Moment of Inertia - parallel axis theorem

In summary, the rods have different masses and they each have a different moment of inertia about their centre of mass. Because of this, the system as a whole has a different moment of inertia.
  • #1
ph123
41
0
A thin, uniform rod is bent into a square of side length "a". If the total mass is M, find the moment of inertia about an axis through the center and perpendicular to the plane of the square. Use the parallel-axis theorem.


According to Parallel Axis theorem:

I = I(cm) + Md^2

The distance across the diagonal of the square (corner to corner) is 1.4142...a (= to sqrt(2a^2)). The distance bisecting the square through the center of mass is a/2.

I = m(a/2)^2 + m(0.7071...a)^2
I = (1/4)ma^2 + (1/2)ma^2
I = (3/4)ma^2

This answer isn't right. I think I need to integrate since every point on the square is a different distance from the rotational axis through the center of mass. But there is not other axis specified in the problem, so I'm not sure as to which parallel axis I need to derive an expression for. Any ideas?
 
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  • #2
You are off by a multiplicative factor but I am not sure what it is...
 
  • #3
You want to find the moment of inertia about an axis through the centre, the calculation you did says you know what this is! The geometrical centre of a square is its centre of mass.

Do you have to use the parallel axis theorem? If not, I would consider each side of the square individually. Find the inertia about the centre of mass due to one side by integrating. Find the distance of a general point on the length as a function, then integrate. The other 4 sides should be the same by symmetry.
 
  • #4
Alright. So, first of all, you should know that the moment of inertia of a rectangular plate, axis through the center of the plate is:

I = (1/12)*M*(a^2+b^2)


1. Simply, for a square it would just be (a^2+a^2) or (2a^2)
2. Then, we find moment of inertia from the parallel-axis theorem.
3. Finally, take the moment of inertia from the parallel-axis theorem and subtract.
(so, parallel-axis - moment of inertia)
4. That's it, that should be the answer.
 
  • #5
Hi highcoughdrop,

highcoughdrop said:
Alright. So, first of all, you should know that the moment of inertia of a rectangular plate, axis through the center of the plate is:

I = (1/12)*M*(a^2+b^2)


1. Simply, for a square it would just be (a^2+a^2) or (2a^2)
2. Then, we find moment of inertia from the parallel-axis theorem.
3. Finally, take the moment of inertia from the parallel-axis theorem and subtract.
(so, parallel-axis - moment of inertia)
4. That's it, that should be the answer.


I don't think this is right. This is not a rectangular plate, it is a set of 4 thin rods. So in your formula we would have to set b=0 for each rod. Then use the parallel axis theorem for each rod and finally add the four results together.

If you believe your procedure gives the right answer anyways, please post some more details. I don't think I understand what you are doing in steps 2 and 3 for this problem.
 
  • #6
Actually all your answers are wrong.
Mass of each rod =M/4
moment of inertia for each rod through its centre =(1/12)(M/4)(a^2)=(M/48)a^2
Use parallel axis theorem for each rod
I=(M/48)a^2 + (M/4)(a/2)^2=(1/12)Ma^2
For 4rod(the whole system)
I=(4)(1/12)Ma^2=(1/3)Ma^2

I taught mant students this type of problems
 

1. What is the Moment of Inertia?

The Moment of Inertia is a physical property of a rotating object that describes its resistance to changes in its rotational motion. It is similar to mass in linear motion, but for rotational motion.

2. What is the Parallel Axis Theorem?

The Parallel Axis Theorem states that the moment of inertia of an object about an axis parallel to its center of mass is equal to the moment of inertia about the object's center of mass plus the product of the mass of the object and the square of the distance between the two axes.

3. How is the Moment of Inertia calculated using the Parallel Axis Theorem?

The Moment of Inertia using the Parallel Axis Theorem is calculated by adding the moment of inertia of the object about its center of mass to the product of its mass and the square of the distance between the center of mass and the parallel axis. This can be expressed as I = Icm + md2.

4. What is the significance of the Parallel Axis Theorem in rotational motion?

The Parallel Axis Theorem is significant in rotational motion because it allows us to calculate the moment of inertia of an object about any axis parallel to its center of mass. This is useful in determining the rotational motion of complex objects or objects with irregular shapes.

5. Are there any limitations to the Parallel Axis Theorem?

Yes, the Parallel Axis Theorem is only applicable for objects with a fixed axis of rotation. It does not account for objects with non-fixed axes of rotation or objects with changing moments of inertia. Additionally, it assumes that the object is rigid and has a constant density throughout.

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