Name_________________________________ Date__________ Partner(s)_______________________________
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Physics 21 Laboratory
Rotational Dynamics
Purpose: To determine the moment of inertia of a system of concentric disks constrained to rotate about
its axis of symmetry. The moment of inertia is found by two methods and the results are compared.
1)
In the first method, the moment of inertia is determined theoretically by applying the formula for the
moment of inertia of a single disk to each of four disks, and adding the results.
2)
In the second method, the moment of inertia is determined experimentally by measuring the
acceleration produced by a falling mass exerting a constant torque on the body.
Finding the Moment of Inertia by the First Method:
1)
Write the equation for the moment of inertia of a uniform, solid disk
rotating about its axis of symmetry.
mass, M
R
W
2)
Write the equation for the moment of inertia of four concentric disks rotating about its axis of symmetry.
The masses are M1, M2, M3, and M4 with radii R1, R2, R3, and R4, respectively.
3)
Write the equation for the mass, M, of a solid disk in terms of its density , radius R, and width W. (See
the dimensions of the disk in question 1) above.)
4)
Use the vernier caliper and measure the diameters and the widths of the four cylinders that make up the
concentric disks of the rotating system.
Disk 1
Disk 2
Radius, R (cm)
Width, W (cm)
1
Disk 3
Disk 4
5)
Use the density of aluminum, 2.70 g/cm3, to calculate the mass of each disk, and express the masses in
kilograms. Show your calculations.
Disk 1
Disk 2
Disk 3
Disk 4
Mass, M (kg)
6)
Now calculate the moment of inertia of each disk and express your result in kg m2. Also report the total
moment of inertia of the system. Show your calculations.
Disk 1
Disk 2
Moment of
Inertia, I
(kg  m 2 )
2
Disk 3
Disk 4
Total
Finding the Moment of Inertia by the Second Method:
A string is wrapped around one of the cylinders of the four cylinder
system. The radius of the disk around which the string is wrapped is R. A
mass m is attached to the other end of the string and is allowed to fall,
unwinding the string, and causing the system to rotate.
1)
Draw the free-body diagram of the block of mass m, indicate the
positive direction, and write the F = ma equation for the mass.
m
2)
Draw the free-body diagram for the system of disks. Show the direction of the positive torque, and
write the  = I equation for the system. (The value of the moment of inertia I is for the entire system –
the four cylinders together.)
3)
Explain how the acceleration a of the block of mass m downwards is related to the angular acceleration
 of the system of cylinders, and write an equation that relates the two.
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4)
Take the three equations you got from above [from 1), 2), and 3)] and solve them simultaneously for the
moment of inertia of the disk system I, in terms of m, R, a, and g (eliminate T and ). Notice that you
need to know m, R, a, and g in order to determine the moment of inertia of the disk system.
5)
In order to determine the acceleration of mass m, you release it from rest and allow it to drop
a known distance d. By determining the time t that it takes to fall that distance, the
acceleration a of the mass can be found. Derive an equation for the acceleration a in terms of
d, t, and g.
m
d
6)
Set up the equipment and the moment of inertia apparatus as shown in the
diagram. You need the following:
Table clamp
Heavy metal rod
Moment of inertia apparatus with clamp
Light string – approximately 1.5 meters long
Small piece of masking tape
50-gram weight hanger
50-gram weight
Digital timer
2-meter long meter stick
4
d
7)
Use a small piece of tape and attach the end of the string to the largest radius disk. Tie a small loop at
the other end of the string and attach the 50-gram weight hanger. Wind the string up on the disk
making sure that the string does not overlap itself. Continue until the bottom of the weight hanger is at
some convenient height above the ground (>1 meter). Record the mass m that is being dropped, the
distance d that it is dropping, and the radius R around which the string is wrapped. Release the mass,
and use the digital timer to measure the time it takes the mass to drop to the floor. Repeat this
measurement at least four more times.
You are to repeat this procedure, but in this case the string is attached to the disk with the second largest
radius, and the falling mass will be 100-grams.
String Wrapped on Largest Radius
Disk
String wrapped on Second Largest
Radius Disk
Time Falling (sec)
Time Falling (sec)
Radius, R (m)
Falling Mass, m (kg)
Distance Falling, d (m)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Average Time
8)
Use the data above from the “String Wrapped on Largest Radius Disk” to calculate a value for the
moment of inertia of the system of disks.
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9)
Use the data above from “String Wrapped around Second Largest Radius Disk” to calculate a value for
the moment of inertia of the system of disks.
Results: Report your three values for the moment of inertia of the system of disks in the table below.
Moment of Inertia of the System of Disks (kg m2)
Application of the Formula for
the Moment of Inertia of a
System of Disks
Experimental Method with
String Attached to Largest
Radius Disk
Experimental Method with
String Attached to Second
Largest Radius Disk
Conclusions:
If you made careful measurements of the dimensions of the disks and were careful in your calculations,
then the first value for the moment of inertia of the system of disks should be very accurate. However,
the two experimental values may be slightly off because of uncontrollable experimental factors. For
example, the measurement of the distance the mass m falls and the timing of the mass falling are
controllable – you just need to be very careful when making these measurements. However, friction
exists between the axle and the disk system, and you can’t control this – it’s there. Here’s the question:
Do your experimental results reflect the existence of this frictional effect? Please explain.
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