Moments of Inertia object problem

In summary, the three objects of equal mass but different shapes - a solid disk, a thin ring, and a thin hollow square - have different moments of inertia when rotated around their respective centers of mass. The disk has a moment of inertia of 1/2MR^2, the ring has a moment of inertia of MR^2, and the square can be calculated using the moment of inertia of a rod through its center and the parallel axis theorem. The moments of inertia can be ranked from greatest to least as follows: disk, ring, square.
  • #1
Momentum09
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Homework Statement



Consider 3 objects of equal masses but different shapes: a solid disk (radius R), a thin ring (radius R), and a thin hollow square (side 2R). The ring and the square are hollow and their perimeters carry all the mass, but the disk is solid and has uniform mass density over its whole area. Compare the three objects' moments of inertia when rotated around their respective centers of mass. Rank their moments of inertia from greatest to least.

Homework Equations



I know that for the disk, I = 1/2MR^2, and for the ring, MR^2.

The Attempt at a Solution



I don't know about the hollow square...can someone please give a hint? Thanks!
 
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  • #2
To do the square look up the moment of inertia of a rod through it's center and then use the parallel axis theorem. Four times.
 
  • #3
PS. If you are clever you don't really have to compute the moment of inertia of all of these things. Think about the definition of moment of inertia. The disks mass is inside of the radius of the ring. The rings mass is inside the square.
 
  • #4
I got it! Thank you so much!
 

1. What is the formula for calculating the moment of inertia of an object?

The formula for calculating the moment of inertia of an object is I = mr², where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation to the object.

2. What is the significance of moments of inertia in physics?

Moments of inertia play a crucial role in rotational motion and are used to measure an object's resistance to changes in its rotational motion. They are also important in determining an object's stability and its ability to maintain its orientation while rotating.

3. How does the distribution of mass affect an object's moment of inertia?

The distribution of mass in an object directly affects its moment of inertia. Objects with more mass concentrated farther from the axis of rotation have a larger moment of inertia, while objects with more mass closer to the axis have a smaller moment of inertia.

4. Can the moment of inertia of an object change?

Yes, the moment of inertia of an object can change if its mass or distribution of mass changes. For example, if an object's mass is moved farther from the axis of rotation, its moment of inertia will increase.

5. How do you calculate the moment of inertia for complex objects?

For complex objects with irregular shapes, the moment of inertia can be calculated by breaking the object down into smaller, simpler shapes and using the parallel axis theorem to determine the moment of inertia for each individual shape. These values can then be added together to calculate the total moment of inertia for the complex object.

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