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DYNAMICS - CASE STUDY SOLUTION


Dimensions
 

First find the moment of inertia of the slender rod about a line L passing through its center of mass.

"Slender" means the length is much greater than the width or depth.

Consider a differential element of length dr at a distance r from the center of mass. The mass of the element is equal to the product of its density and volume:

     dm = ρ dv = ρ A dr

Substitute this into equation for the moment of inertia of the rod about L,

     

Note that the mass of the rod is given by

     mr = ρ Vrod = ρ A Lr

thus

     Irod = 1/12 mr Lr2

This is the same result listed in the Sections Appendix. for a slender rod.

     

Dimensions
 

Now the parallel axis theorem can be used to calculate the moment of inertia of the rod about the line Lo passing through the point 0,

     Io-rod = Irod + dr2 mr

              = 1/12 mr Lr2 + (1/2 Lr)2 mr

             = 1/3 mr Lr2 = 0.3333 (0.5 slugs) (1.2 ft)2

             = 0.2400 slugs-ft2

Next, determine the moment of inertia of the disk about a line L passing through its center of mass, perpendicular to the flat face of the disk. Consider a differential element of width dr at a radius r from the center of mass. The mass of the element is equal to the product of its density and volume:

     dm = ρ dv = ρ td 2 π r dr

Substitute this into the moment of inertia equation to get I of the disk about L,

     

           = 0.5 ρ td π rd4

The mass of the element is equal to the product of its density and volume,

     md = ρ Vd = ρ π rd2 td

     Idisk = 1/2 md rd2

Again, parallel axis theorem can be used to find the moment of inertia of the disk about the line Lo passing through the point 0,

     Io-disk = Idisk + dd2 md

           = 1/2 md rd2 + (Lr + rd)2 md

           = md [1/2 rd2 + (Lr + rd)2]

          = 2.67 [1/2 (0.4)2 + (1.2 + 0.4)2]

         = 7.049 slugs-ft2

     
   

The sum of the moment of inertia of the disk and rod to will give the total moment of inertia of the pendulum about Lo as

     Io-pendulum = Io-rod + Io-disk

         = 7.289 slugs-ft2

     
   
 
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