Calculate the moment of inertia of this body

In summary, the moment of inertia for a cube built out of 12 rods can be calculated by using the parallel axis theorem and considering the perpendicular distance of the 8 rods from the axis passing through the midpoint of the cube. The remaining 4 rods, treated as cylinders, contribute to the overall moment of inertia by neglecting the radius. The direction of the rotational axis does not affect the calculation as all axes have the same moment of inertia due to the symmetry of the cube.
  • #1
leynat
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1
Homework Statement
12 thin, homogenous rods, each of mass m and length l are welded together at the endpoints so they form a cube. Calculate the moment of inertia of this body with respect to an axis through its midpoint.
Relevant Equations
I = I0 + md^2
I use the moment of inertia I = 1/12ml2 for an axis perpendicular and passing through the center of mass of a rod.

In a cube built out of 12 rods I have 8 rods at a perpendicular distance l/2 from the axis through the midpoint of a cube. These 8 rods contribute the moment of inertia I1 = 8(1/12ml2 + m(l/2)2) according to the parallel axis theorem:
I = I0 + md2

What about the 4 remaining vertical rods? They are parallel to the axis passing through the midpoint of a cube. If I consider them cylinders then they contribute 4(1/2mr2+(m(l/2)2) to the overall moment of inertia? And by neglecting r2 I get 4(m(l/2)2)?
 
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  • #2
leynat said:
What about the 4 remaining vertical rods? They are parallel to the axis passing through the midpoint of a cube. If I consider them cylinders then they contribute 4(1/2mr2+(m(l/2)2) to the overall moment of inertia? And by neglecting r2 I get 4(m(l/2)2)?
A.f.a.i.k., correct. The problem may be in what you did not clearly stated the direction of rotational axis. You can pull a multiple axes through "midpoint of the cube"
 
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  • #3
trurle said:
A.f.a.i.k., correct. The problem may be in what you did not clearly stated the direction of rotational axis. You can pull a multiple axes through "midpoint of the cube"
Thank you. Obviously, I didn't make it clear but I meant the axis perpendicular to the base and passing through the midpoint.
 
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  • #4
leynat said:
4(1/2mr2+(m(l/2)2)
How far are those 4 rods from the axis?
 
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  • #5
trurle said:
A.f.a.i.k., correct. The problem may be in what you did not clearly stated the direction of rotational axis. You can pull a multiple axes through "midpoint of the cube"
This actually does not matter. As can be argued by the symmetry of the cube, the moments of inertia around all axes are the same. You can therefore choose your axis in such a way that the moment of inertia becomes easy to compute.
 
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Related to Calculate the moment of inertia of this body

What is the moment of inertia?

The moment of inertia of a body is a measure of its resistance to rotational motion. It is the sum of the products of each particle's mass and its distance from the axis of rotation squared.

How is the moment of inertia calculated?

The moment of inertia is calculated by integrating the mass of each particle in the body with respect to its distance from the axis of rotation squared. The equation is I = ∫r^2dm.

What factors affect the moment of inertia?

The moment of inertia is affected by the mass and distribution of mass within the body, as well as the distance of the mass from the axis of rotation. Objects with larger mass and mass distributed farther from the axis have a greater moment of inertia.

Why is calculating the moment of inertia important?

The moment of inertia is an important concept in rotational motion as it helps determine an object's rotational kinetic energy and angular acceleration. It is also used in designing and analyzing rotating machinery and structures.

Are there different methods for calculating moment of inertia?

Yes, there are different methods for calculating moment of inertia depending on the shape and distribution of mass in the body. Some common methods include using the parallel axis theorem or the perpendicular axis theorem to calculate the moment of inertia for more complex shapes.

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