Lab
Rotational Inertia
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In this lab we will be using the Rotational Inertia Demonstrator made by Arbor Scientific.
This device has a central, low friction, pulley with three different radii from which strings can be tied. It also has four thin rods that form spokes on which masses can be placed at virtually any desired position.
The moment of inertia of the central pulley, independent of the four rods and their moveable masses, is given by the manufacturer as 0.00058 kg m
^{2 }
. All other measurement will be taken as the experiment is completed.
In each experiment, we will ultimately be calculating the moment of inertia of the pulley and its moveable masses. We will deal with rotational (angular) kinematics properties and their relationship to linear (translational or tangential) properties of a falling mass as well as potential gravitational energy, translational kinetic energy, and rotational kinetic energy.
Refer to the following information for the next four questions.
Before we collect any further data, we need to take measurements that will allow us to directly calculate the moment of inertia of the entire pulley assembly: central pulley, rods, and moveable masses.
mass of a single rod (in kg)
length of a single rod measured from the center of the pulley (in meters)
mass of a single moveable mass (in kg)
radius of the moveable masses measured from the center of the pulley (in meters)
Refer to the following information for the next question.
Phase I. In this part of the experiment we will wrap the string around the smallest radius of the pulley, 0.0202 meters.
What is the mass (in kg) of the hanging mass?
You will complete three trials using a motion detector as well as the rotational inertia demonstrator. From each trial you need to record the initial time, initial height, final time, and final height. Then you will calculate the time interval and the distance required for the mass to reach the ground.
initial time
initial height
final time
final height
time interval
distance
linear acc
Trial
(sec)
(meters)
(sec)
(meters)
(sec)
(meters)
(m/sec
^{2}
)
1
2
3
Analysis of first radius
What was the average linear acceleration of your falling mass for all three trials?
Which trial's acceleration came closest to this value?
1
2
3
For the remaining calculations in this section, use the data for the acceleration, time interval spent falling, and the distance fallen from the trial chosen in the previous question.
Calculate the final linear velocity of the falling mass in m/sec.
Calculate the initial potential gravitational energy of the falling mass in Joules.
Calculate the final translational kinetic energy of the falling mass in Joules.
Calculate the final angular velocity of the pulley with its rods and moveable masses in rad/sec.
Refer to the following information for the next question.
Phase II: In this part of the experiment we will wrap the string around the largest radius of the pulley, 0.03852 meters.
You will complete three trials using a motion detector as well as the rotational inertia demonstrator. From each trial you need to record the initial time, initial height, final time, and final height. Then you will calculate the time interval and the distance required for the mass to reach the ground.
initial time
initial height
final time
final height
time interval
distance
linear acc
Trial
(sec)
(meters)
(sec)
(meters)
(sec)
(meters)
(m/sec
^{2}
)
1
2
3
Analysis of second radius
What was the average linear acceleration of your falling mass for all three trials?
Which trial's acceleration came closest to this value?
1
2
3
For the remaining calculations in this section, use the data for the acceleration, time interval spent falling, and the distance fallen from the trial chosen in the previous question.
Calculate the final linear velocity of the falling mass in m/sec.
Calculate the initial potential gravitational energy of the falling mass in Joules.
Calculate the final translational kinetic energy of the falling mass in Joules.
Calculate the final angular velocity of the pulley with its rods and moveable masses in rad/sec.
Refer to the following information for the next eleven questions.
Data Analysis
For the first radius, what was the pulley's angular acceleration in rad/sec
^{2}
?
For the first radius, what was the tension, in N, in the string as the mass was falling?
For the first radius, what was the torque, in mN, produced by the tension in the string as the mass was falling?
For the first radius, what was the experimental moment of inertia, in kg m
^{2}
of the complete pulley assembly?
For the first radius, what was the final rotational KE, in J, of the pulley with its rods and moveable masses?
For the second radius, what was the pulley's angular acceleration in rad/sec
^{2}
?
For the second radius, what was the tension, in N, in the string as the mass was falling?
For the second radius, what was the torque, in mN, produced by the tension in the string as the mass was falling?
For the second radius, what was the experimental moment of inertia, in kg m
^{2}
of the complete pulley assembly?
For the second radius, what was the final rotational KE, in J, of the pulley with its rods and moveable masses?
Based on the data collected on the thin rods and moveable masses, what should have been the theoretical moment of inertia, in kg m
^{2}
, for your complete pulley assembly?
Refer to the following information for the next four questions.
At this point we are now ready to make some conclusions regarding our values. The Law of Conservation of Energy states that mechanical energy sould be conserved in a system in the absence of friction. In our trials, we would compare the gravitational potenial energy of the falling mass at the start of a trial to the total kinetic energy of the mass and the rotating pulley at the end of the trial.
For the small radius, what was your value for the sum of the two final kinetic energies: KE
_{translational}
+ KE
_{rot}
?
For the large radius, what was your value for the sum of the two final kinetic energies: KE
_{translational}
+ KE
_{rot}
?
Which radius had the closest match to the original PE of the mass?
small
large
both matched equally well
Did this experiment confirm that mechanical energy is conserved in a frictionless system? Elaborate.
Refer to the following information for the next three questions.
In each set of trials, you were asked to calculate the moment of inertia of the complete pulley assembly. We also calculated the theoretical moment of inertia by measuring the masses of the thin rods and moveable masses as well as their lengths and radii. You are now going to calculate three errors for your experiment.
What was your average experimental value for the moment of inertia of the complete pulley assembly?
Calculate a percent difference between the two experimental values.
Calculate a percent error between your average expeimental value and the calculatd value for the moment of inertia.
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Tension Cases: Four Special Situations
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Torque: An Introduction
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Work
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Work and Energy
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Torque Beams
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Work and Energy
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Cliffs
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Elliptical Orbits
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Falling Sticks
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Gravitation #2
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Ramps
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Rolling Cans
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Rolling Spool
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