Physics Laboratory 8

Analyzing a mechanical system

Objectives:

In this laboratory students will analyze a mechanical system.  The linear and angular positions, velocities, and accelerations of its components  will be measured using a Pasco Rotary Motion Sensor (RMS).  

The RMS is a bi-directional position sensor.  It contains an optical encoder, which gives a maximum of 1440 counts per revolution (360 degrees) of the RMS shaft.  The resolution can be set to 360 or 1440 times per revolution (1 degree or ¼ degree).  The direction of the rotation is also sensed.  A 3-step pulley keys into the rotating shaft and can be mounted on either side of the shaft.

A thin rod with two brass masses or an aluminum disk and a steel ring is mounted to the rotary motion sensor.  A string is wrapped around one groove of the 3-step pulley and one end of the string is connected to a hanging mass.  This string is guided over a pulley.  As the hanging mass falls from rest its  position, velocity, and acceleration are measured as a function of time.  The  angular position, velocity, and acceleration of the 3-step pulley and the attached apparatus are also measured as a function of time.  Students will determine the moment of inertia of the brass masses and steel ring from these measurements.

Method 1

When analyzing a conservative mechanical system, one can proceed by finding all internal and external forces and torques acting on the system and the resulting linear and angular accelerations.  The equations of kinematics then yield  the linear and angular velocities and positions as a function of time.

The measured acceleration a of the hanging mass m yields the tension T in the string.

mg-T=ma

 

The tension T yields the torque t acting on the apparatus.

t=rT

Here r is the radius of the chosen pulley about which the thread is wound.

The measured angular acceleration a yields the moment of inertia of the apparatus.

I=t/a

 

Method 2

One can also analyze a conservative system using conservation of mechanical energy.  The sum of the potential energies and rotational and translational energies of the system is a constant of motion.
The measured position of the hanging mass at some time t yields its change in potential energy.

DU=mgDy

Since the mass is falling, Dy and DU are negative.

 

The measured velocity of the hanging mass at that same time t yields its change in kinetic energy, since the mass starts from rest.

DK=(1/2)mv2

 

The measured angular velocity of the rotating apparatus at that same time t yields the change in its rotational kinetic energy, since the apparatus is initially at rest.

DKrot=(1/2)Iw2

 

Conservation of mechanical energy now yields the moment of inertia.

DU+DK+DKrot=0
I=(2/w2)(-mgDy-(1/2)mv2)

Students will compare their dynamically obtained value of the moment if inertia to the value they obtain when they determine the mass and dimensions of their apparatus.

Procedure:

A string is tied to the 3-step pulley and wound around the medium pulley.  The radius of the medium pulley is 1.43cm.
The other end of the string is guided  over the clamp-on  pulley and weight (0.5N) is attached to it.
An object is mounted onto the 3-step pulley.  Initially the object and the 0.5N weight have no kinetic energy.  As the weight is falling, it gains translational kinetic energy while the object gains rotational kinetic energy.
The linear position and velocity of the weight and the angular position and velocity of the rotating object are measured as a function of time.

Rod and brass masses

A rod is mounted on the RMS without the brass masses attached to its ends.  The linear position and velocity of the weight and the angular position and velocity of the rod are measured as a function of time and stored in the file rod.xls.
Right click on the link to rod.xls and save the target to your local disk drive.  Then open the file in Excel.  Sheet1 contains the the data for the rod.
Create graphs of velocity and angular velocity versus time.
Examine the graphs. 
Calculate the moment of inertia of the system using method 1.

Method 1
From the slope of the velocity versus time graph determine the acceleration a.
From the slope of the angular velocity versus time graph determine the angular acceleration a.
Calculate T=mg-ma  (Here m is the mass of the falling weight.)
Calculate t=rT.
Calculate the moment of inertia I1=t/a of the rod and the pulley.

Calculate the moment of inertia of the system using method 2.

Method 2
At the time t when the position is approximately 0.35m (i.e. when Dy=-0.35m) find the velocity v and the angular velocity w.
Calculate the moment of inertia I1=(2/w2)(-mgDy-(1/2)mv2) of the rod and the pulley.

Brass pieces are attached to each end of the rod, a distance R= 0.181m from the center of the rod.  The experiment is repeated.  Sheet2 contains the the data for the rod and the brass pieces.
Calculate the moment of inertia I2 of the rotating system using method 1.
Calculate the moment of inertia I2 of the rotating system using method 2.
Now find the moment of inertia of the brass masses I=I2-I1.
Fill in a table as show below.  In this table mhanging is the mass of the 0.5 N weight.

 

Method 1 Method2 Method 3
  rod only rod + brass brass only   rod only rod + brass brass only   brass
mhanging (kg)     x mhanging (kg)     x Mtot (kg) 0.148
a (m/s2)     x Dy(t) (m)     x R (m) 0.181
a (rad/s2)     x v(t) (m/s)     x I (kgm2)  
r (m) 0.0143 0.0143 x w(t) (rad/s)     x    
T (N)     x I (kgm2)          
t (Nm)     x            
I (kgm2)                  
 

The mass Mtotal of both brass pieces is 0.148 kg.  The moment of inertia of a point mass M a distance R from the center of rotation is I=MR2.  The moment of inertia of the two brass masses (treated as point masses) is therefore I=MtotalR2, where Mtotal is the mass of both brass masses.  

Calculate the moment of inertia of the brass masses.

Aluminum disk and steel ring

An aluminum  disk is mounted on the RMS.  The linear position and velocity of the weight and the angular position and velocity of the disk are measured as a function of time and stored in the file disk.xls.
Right click on the link to disk.xls and save the target to your local disk drive.  Then open the file in Excel.  Sheet1 contains the the data for the disk.
Create graphs of velocity and angular velocity versus time.
Examine the graphs. 
Calculate the moment of inertia of the system using method 1.
Calculate the moment of inertia of the system using method 2.
A steel ring is placed on top of the disk.  The experiment is repeated.  Sheet2 contains the the data for the disk and the ring.
Calculate the moment of inertia I2 of the rotating system using method 1.
Calculate the moment of inertia I2 of the rotating system using method 2.
Now find the moment of inertia of the steel ring I=I2-I1.
Fill in a table as show below.  In this table mhanging is the mass of the 0.5 N weight.

 

Method 1 Method2 Method 3
  disk only disk + ring ring only   disk only disk + ring ring only   ring
mhanging (kg)     x mhanging (kg)     x M (kg) 0.45
a (m/s2)     x Dy(t) (m)     x R1 (m) 0.0275
a (rad/s2)     x v(t) (m/s)     x R2 (m) 0.0375
r (m) 0.0143 0.0143 x w(t) (rad/s)     x I  
T (N)     x I (kgm2)          
t (Nm)     x            
I (kgm2)                  
 

The mass M of the steel ring is 0.45 kg.  Its inner radius is R1=0.0275m and its outer radius is R2=0.0375m.  The moment of inertia of a ring about its center of mass is given by I=(1/2)M(R12+R22). 

Calculate the moment of inertia of the ring.

Data Analysis:

Examine your tables.

Do the moments of inertia of the brass masses or the steel ring obtained using method 1 or method 2 agree with each other?  What is the percentage difference between the two values? 
Which method do you believe yields the more accurate value and why?
Does your dynamically obtained value of the moment of inertia agree with the the value you obtain using the mass and dimensions of your apparatus?  What is the percentage difference?
 

Open Microsoft Word and prepare a report using the template shown below.

Name:
E-mail address:

Laboratory 8 Report

 

In a few sentences summarize the experiment.
Show the two tables containing your results.
Answer the questions posed in the Data Analysis section  in full sentences.

Save your Word document (your name_lab8.doc) and attach it to an e-mail message to mbreinig@utk.edu.