Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Rotational Motion and Moment of Inertia
Purpose:
To determine the rotational inertia of a disc and of a ring and to compare these with the
theoretical values.
Equipment:
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Rotating Table, Disc, Ring
Hooked Mass Set
Long Rod
Right Angle Clamp
Cylindrical Rod Clamp
Table Clamp
Smart Pulley/Photogate
Stop Watch
2-meter Stick
Vernier Calipers
Carpenter’s Level
String
Theory:
By now you’ve probably figured out that every variable that was defined when talking
about linear motion has an analogue in rotational motion. Instead of distance traveled, d,
we have angle turned, θ. Rather than speaking simply of velocity, v, we talk of angular
velocity, ω. Every aspect of linear motion has its partner in rotational motion. And not
only are they analogous, but they’re related, too! …usually by some power of the radius
of the circle.
In fact, here’s how it goes…
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Linear
Variable
Symbol
Angular
Variable
Symbol
Relationship
distance
d
angle
θ
d=θr
velocity
v
ω
v=ωr
acceleration
a
α
a=αr
force
F
τ
τ = r×F
mass
m
I
I = ∫ r 2 dm
momentum
p
L
L = r×p
angular
velocity
angular
acceleration
torque
rotational
inertia
angular
momentum
Figure 1 Relationships between Linear and Angular (Rotational) Variables
The same basic equations of linear motion can be used for rotational motion with a
simple change of variables:
x f = xi + vo t + 12 at 2
v f = vo + at
→
θ f = θ i + ω o t + 12 αt 2
ω f = ω i + αt
Eq. 1
The difference between angular and linear motion, and the relation between them, is
something that most students are able to understand fairly easily. Almost everyone has
played on a playground merry-go-round, and has noticed the difference in velocity
between standing at the center and standing at the edge. The confusion begins to set in
with discussions of torque and rotational inertia, the questions being specifically why do
we need to differentiate between the simple Force and Mass?
When discussing linear motion, it is assumed that all objects are “point objects”
whose mass is centered symmetrically about a single point, and they have no size to
speak of – that is, all objects are shaped essentially like very small balls that don’t spin.
Most objects, however, don’t actually fit this category. Most objects (even most balls)
are extended objects – they have a measurable size and they may not be totally
symmetric. When a force is exerted on an extended object, it matters not only how big
the force is, but also where on the object it is applied. Imagine a bar on a table:
v
F
F
F
Figure 2 The application of a force, and the resulting motion of a bar
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
If the force is applied directly to the center of the object, it will translate linearly across
the table. However, a push on either side of the center will cause a rotation of the object.
When a force is applied away from the axis of rotation (in this case, the center of
mass) it causes the object to rotate. This is what we call a torque. The torque is defined
as
τ = r×F
Eq. 2
where F is the force applied and r is the distance from the axis of rotation to where the
force is applied. Remember that the definition of a cross product is:
A × B =| A | | B | sin θ
where θ is the angle between vectors A and B. Thus,
τ = r F sin θ
Eq. 3
This means that the largest torque occurs when r and F are perpendicular to each other
(sinθ = 1) and the smallest torque is when r and F are parallel (sinθ = 0). As an example,
think of using a wrench to turn a bolt:
r
r
F
r
F
Smaller torque
τ = |r| |F| sinθ
where sinθ <1
so τ < |r| |F|
Largest torque
τ = |r| |F| sinθ
where sinθ = 1
so τ = |r| |F|
Smallest torque
τ = |r| |F| sinθ
where sinθ = 0
so τ = 0
Figure 3 Torque: Changing the Direction of the Applied Force
Similarly, we can change the radius of rotation:
r
F
r
r
F
largest r → largest τ
F
smallest r → smallest τ
Figure 4 Torque: Changing the Radius of Rotation
If you’ve ever tried to turn a tightened bolt, you probably already know that the best
results come when the force is applied to the end of the handle, in a perpendicular
direction.
What then, about rotational inertia?
Remember Newton’s First Law of Motion, sometimes called “The Law of Inertia”:
Any object will continue in its state of rest or straightline motion unless acted upon by an outside force.
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F
Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
“Any object” really means “any object with mass”, which is usually a safe assumption.
Practically, this means that any object that has mass resists changes in its motion. This
resistance is what is referred to as inertia. Since inertia is a property of all objects having
mass, some go so far as to say that inertia is mass (or, mass is inertia).
Now think back to the wrench again. Imagine holding one end in your hand and rotating
your wrist to swing the other end in a small arc. Now switch ends. Is it easier one way
than the other? It should be! You should find that it is easier to swing the wrench when
you are holding the rounded (clamping) end. It is harder when you allow that same
(rounded) end to swing.
NOTE: If you have never actually done this, or have forgotten what it’s like, ask your
lab instructor or lab tech for a wrench to try it out.
Why would this be? The wrench has the same mass no matter how you hold it,
and if you exert the same force, it seems that it should accelerate in the same manner.
You have just discovered the reason for defining an object’s Rotational Mass!!!
Rotational inertia takes into account not only the total mass of an object, but how the
mass is distributed around an axis of rotation, and just as mass tells us how much
resistance a force will meet, rotational mass (or moment of inertia) tells us how much an
object will resist rotating (or changing its rotation). A single object’s rotational inertia
will change depending on where the pivot point (or axis) is placed. It is easier to turn an
object when most of the mass is located near the pivot point, thus this configuration has a
lower rotational inertia. If most of an object’s mass is located far away from the pivot
point, the rotational inertia is larger, and it will be more difficult to rotate.
Rotational mass can be defined in two (equivalent) ways:
I = ∫ mr 2 dV
a
or
I = ∫ r 2 dm
Eq. 4a and b
b
The expression you use depends mostly on what proves easiest for the particular
situation. Most often, you will not need either, for the geometry of the system will be
simple enough to use some combination of the pre-defined moments of inertia.
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Hoop or cylindrical shell
Hollow cylinder
I = MR
I = 12 M R12 − R22
(
2
)
R1
R
R2
Solid cylinder or disk
Rectangular plate
I = MR
I = 121 M a 2 + b 2
1
2
(
2
)
R
b
a
Long thin rod
Long thin rod
I = 121 ML2
I = 13 ML2
L
L
Solid sphere
Thin spherical shell
I = MR
I = 32 MR 2
2
5
2
R
R
Figure 5 Various Solid Figures and their Moments of Inertia
When we want to quantify the relation between the force exerted on an object and its
mass (or inertia) we use Newton’s Second Law of Motion:
F = ma .
Eq. 5
To quantify the relation between the torque on an object and its rotational mass (or
inertia), we simply replace each part of Newton’s Second Law with its rotational
analogue:
τ = Iα
.
Eq. 6
Thus, we can perform an experimental test. The equations in the chart above allow
calculation of the theoretical rotational mass of a particular disk or ring. By applying a
known torque to the object and measuring the angular acceleration, we can determine an
experimental rotational mass, and compare the two.
1
1
The theory for this lab was written by Jennifer LK Whalen
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Experiment:
1. Set up equipment as shown in the sketch below. Let m = 200 g.
smart pulley-photogate
m
to LabPro…
h
Rotating
Table and
Disc
smart pulley
Figure 6 Basic Experimental Set-Up
2. The experiment will be performed under three configurations: Configuration I will
include an empty rotating table (no disc nor hoop), Configuration II will include the
disc on the table (no hoop), and Configuration II will include the hoop on the table
(no disc).
3. For the first run, (m = 200 g, empty table) measure the distance, h the weight falls and
the time, t, (using a stopwatch) it takes the weight to fall that distance. Determine the
acceleration from the relation: h = ½ at2.
4. Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt
power supply into the side of the interface. Shortly after plugging the power supply
into the outlet, the interface will run through a self-test. You will hear a series of
beeps and blinking lights (red, yellow, then green) indicating a successful startup.
5. Attach the LabPro to the computer using the USB cable. The LabPro computer
connection is located on the right side of the interface. Slide the door on the
computer connection to the right and plug the square end of the USB cable into the
LabPro USB connection.
6. Connect the Smart Pulley/Photogate to the DIG/SONIC1 port of the LabPro. Remove
the cable with the Phono plug, and connect the Photogate Cable with a BritishTelecom plug on one end.
7. Start the program Logger Pro software. If you have the suggested sensor attached to
the LabPro in the suggested port, then you should see the appropriate graphs and
tables.
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
8. Once Logger Pro 3 is open, click on Experiment > Set Up Sensors > LabPro 1.
Click on the photogate icons, and verify that “Motion Timing” is selected under
“Current Calibrations”. In the same menu, choose “Set Distance or Length…”, make
sure that “Smart Pulley (10 Spoke) in Groove” is selected. The program calculates
the acceleration and velocity of the falling mass by treating the pulley as a picket
fence with the “proper” spacing.
9. Now press Collect and release the mass. Check the results of the computer
measurements against your own measurements of accelerations (from height and
time). The "by hand" values can differ by as much as 20% from the Logger Pro
values.
10. When you are satisfied that you are measuring acceleration correctly, use Logger Pro
to measure the acceleration of the mass as it falls and turns the empty table
(Configuration I). Repeat this measurement two times and record the three different
acceleration values (three different slopes of v vs. t). You can later use the range in
these values to determine your uncertainty range for Iexp.
11. It will later be necessary to determine the frictional torque acting on the system. To
do this, replace the 200 gram mass with a tiny mass. Start with 5 or 10 grams and
increase the mass until the table starts to turn. The minimum mass which will just
start the table rotating is mo. Record this value.
12. Add the disk to the table (this is configuration II.) and repeat Steps 5 and 6. To
overcome the greater inertia of the table, use m = 500 grams. Don't forget to measure
the minimum mass mo that just starts the table + disk rotating.
13. Remove the disk and add the ring to the table (This is configuration III). Repeat Steps
5 and 6 (500 grams may work well for this configuration). Be sure to record the new
value of m and mo.
14. Measure r = (diameter of the shaft of the rotating table / 2).
Analysis:
1. For each configuration, calculate the average acceleration, a, of the mass, m, for that
configuration. Also calculate uncertainty (deviation) in the experimental
acceleration.
2. Using Newton’s 2nd law: ΣF = ma (applied to the hanging mass) and it’s rotational
analog
∑ τ = Iα = rT
Eq. 7
(where T is the tension in the string), solve for I by eliminating T (show your work) to
get
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
I exp
r 2 m( g − a )
=
a
Eq. 8
3. Calculate the rotational inertia I, using Eq. 8 for the three configurations:
I1 = Itable
I2 = Itable + Idisc
I3 = Itable + Iring
4. Calculate:
Idisc = I2 - I1
Iring = I3 - I1
5. Calculate the theoretical values of Idisc and Iring using the formulas in the text. Note
that the disk has a weighted "plug" at the center to replace the missing mass in the
hole (so it can be considered a solid disk). Also note that the "ring" is actually a
hollow cylinder with inner and outer radii (Do you think if makes any difference in
the answer in this case if you use the formula for a ring rather than that of a hollow
cylinder?).
6. Now, if Iexp does not equal Itheory then part of the problem might be frictional torques
acting on the system. Assuming that this frictional force prevents the shaft from
rotating, we can measure it directly by hanging a small weight mo. This weight will
be the maximum value that can be hung without the table rotating.
T = mo g
r
Shaft
Ff
Figure 7 Finding the Force of Rotating Friction
Since the system is now in equilibrium, we find that τ net = 0 and m0 gr − F f r = 0 .
Also, if we call F f r = τ f , then:
m0 gr = τ f .
Eq. 9
7. Determine the value τf for each configuration and record. Now go back to Step 2 of
the analysis and include τf in your determination for Iexp. Show that the Corrected
value is:
r 2 m( g − a ) τ f r
Iexp =
a
a
Where τf = mogr.
8. Calculate the corrected values for Iexp for each configuration.
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
9. Using the corrected values repeat Step 4.
10. Determine the total uncertainty in the experiment: i.e., theory + exp.
11. Compare the theoretical values to the experimental values of Idisc and Iring (are the %
differences within the uncertainty?) If not, why not?
Results:
Write at least one paragraph describing the following:
• what you expected to learn about the lab (i.e. what was the reason for conducting the
experiment?)
• your results, and what you learned from them
• Think of at least one other experiment might you perform to verify these results
• Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in this
laboratory.
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. If you do not turn in this page with your instructor’s signature with
your lab report, you will receive a 5% point reduction on your lab grade. How you divide
clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Data Tables
hand check:
distance, h: _________
time, t: _________
determination of acceleration:
computer measurement of acceleration: _________
percent difference between hand and computer measurements: _________
Derivation of Eq. 8:
Derivation of corrected form of Eq. 8:
Configuration I: Empty Table
hanging mass, m: _________
acceleration, a1: _________
average acceleration, aavg: _________
acceleration, a2: _________
uncertainty in acceleration, δa: _________
acceleration, a3: _________
minimum mass which will turn table, mo: _________
rotational inertia from Eq. 8, I1:
torque due to friction, τf: _________
Corrected value of I1:
total uncertainty:
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Configuration II: Table + Disk
hanging mass, m: _________
mass of disk: _________
acceleration, a1: _________
average acceleration, aavg: _________
acceleration, a2: _________
uncertainty in acceleration, δa: _________
acceleration, a3: _________
minimum mass which will turn table, mo: _________
rotational inertia from Eq. 8, I2:
experimental rotational inertia of disk, Idisk, exp:
theoretical rotational inertia of disk, Idisk, theory:
torque due to friction, τf: _________
corrected value of I2:
corrected value of Idisk, exp:
total uncertainty:
percent difference, theory and experiment:
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Physics 2A
Rotational Motion and Moment of Inertia
©
2003 Las Positas College, Physics Department Staff
Configuration III: Table + Ring
hanging mass, m: _________
mass of ring: _________
acceleration, a1: _________
average acceleration, aavg: _________
acceleration, a2: _________
uncertainty in acceleration, δa: _________
acceleration, a3: _________
minimum mass which will turn table, mo: _________
rotational inertia from Eq. 8, I3:
experimental rotational inertia of ring, Iring, exp:
theoretical rotational inertia of ring, Iring, theory:
torque due to friction, τf: _________
corrected value of I3:
corrected value of Iring, exp:
total uncertainty:
percent difference, theory and experiment:
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