Download Lab Writeup Moment of Inertia

Moment of Inertia
LBS 164L
Purpose
In this experiment, you will compute the moment of inertia of a simple rigid body from
its mass distribution and compare that calculation with a measurement derived through an
angular acceleration due to an applied torque.
Theory
If we apply a single, unbalanced force, F, to an object, the object will undergo linear
acceleration, a, which is determined by the force and the mass, m, of the object. The
mass is a measure of the object’s resistance to changing velocity, its inertia. This
relationship is written
 F  ma .
If we apply a single, unbalanced torque,, an object will undergo angular acceleration,,
which is determined by the torque and the moment of inertia, I, of the object. The
moment of inertia is a measure of the object's resistance to changing its angular velocity,
its moment of inertia. This relationship is written
   I .
The torque can be written in terms of a force, F, and the perpendicular distance, r , which
is also called the moment arm. This relation is given by
   Fr
Every rigid object has a definite moment of inertia about any given axis of rotation.
Some examples are
 Point mass of mass m on a weightless rod of length r
2
 I  mr
 Two point masses, m1 and m2, each on a weightless rod of length r1 and r2
2
2
 I  m1r1  m2 r2
r2

Thin rod of mass m
and length L rotated
about its center
1
2
 I
mL
12
In this experiment, the
equipment will be set up as
follows:
r2
M2
For the hanging mass we have
the equation
  Fy mhg  T  m ha
M1
T
d
Smart
Pulley
T
For the rotating system we have
m hg
Moment of Inertia Lab Write-up
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
   Fr

T
d 
 I
2 
We also remember that
a 2a
  
r d
so we get
Td
2a

I
2
d
Solving both equations containing T and equating we get
4aI
 m hg  mh a  2
d
So we obtain the moment of inertia in terms of the linear acceleration of the hanging mass
m (g  a)d 2
 I h
4a
The moment of inertia of the rotating system is
 I  I support  I masses
or
 I masses  M1r12  M 2r22
In this experiment you will measure the moment of inertia of the support first. Then you
will add the two masses at various radii and measure the moment of inertia of the
resulting rigid body. You will then compare your measurement with the theoretical
expectation.
Equipment Needed
Macintosh computer
PASCO CI-6560 Signal Interface II
PASCO ME-9387 Smart Pulley
LBS Moment of Inertia Device
String. Weights, clamps
Science Workshop, KaleidaGraph, Microsoft Word, Microsoft Excel
The equipment you will need has been assembled and tested. To make sure that
everything is in order, carry out the following checks:
 Make sure the Signal Interface is turned on,
 Make sure that the Smart Pulley is connected correctly to the Signal Interface (cable
into Digital Channel 1.),
 Open theScience Workshop document “Moment of Inertia”.
Measurements
Before beginning, make the following measurements:
 Measure d, the diameter of the drum. You can do this by measuring the
circumference via the length of string which goes around the drum exactly
once,
 Determine the mass of the hanging mass and additional masses as needed,
 Determine the mass of the movable weights, M1, and M2.
Moment of Inertia Lab Write-up
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Record these values in a copy of the Excel spreadsheet file “Moment of Inertia
Spreadsheet”.
The linear acceleration of the hanging mass is measured using the Smart Pulley. One
measures the acceleration as in previous experiments by taking the slope of v versus t,
selecting only the data where the mass is accelerating freely. The moment of inertia is
then determined using the formula given above.
Procedure
1. Remove the masses M1, and M2 and measure the moment of inertia of the support
structure alone.
2. Make a series of three measurements of the moment of inertia of the system with the
masses installed at different distances from the center of rotation.
3. Compare your measured results for Imasses with the calculated values using the
spreadsheet calculations.
Questions
1. Did your measured and calculated results for Imasses agree? If not, can you think of
reasons why they might not agree?
2. Make an estimate of the moment of inertia of the support structure and compare it
with your measured value.
Moment of Inertia Lab Write-up
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Constants
M 1 (kg)
M 2 (kg)
g (m/s^2)
d (m)
S upport measurement
mh
a (support)
I (support)
(kg)
(m/s^2)
kg*m^2
#DIV/0!
mh
(kg)
r1
(m)
Total system measurement
a (system)
I (system)
(m/s^2)
kg*m^2
#DIV/0!
#DIV/0!
#DIV/0!
I (masses)
kg*m^2
#DIV/0!
#DIV/0!
#DIV/0!
Predictions
r2
I (predicted) I (difference)
(m)
kg*m^2
kg*m^2
0.00E+00
#DIV/0!
0.00E+00
#DIV/0!
0.00E+00
#DIV/0!
Moment of Inertia Lab Write-up
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