Moment of inertia for thin uniform rectangle/hexagon

In summary, the problem involves a rectangle split into four triangles, with the long side labeled a, the short side labeled b, and the inner angles of the triangles labeled \theta_{1} and \theta_{2}. The equation given is I = (1/12)M(a^{2} + b^{2}), and the attempt at a solution involves proving that I = (1/12)M(a^{2}b^{2}) is the moment of inertia for the full mass. This can be done by writing down h and \theta in terms of a and b, using the fact that m(triangle) = (M/ab)*(area of triangle), and possibly utilizing the Parallel Axis theorem.
  • #1
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Homework Statement


the first problem is i have a rectangle split into four triangles the long side labeled a the short side labeled b and the inner angles of the triangles are [tex]\theta[/tex] [tex]_{1}[/tex] and [tex]\theta[/tex] [tex]_{2}[/tex]
(since the traingles are made from a rectangle their are 2 sets of triangles with different [tex]\theta[/tex])
also not [tex]\theta[/tex] is the angle towards the center of the mass not an outer ridge angle(all together 4 [tex]\theta[/tex] representing all 4 triangles)



Homework Equations




im given the eqaution
I=moment of inertia= [tex](1/12) [/tex]M(a[tex]^{2}[/tex] + b[tex]^{2}[/tex])
and also earlier in class worked out for isosceles trianlges
I=([tex]1/2)[/tex]M[1+([tex]1/3[/tex])tan[tex]^{2}[/tex]((1/2)[tex]\theta[/tex])]h[tex]^{2}[/tex]

The Attempt at a Solution


given those 2 equations i have to prove that I=[tex](1/12) [/tex]M(a[tex]^{2}[/tex]b[tex]^{2}[/tex]) really gives the moment of inertia for full mass.
I started working backwards on the problems replacing M with [tex]\sigma h^{2} tan((1/2)\theta[/tex])
being that [tex]\sigma = M/A [/tex] and A = h[tex]^{2}[/tex][tex]\ tan((1/2)\theta[/tex])
btw M=total mass of object, A=total area
(sorry trying my best to not confuse on problem)
since the height of one triangle is 1/2b or 1/2a depending on which triangle picked
i also replaced the terms for a[tex]^{2}[/tex] and b[tex]^{2}[/tex]

but I am at a lost at this point and don't know where to go past this
i figured if i reverse engineered the problem i may be able to figure the 4 triangles relationship to the total moment of inertia so then i could return the sumed up eqaution into the original eqaution for an isosceles triangle so i can understand how to use the triangles for a hexagon and octagon.

please help if possible with this problem.
 
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  • #2
When you specify a moment of inertia for an object, you need to also specify the axis about which the moment is calculated.

You basically need to write down h and [itex]\theta[/itex] in terms of the sides a,b, and use the fact that m(triangle) = (M/ab)*(area of triangle). If necessary you may need to use the Parallel Axis theorem to have all moments about the same point. Then it's just plugging and adding up the 4 moments to get the total.
 

1. What is moment of inertia for a thin uniform rectangle/hexagon?

The moment of inertia for a thin uniform rectangle/hexagon is a measure of its resistance to rotational motion. It is a property that describes how the mass of an object is distributed around its axis of rotation.

2. How is moment of inertia calculated for a thin uniform rectangle/hexagon?

The moment of inertia for a thin uniform rectangle/hexagon can be calculated using the formula I = (1/12) * m * (a^2 + b^2), where m is the mass of the object and a and b are the length and width of the rectangle/hexagon, respectively.

3. What is the unit of measurement for moment of inertia?

The unit of measurement for moment of inertia is kilogram square meter (kg*m^2).

4. How does the shape of a thin uniform rectangle/hexagon affect its moment of inertia?

The shape of a thin uniform rectangle/hexagon can greatly affect its moment of inertia. A longer and narrower shape (such as a rectangle) will have a larger moment of inertia compared to a shorter and wider shape (such as a hexagon) with the same mass.

5. Why is moment of inertia important in physics?

Moment of inertia is an important concept in physics because it helps us understand and predict the behavior of objects in rotational motion. It is used in various applications such as designing machines, calculating the stability of structures, and predicting the motion of celestial bodies.

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