Moment of inertia for inclined plates

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SUMMARY

The moment of inertia (MOI) for a rectangular plate inclined at an angle to its axis passing through the centroid can be calculated by projecting the plate along the plane perpendicular to the axis of rotation. The MOI of this projection can then be determined using standard techniques. It is essential to recognize that the moment of inertia is a tensor quantity, and the transformation law differs from scalar quantities. When the product of inertia is zero, the formula I = Ixx cos²(θ) + Iyy sin²(θ) applies, highlighting the importance of the axis orientation in calculations.

PREREQUISITES
  • Understanding of moment of inertia and its tensor nature
  • Familiarity with projection techniques in mechanics
  • Knowledge of angular momentum and its relation to moment of inertia
  • Basic skills in calculus for direct integration methods
NEXT STEPS
  • Study the transformation laws for tensor quantities in mechanics
  • Learn about the implications of non-principal axes on angular momentum calculations
  • Explore advanced integration techniques for calculating moment of inertia
  • Investigate applications of moment of inertia in rotational dynamics
USEFUL FOR

This discussion is beneficial for mechanical engineers, physics students, and anyone involved in rotational dynamics or structural analysis of inclined plates.

abhisheak
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What is the moment of inertia of a rectangular plate making an angle except right angle or multiple of right angle with it's axis which is passing through the centroid of the plate?
 
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Just take the projection of the rectangular plate along the plane perpendicular to the axis of rotation. And now find the MOI of the projection using regular techniques. That will give you the MOI.

This works because displacing any component of a system of particles , parallel to the axis , cannot change the MOI.
 
do it in ordinary way by direct integration.you should know that the quantity is a tensor and transformation law for it is different.if product of inertia is zero then

I=Ixxcos^2(theta)+Iyysin^2(theta)
 
Thanks for your answers, i shall work on it ...
 
Keep in mind that you are dealing with an axis that isn't a principal axis of that body. So if you try to find angular momentum using L=Iω, for example, you'll get an incorrect result. It is adequate for computing angular acceleration from torque or rotational kinetic energy, however.
 
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could anyone please work this for me?
 
please guide me in doing this ... got mad in finding out
 

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