Moment inertia - theoretical problem

In summary, the conversation discusses how to calculate the moment inertia of a hollow cylinder with inner radius a and outer radius b, for the center axis. The solution involves splitting the cylinder into two parts and subtracting the overlapping areas. The formula for the moment of inertia of a solid cylinder is used, and it is mentioned that for a composite body, the total moment of inertia is the sum of the individual moments of inertia.
  • #1
silenzer
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Moment inertia -- theoretical problem

Homework Statement



Show that the moment inertia of a hollow cylinder with inner radius a and outer radius b is (1/2)*M*(a^2+b^2), calculated for the center axis.

Homework Equations



I know that the moment inertia of a non-hollow cylinder is I = (1/2) MR^2, but I don't know the moment inertia of a hollow one.

I = MR^2, generally.

Krotation = (1/2)Iw^2

The Attempt at a Solution



I'm not entirely certain on where to start. Should I split the cylinder into two parts and work from there?
 
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  • #2
silenzer said:
Should I split the cylinder into two parts and work from there?
Yes. Consider overlapping cylinders. Subtract!

Be careful with the mass.
 
  • #3
I'm sorry but I'm still having problems... I googled the problem and all of the solutions use definite integration. Can I solve this problem without using that?
 
  • #4
silenzer said:
I'm sorry but I'm still having problems... I googled the problem and all of the solutions use definite integration. Can I solve this problem without using that?
All you need to know is the formula for the moment of inertia of a solid cylinder--which you already know.

Hint: For a composite body with parts a and b, Itotal = Ia + Ib.
 
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  • #5
Sorry for the late reply. I did it! :D Thanks a lot.
 
  • #6
silenzer said:
Sorry for the late reply. I did it! :D Thanks a lot.
:thumbs:
 

Related to Moment inertia - theoretical problem

1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is the property of an object that determines how difficult it is to change its rotational velocity.

2. How is moment of inertia calculated?

Moment of inertia is calculated by multiplying the mass of the object by the square of its distance from the axis of rotation. The further away the mass is from the axis of rotation, the higher the moment of inertia will be.

3. What are the units of moment of inertia?

The SI unit for moment of inertia is kilogram square meters (kg•m^2). However, it can also be expressed in other units such as gram square centimeters (g•cm^2) or pound square feet (lb•ft^2) depending on the system of measurement used.

4. How does moment of inertia affect an object's rotational motion?

Objects with a higher moment of inertia will have a harder time changing their rotational motion. This means they will require more force to start rotating or to stop rotating, and will also have a slower rate of rotation. Objects with a lower moment of inertia will be easier to rotate and will have a faster rate of rotation.

5. What are some real-life applications of moment of inertia?

Moment of inertia is an important concept in physics and engineering, and it has various real-life applications. Some examples include calculating the stability of structures, designing vehicles and machinery, understanding the dynamics of rotating objects like spinning tops or gyroscopes, and predicting the behavior of celestial objects such as planets and stars.

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