Radius of Gyration and moment of inertia

In summary, the conversation discusses finding the moment of inertia for a circle shape using the equation r=sqrt(I/A), where A is the area and I is the moment of inertia. The person initially had trouble finding I and mentioned trying to find the centroid of the shape. Another person suggests a better approach using the parallel axis theorem and provides the formula for finding I_inner. The final solution involves using the moment of inertia equation for a uniform disk and using the parallel axis theorem to account for the inner disc not being centered on O. The final value of I is found to be 5.66.
  • #1
SteelDirigibl
40
0

Homework Statement


Screenshot2011-04-22at53716PM.png



Homework Equations


i guess its r=sqrt(I/A)
where A is the area of the circle thing and I is the moment of intertia.

The Attempt at a Solution



I guess I'm just having trouble getting I. A is 33pi
 
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  • #2
what have you tried?
 
  • #3
finding the centroid of the shape.
 
  • #4
were you able to find that? I think there is a better approach: I = I_outer - I_inner . (be careful that everything is calculated about O).
 
  • #5
how do I find I_inner ?
 
  • #6
The moment of inertia of a uniform disk around its center is 1/2MR^2. However, the inner disc (the empty region) is not centered on O, but on C. You have to use the parallel axis theorem to account for this.
 
  • #7
so according to that...

1/2*49pi*7^2 [I_outer] - (1/2*16pi*4^2+16*1)[I_inner]=3319

3319/(33pi)=32

sqrt(32)=5.66

And that was right...
thanks guys. I understand it a little better now..
 

1. What is the definition of radius of gyration?

The radius of gyration is a measure of how far an object's mass is distributed from its axis of rotation. It is the distance from the axis at which the object could be concentrated without changing its moment of inertia.

2. How is radius of gyration related to moment of inertia?

The radius of gyration and moment of inertia are directly related. The moment of inertia is equal to the mass of the object multiplied by the square of the radius of gyration.

3. How do you calculate the radius of gyration for a given object?

The radius of gyration can be calculated by dividing the moment of inertia of an object by its mass and then taking the square root of the result.

4. Why is the radius of gyration important in structural engineering?

The radius of gyration is important in structural engineering because it is used to determine the stability and stiffness of a structure. A lower radius of gyration indicates a more compact and stable structure, while a higher radius of gyration indicates a less stable and more flexible structure.

5. Can the radius of gyration be changed?

Yes, the radius of gyration can be changed by altering the distribution of mass in an object. For example, by moving the mass closer to the axis of rotation, the radius of gyration will decrease, making the object more stable.

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