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Rotational Dynamics

Calculus Based Section: Rotational Inertia of Solid Bodies

Problems

Problems

In our study of rotational dynamics we skipped over exactly how to calculate the rotational inertia of a solid body. The process for calculating this quantity is quite complicated, and requires quite a bit of calculus. Thus we devote a section to calculating this quantity.

Consider a small section of a rod, a radius r from the axis of rotation, and with a mass δm , as shown below:

Figure %: A small piece of mass on a rod being rotated about an axis.
Because the volume of the section of the rod is sufficiently small, we can calculate the moment of inertia of this single piece: I = δmr 2 . To find the moment of inertia of the entire rod, we sum over all pieces of a similar size that compose the rod:

I r k 2 δm k

To get an exact answer for the moment of inertia, we take the limit as the δm gets smaller; as the rod is broken up into more and more pieces. Thus:


I = r k 2 δm k  
  = r 2 dm  

This integral equation is the basic equation for the moment of inertia of a solid body.

Even with this equation, it is quite difficult to calculate the moment of inertia of a solid body. We will go through an example to show how it is done. Let us simply return to the example of the solid rod of length L, and mass M, rotated about its center, as shown below.

Figure %: A small piece of mass, dm, of length dx on a rod being rotated about an axis.
Let us denote the cross sectional area of the rod by A. Thus the volume of the small element of mass, dV = Adx , where dx is the length of the small element of mass. Thus, if we denote the density of the rod by ρ , then we can describe dm in terms of dx :

dm = ρdV = ρAdx

However, we can also express ρ in terms of measured quantities: ρ = M/V = M/AL . Thus we can plug all this into our integral equation:


I = r 2 dm  
  = x 2(ρAdx)  
  = x 2( Adx)  
  = x 2 dx  

Thus we now have an integral we can evaluate. We simply need to determine the limits. If we denote the axis of rotation to be at x = 0 , then we simply integrate from -L/2 to L/2:


I = x 2 dx  
  = []-L/2 L/2  
  = ML 2  

This is the equation for the moment of inertia of a thin rod, and it agrees with measured values.

In general, the moment of inertia of a solid body varies with MR 2 , where R is the measure of the radius, or length of a given object. To find the exact value of the moment of inertia, however, the complicated calculus is required.

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