AP Physics C Mathy Unit 10 Rotational Motion Lab #7 Rotational Inertia & Torque Please do not write on this lab protocol Objectives
• To familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid
body as mass plays in the description of linear motion
• To investigate how rotational inertia can be calculated using linear terms
APPARATUS
MATERIALS
PVC Rotating System
Spring scale
Pasco Motion Sensor
Meter Stick
2 pulleys
string
Cardboard Reflector
Motion Sensor
4 separate hook masses
Ring Stand
C clamp
THEORY
If we apply a single unbalanced force, F, to an object, the object will undergo a linear acceleration which is determined by the
unbalanced force acting on the object and the mass of the object. A force that is applied to an object in a way that will cause rotational
motion upon its axis at radius (r ) is called a torque. By wrapping a string around the base of the PVC apparatus, the force of tension
will apply a torque to the system and create rotational motion.
A simplified version of the lab set up For this lab you need to know how much torque a string wrapped around a PVC pipe base applies to the system. This is easy to
calculate: it is just the tension in the string multiplied by the radius of the pulley, positive if the twist on the pulley is in the counterclockwise direction and negative if the twist on the pulley is in the clockwise direction. See Figure 1.
The rotating system and whatever is mounted in it has a total moment of inertia I. If the mass of the pulley is negligible, the tension in
the string T is constant and the torque on the rotating apparatus is
τ = rT,
or,
Iα = rT,
where a is the angular acceleration of the spindle. If the string does not stretch, the tangential acceleration (a) of a point on the edge of
the spindle is equal to the acceleration of the hanging mass. Moreover, a = rα so that
I(a/r) = rT
so that
Ia=r2T
Therefore, the rotational Inertia I can be seen as a function of the following equation: I= r2T/a Thus, by measuring the acceleration of the hanging mass m we can compute the moment of inertia of whatever is rotating if we know
the force of tension and the radius of where that force is applied. However, the rotational inertia of the PVC system can also be calculated as an ideal theoretical value without knowing these variables. Every rigid object has a definite moment of inertia about any particular axis of rotation. Here are a couple of examples of
the expression for I for two mass objects:
One point mass m on a weightless rod of radius r (I = mr2):
Two point masses on a weightless rod (I = m1r12 + m2r22 ):
To illustrate we will calculate the moment of inertia for a mass of 2 kg at the end of a mass less rod that is 2 m in length (object #1
above):
I = mr2 = (2 kg) (2 m)2 = 8 kg*m2 because there is a mass on the other side, apply the equation again and sum
these values.
Procedure:
1. Set up the apparatus as shown on a lab table where the mass will be free to fall to the ground. Ring clamp down the base stand
for stability.
2. On the computer at your lab station, open up Data Studio and plug in a Pasco Motion Detector. In the setup icon, select the
velocity function and drag this down to the table icon so it will tally velocity as a function of time v(t). Set the motion sensor
on the floor under the mass that is to be lowered by the PVC apparatus.
3. Wind the string around the PVC apparatus so that there is no slack in the line that connects the pulleys and mass. You must
now find a mass that will allow the system to rotate. To overcome friction in the system, add small masses and find the exact
moment where the system will start moving. Record this mass, but do not include in your mass tally in your data table. Next,
have one partner hold the spring scale upright at the other end in a stationary position as shown in the illustration. At this
point in time , measure the height off the floor at which the falling mass will start. Record this in the data section. You will
start each trial at this starting position.
4. In your data section, you will need a table to keep track of the Tension, radius, mass, and acceleration created by the falling
mass. The radius is the measured cross sectional distance in the PVC pipe stem. It will serve as a constant. In this lab, you
will need to change the mass four times using increasing increments of your choice.
5. With one partner holding the PVC system stationary, start the motion detector and allow the mass to fall towards the motion
sensor. Do not let it hit the motion sensor. You only need enough data to accurately gauge the change in velocity. Keep track
of the final height achieved by the mass. Record this and keep it consistent throughout the lab this trial created a velocity –
timetable. You may cut and paste this table and place it into your data section or reproduce it by hand.
6. Repeat step 5 with three additional increases in mass.
7. Open Excel and create a velocity(y) vs time(t) graph for each trial. The slopes of these graphs go in the data table in the
acceleration column. Cut and paste these graphs and place them in your data section.
CALCULATIONS
1. In Excel, create another graph that will plot r2T/a. List what the slope of this graph indicates and its units.
2. Compare the theoretical Rotational Inertia for this system with the experimental one you obtained.
3. Calculate the maximum Rotational Kinetic Energy of the rotating apparatus. (Hint v=rω)
4. Create a conservation of Energy equation that will describe the initial and final energies. Use this equation to solve for the
rotational inertia I for the PVC system.
5. Compare the value from question #4 to your ideal value.
6. In your conclusion, describe which method for calculating rotational inertia was most successful (graphing vs energy
equation).
*The term “compare” always entails using a percentage of error equation.
CONCLUSION
Consult your formal lab write up for this portion