Inertia matrix of a homogeneous cylinder

In summary, the conversation is discussing the derivation of the equations (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from the integration of y'^2 + z'^2 and x'^2 +y'^2. The conversation includes discussions on the use of coordinates and the integration process. The final solution involves using the value of ##\rho## in terms of the total mass ##m## and integrating over the appropriate boundaries.
  • #1
influx
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Homework Statement


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Homework Equations


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The Attempt at a Solution


What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly? Could someone point me in the right direction?

Thanks
 
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  • #2
Did you try actually computing the integrals? What did youget? Please show your work.
 
  • #3
Orodruin said:
Did you try actually computing the integrals? What did youget? Please show your work.

What are y'^2 and z'^2? I don't know what to sub in for these (not sure what they represent?) so can't do the integral until I know them.
 
  • #4
They are coordinates as shown in the figure.
 
  • #5
Orodruin said:
They are coordinates as shown in the figure.
I understand that but I mean how would one integrate those terms with respect to m? They aren't constants?
 
  • #6
They depend on where in the body you are.
 
  • #7
Orodruin said:
They depend on where in the body you are.
Sorry for the late reply. I am still confused. I just want to know how they (mathematically) got from the left hand side to the right. I don't understand how integrating the left hand side yields the right hand side.
 
  • #8
Use ##dm = \rho \, dx'dy'## and integrate over ##x'## and ##y'## with the appropriate boundaries. The value of ##\rho## in terms of the total mass ##m## can be inferred by computing the volume ##V## of the body and using ##\rho V = m##.
 

What is the inertia matrix of a homogeneous cylinder?

The inertia matrix of a homogeneous cylinder is a mathematical representation of the distribution of mass and rotational inertia of the cylinder about its center of mass. It is a 3x3 symmetric matrix that contains information about the cylinder's moment of inertia around its principal axes.

How is the inertia matrix of a homogeneous cylinder calculated?

The inertia matrix of a homogeneous cylinder can be calculated using the formula I = [Ixx, Ixy, Ixz; Iyx, Iyy, Iyz; Izx, Izy, Izz], where Ixx, Iyy, and Izz represent the moments of inertia around the principal axes, and Ixy, Ixz, Iyx, Iyz, Izx, and Izy represent products of inertia. These values can be calculated using integrals and geometric properties of the cylinder.

What is the significance of the inertia matrix of a homogeneous cylinder?

The inertia matrix of a homogeneous cylinder is significant because it provides important information about the cylinder's resistance to changes in its rotational motion. It is used in various engineering and physics applications, such as designing and analyzing rotating machinery and predicting the behavior of objects in motion.

How does the inertia matrix of a homogeneous cylinder differ from that of other shapes?

The inertia matrix of a homogeneous cylinder differs from that of other shapes because it is dependent on the cylinder's dimensions and mass distribution. While the inertia matrix of a homogeneous sphere or cube will be the same regardless of its orientation, the inertia matrix of a cylinder will change depending on its orientation relative to the principal axes.

Can the inertia matrix of a homogeneous cylinder change?

The inertia matrix of a homogeneous cylinder can change if there is a change in the cylinder's mass distribution or if it is subjected to external forces. However, for a perfect homogeneous cylinder with a constant mass distribution, the inertia matrix will remain the same regardless of its orientation or position in space.

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