Download Moment of Inertia

Fall 2006
Moment of Inertia
Name
Section
Theory
In applying Newton’s Second Law of Motion to rotational motion, it is known that the relation between torque and angular
acceleration depends on both the mass and the distribution of that mass; this relationship is known as the moment of inertia. The
moment of inertia for discrete distributions of mass is defined as
I  mi ri2
and for continuous distributions of mass

I  r 2 dm
Fortunately, the moment of inertia has been calculated and expressed in simpler form for a number of regular bodies. These
values can be found in any physics textbook.
In this experiment, you will use an apparatus that will allow you to measure the moment of inertia of several different bodies
dynamically. These experimentally determined moments will be compared to the theoretical moments of inertia.
The apparatus, shown below, has a rotating cradle into which various regular objects can be placed. A string is wrapped around a
drum under the cradle and runs parallel to the table and over a pulley at the edge of the table. A mass m placed at the end of the
string causes the cradle to rotate as the mass descends to the floor. The acceleration of a of the mass can be determined from
y
1 2
at
2
(1)
where t is the time it takes the mass to fall a vertical distance y. The linear acceleration of the mass is related to the angular
acceleration  of the cradle by
a  r
(2)
where r is the radius of the drum (the string has the same linear acceleration). The rotational analog of Newton’s second law
applied to the apparatus gives
  I
(3)
where  is the torque applied to the drum and I the moment of inertia of the apparatus (cradle + any object in the cradle). But  is
just
  rT
(4)
since the applied force is just the tension T in the string wrapped around the drum. Applying the linear form Newton’s second law
to the hanging mass
mg  T  ma
(5)
gives the last relation needed to derive an expression for experimentally determining I. Starting with Equation 3, then substituting
Equations 4, 5, and 2,
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g 
I  mr 2   1
a 
(6)
Apparatus
Rotational dynamics apparatus, Table clamp, Vertical rod,
String, Pulleys, Stopwatch, Hooked masses, Meterstick, Vernier
caliper, Metal disk, Metal hoop.
Figure 1
Experimental setup.
Figure 2
Dynamical relationships, drawn for clarity; as shown in Figure
1, the axis of rotation of the system is actually parallel to the
plane of the page.
Procedure
Moment of Inertia of the Cradle
1. Make sure that the cradle apparatus is level – the feet are adjustable if needed. Rotate the cradle so that the string unwinds
from the drum. Measure the diameter of the drum with the vernier caliper and record this value in Table 1.
2. Wind the string back around the drum and place a 20g hooked mass at the other end of the string off the table. You will
need to hold the cradle in place once the mass is added or it will begin its descent to the floor.
3. Measure the vertical distance y that the mass m will descend to the floor and record these values.
4. When ready to make a run, let go of the mass and start the timer simultaneously. Stop timing when the mass hits the floor
and record the time.
5. Repeat Step 4 4 more times and determine the average time. Remember to keep the distance y constant!
6. Calculate and record the rate of acceleration a with the average time and Equation 1.
7. Calculate and record the moment of inertia of the cradle itself with Equation 6.
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Moment of Inertia of the Disk
1. Measure the diameter of the disk and record this value, along with the mass of the disk, in Table 2.
2. Using the dimensions you determined in Step 1, calculate and record the theoretical moment of inertia for the disk.
3. Switch to a 100g hooked mass. As in Procedure 1, time the descent of this mass over a distance y 5 times and record these
values, along with the average time.
4. Calculate and record the rate of acceleration a with the average time and Equation 1.
5. Calculate and record the moment of inertia of the disk and cradle combined with Equation 6. Since we are interested in the
moment of inertia of the disk alone, subtract the moment of inertia you found for the cradle itself in Procedure 1 and record
the moment of inertia of the disk.
6. Calculate and record the percent error in the moments of inertia of the disk.
Moment of Inertia of the Hoop
1. Repeat Procedure 2 with the hoop in place of the disk. Record all data in Table 3.
Table 1
Moment if Inertia of the Cradle
Diameter of the drum (m) ____________________
Descending mass (kg) ____________________
Vertical distance (m) ____________________
Trial
Rate of acceleration (m/s2)
Time (s)
1
____________________
2
3
Moment of inertia of the cradle (kgm2)
4
____________________
5
Average time (s) _______________
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Table 2
Moment if Inertia of the Disk
Diameter of the disk (m) ____________________
Mass of disk (kg) ____________________
Theoretical moment of inertia of disk (kgm2) ____________________
Descending mass (kg) ____________________
Vertical distance (m) ____________________
Trial
Rate of acceleration (m/s2)
Time (s)
1
____________________
2
3
Moment of inertia of the disk and cradle (kgm2)
4
____________________
5
Average time (s) _______________
Moment of inertia of the disk alone (kgm2)
____________________
Percent error
____________________
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Table 3
Moment if Inertia of the Hoop
Diameter of the hoop (m) ____________________
Mass of hoop (kg) ____________________
Theoretical moment of inertia of hoop (kgm2) ____________________
Descending mass (kg) ____________________
Vertical distance (m) ____________________
Trial
Rate of acceleration (m/s2)
Time (s)
1
____________________
2
3
Moment of inertia of the hoop and cradle (kgm2)
4
____________________
5
Average time (s) _______________
Moment of inertia of the hoop alone (kgm2)
____________________
Percent error
____________________
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Pre-Lab: Moment of Inertia
Name
Section
1. What is the formula for calculating the moment of inertia (about the central axis) of a solid disk? This will be in terms of the
dimensions of the disk.
2. What is the formula for calculating the moment of inertia (about the central axis) of a hoop? This will be in terms of the
dimensions of the hoop.
3. What are the SI (mks) units of the moment of inertia?
4. Explain what each of the variables (I, m, r, g, and a) in Equation 6 represents.
5. A uniformly accelerated object descends (from rest) a vertical distance of 132.7cm to the floor in a time of 7.13s. At what
rate was it being accelerated?
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