with a graph of the velocity versus time for the free fall. Once done,
determine the acceleration for the run (it will be the slope of the graph).
e.) Do a second run repeating Procedure d to be sure your data is good. That
is, the acceleration for your second run should be approximately the same as for
your first run. If it is nowhere close, repeat Procedure d until you get consistency.
Once you get consistency, leave this data in your computer and go on to Part C.
Part C: (data to determine Iα)
f.) To determine the net torque using Iα, we need both the disk-system's moment of inertia and its angular acceleration. The angular acceleration is easy the
computer has given us the linear acceleration of a point on the hub's perimeter
(this is the same as the acceleration of the string and, hence, the hanging mass-that information was determined in Part e above). Determining the moment of
inertia is a bit trickier as we will have to use CONSERVATION OF ENERGY to
do so (that means we are going to need velocities and a distance fallen). The
procedure, along with an explanation of the trickiness, follows.
g.) The conservation of energy equation requires knowledge of the hanging
mass's velocity at TWO points in time (call these velocities v1 and v2) along with
the distance fallen (that is, d = d2 - d1) between those two points in time (you will
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L-14
also need to know the angular velocity of the disk at both points in time, but if you
know v1 and v2, you can use v = rω to determine those values).
h.) To take the data needed to determine the moment of inertia of the disk:
--With a Velocity versus Time graph on the screen, use the slope tool to
determine the hanging mass's acceleration (i.e., determine the slope of the
Velocity versus time graph along a linear section of the graph). Leaving this
information on the graph,
--PRINT the Velocity vs. Time graph for your run.
--Change the graph to a Distance versus Time graph. Remove the
regression line you used on the Velocity versus Time graph, the print the
graph.
--You now have all the information needed to determine the position
change, the velocity, and the velocity change between any two time points you
should choose.
CALCULATIONS
Part A: (torque from rxF)
1.) Preface: In this section, we want to determine the torque being provided to the disk system by using the vector cross product rxF, where F is the net
force acting on the disk (in this case, the tension T in the string--we have assumed
no friction) and r is the distance between the axis of rotation and the place where
T acts on the disk (this will be the radius of the hub around which the string was
wound). To carry out this cross product, we need T.
To get T:
a.) We know the acceleration a of the hanging mass--the computer
provided us with that number during lab. Use N.S.L. (f.b.d. and all) on the
hanging mass to determine a general algebraic expression for the tension T
in the string during freefall. Box your general equation.
b.) Put numbers into the equation derived above to determine a numerical value for the tension T in the line during the freefall.
2.) Use rxF to determine the net torque acting on the rotating system (I'd
suggest you draw an f.b.d. on the disk, just to be sure you know what you are doing). Ignore friction and call this calculated value ΓrxF.
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Part B: (torque from Iα)
3.) The rotational counterpart to Newton's Second Law states that the net
torque acting on a system must equal the moment of inertia times the angular
acceleration α, or Iα . We would like to check this out. To begin:
a.) Use a = Rα to determine the angular acceleration of the rotating
disk as the hanging mass falls.
b.) Use conservation of energy and the assumption that the system was
essentially frictionless to derive a general algebraic expression for the
moment of inertia I of the disk. This should be in terms of the hanging
mass mh, a distance dropped d between times t1 and t2 (it is your choice as
to what these times will be--you will take all the data necessary off your
graphs when it comes time to put numbers into this expression), the hanging mass's velocity v1 and v2 respectively, and the hub's radius r (note: you
should be able to eliminate ω terms using the relationship you've all come
to know and love that relates v and ω). Box your general equation.
c.) Use the relationship derived in Part 3c, determine a numerical
value for your moment of inertia.
4.) Determine the net torque acting on your system using the vector
product Iα. Call this ΓIα.
5.) We are now in a position to decide whether the rotational analog to
Newton's Second Law really works (i.e., if (rxF)net = Iα):
a.) Do a % comparison between ΓrxF and ΓIα.
b.) Did the deviation seem big? If so, explain from whence the
discrepancy is most likely to have come.
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