Lab 9 Moment of Inertia!
!
!
Objective!
The purpose of this lab is to examine how a rotating object behaves when a torque is applied to it.
We will use this knowledge to find the mass of rotating objects.!
The Setup!
A hanging mass will be used to apply a torque on a rotating table.!
axis of rotation
3. torque
1. lever arm
2. applied!
force
to hanging mass
!
Because the force applied is away from the axis of rotation by the distance of the lever arm, it will
produce a torque. Being the only torque means there will be an angular acceleration thus making
the table spin.!
There are two objects in this system. The hanging mass and the rotating table. The hanging
mass travel only vertically so we apply to it Newton’s second law of motion for translation.!
ΣFhanging = mhanginga
⇒ mhanging g −T = mhanginga !
The rotating table only spins, so we apply to it Newton’s second law of motion for rotation. The
lever arm is the radius of the spool of the rotating table. The force is the tension applied to the
spool.!
Στtable = I table α
⇒ RspoolT = I table α !
Since a string connects the hanging mass to the rotating table without slipping, the acceleration is
related to the angular acceleration through the spool radius.!
a = Rspool α !
Replacing the tension and the angular acceleration in these three equations gives us the
acceleration of the hanging mass.!
a=
mhanging g
mhanging +
I table
!
2
Rspool
Since all of the variables in this equation are constants, the acceleration is a constant as well. To
measure the acceleration, we are going to look at the hanging mass. Constant-acceleration
kinematics tells us this.!
1
h = at 2
2
⇒ a=
2h
t2
!
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Combining all of the above, the moment of inertia of the rotating table is this.!
⎛ gt 2
⎞
I tabke = ⎜⎜⎜
− 1 ⎟⎟⎟mhanging R 2spool !
⎟⎠
⎜⎝ 2h
When an object is added onto the rotating table, its moment of inertia is added to the rotating
table. By measuring the total moment and subtracting the table’s moment of inertia, the moment
of inertia of the object can be found.!
I added
object
= I total − I table !
The mass of the object is found using the appropriate formula for the geometry of the extra object.
For reference, the moment of inertia of the two objects we are measuring are listed here, a thick
ring and a disk.!
1
2
= M ring Rinner
2
I thick
ring
I disk
1
2
= Mdisk Rdisk
2
(
ring
2
+ Router
ring
)
!
Experiment: Moment of Inertia of the Table!
Set up the experiment as shown above with nothing on the rotating table. Give the hanging mass
as much fall space as possible. Use the foam to soften the landing. Record this distance. Use
an amount of hanging weight such that the fall time is at most 4 seconds. Release the hanging
weight and measure the time it takes the hanging weight to land on the foam. Do this a total of 5
times. Calculate the average drop time.!
Measure the spool diameter of the rotating table using calipers then calculate the spool radius.
Use the equations above to calculate the moment of inertia of the table.!
Experiment: Mass of an Object!
You will be assigned an object, the ring or the disk. Place it onto the rotating table. Measure the
moment of inertia your object using the method above. Adjust the hanging weight such that the
fall time is from about 4 seconds (±0.2 second) to about 10 seconds in steps of about 0.5 second.
Once you have the correct hanging weight, repeat the drop three times to get an average fall time
over the same height.!
Record the hanging weights, the fall heights, and the average fall times. !
Measure the physical parameters of the objects. Calculate the masses of your object from the
moment of inertia. Graph the mass of the object (y) as a function of the hanging weight (x).!
Weigh the object directly using a balance. Graph this constant value on the plot as well. This is
your accepted value.!
Conclusion!
Report your results for the table by itself and for the object you are assigned. !
What hanging weight and fall time produces the best result and is there a trend based on the
hanging weight?
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