Determination of moment of inertia of a uniform rectangular bar and a connecting rod using the trifilar suspension, and by swinging the connecting rod as a compound pendulum.

Results : 

For the uniform rod (using trifilar suspension)                    I =

For the uniform rod (theoretically)                     I =

For the connecting rod (using trifilar suspension)       I =

For the connecting rod (1st suspension)                        I =

For the connecting rod (2nd suspension)                        I =

Discussion :

        The equation 1 is accurate only for oscillations with small angular displacements. If this displacement is considerably large an error could occur in the final results due to this reason. And also when using the trifilar suspension the center of mass of the object under consideration should be right over the platform. Other wise a additional moment of inertia of the system will cause a increase of the result, actually expected. When measuring the time an error is always possible in the measurement due to differences in human reaction time.

Analysis of Error,

When I = 0 and M = 0,

From equation 1,        

Error in   ,  

        =

        =  7.62 x10-3

                        = 7.62 x10-3 x

                        =

For the uniform rod,

        

Let           m = ( M + Mo )

Then,            m =  (1125 + 1954.8 ) x 10-3 kg

        =  3.080 kg

Error in m  is,    

                                       

                                       =  1.01 x 10-6 kg

From equation 1,
        

Let                              

 Error in,              

        

                                                                                                 =  kgm2        

           = 5.1 x10-5 x

                                        =        

Error in I  ,          

               

         kgm2

                                     =

For the theoretical calculation of I for the uniform rod,

From equation 2,

        

Let  p = I2 + w2 = (0.25452 +.0385 2) m2

Then error in p,      

         m2

         m2

Now error in  I,        

        

         kgm2

                                   = 

For the connecting rod,

When using the trifilar suspension,

        

Let           m = ( M + Mo )

Then,            m =  (1848.52 + 1954.8 ) x 10-3 kg

        =  3.8033 kg

Error in m  is,    

                                       

                                       =  1.0001 x 10-6 kg

From equation 1,
        

Let                              

 Error in,              

        

                                                                                                 =  kgm2        

           =

                                        =        

Error in I  ,          

               

         kgm2

                                     =

Compound pendulum method for con-rod.

When l = 236 mm,

Error in I ,

        

        

                          =  

        =  

Let Y=,    

                   

                =        

        =  

Error in ,              

             

        

When l = 100 mm,

Error in I ,

        

        

                          =  

        =  

Let Y=,    

                   

                =        

        =  

Error in ,              

              

        

Appendix :

For the trifilar suspension the periodic time is given by,

         

                

where,

        L        -        length of the suspension wires,

r          -        radius from the center of the platform to the attachment points of  the suspension wires,

        Io        -        moment of inertia of platform,

        Mo        -        mass of platform,

        I        -        moment of inertia of body,

        M        -        mass of body,

        G        -        acceleration due to gravity.

Raw data,

For the oscillations of the trifilar suspension,

     

 From Equation 1,

           

        

                               

   

Dimensions for uniform rod,

For the oscillations of the system rod and the trifilar suspension,

        

From Equation 1,

        

               

           

Theoretically,

                          Equation 2

           

 

        

        

For the connecting rod,

Using Equation 1 for the system of trifilar suspension and the connecting rod,

             

             

        

When the connecting rod was suspended by a end let to swing the moment of inertia of the rod about the axis through the point of suspension is given by,

        ……………. Equation 3

Where,

        T        -        periodic time

        M        -        mass

        G        -        acceleration due to gravity

        l        -        distance from the point of suspension.

Using the parallel axes theorem,

        Iy1y1 =  Iyy + Md2             

        Iyy      =  Iy1y1 - Md2………….Equation 4

When l = 236 mm,

        

From equation 3,

        

        

        

From Equation 4,

        

        

        

When l = 100 mm,

        

From equation 3,

        

        

        

From Equation 4,