Principle Axes and Euler's Equation

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SUMMARY

The discussion focuses on calculating the force on bearings supporting a flat rectangular plate of mass M and dimensions a and 2a, rotating with angular velocity w about an axle through its diagonal corners. The correct force is derived as F = ma w² / (10√5). Participants emphasize the importance of using Euler's equations to relate angular velocities and moments of inertia, specifically by determining the moments of inertia in the system of the three principal axes to compute angular momentum L(t) and the resulting torque.

PREREQUISITES
  • Understanding of Euler's equations in rotational dynamics
  • Knowledge of moments of inertia for different geometries
  • Familiarity with angular momentum concepts
  • Ability to apply principles of torque in mechanical systems
NEXT STEPS
  • Study the derivation of moments of inertia for various shapes, particularly rectangular plates
  • Learn how to apply Euler's equations to dynamic systems
  • Explore the relationship between angular momentum and torque in rotational motion
  • Investigate the use of principal axes in calculating inertia tensors
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Students and professionals in mechanical engineering, physics enthusiasts, and anyone involved in the analysis of rotational dynamics and mechanical systems.

Ed Quanta
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A flat rectangular plate of Mass M and sides a and 2a rotates with angular velocity w about an axle through two diagonal corners. The bearings supporting the plate are mounted just at the corners. Find the force on each bearing.

I am not sure how to find force using Euler's equations since they just relate angular velocities and moments of inertia. The answer is supposed to be F=maw^2/10*sqrt5. Anyone know how I use the 3 principle axes to solve this?
 
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Yes, you relate angular velocity and moment of inertia, but that's the point! Because now you can find the m. of in. as a function of time, L(t). From that, you compute dL/dt which is equal to the exterior torque produced by the two forces on the bearings.

Bruno
 
Sorry, I've made a mess with moments of inertia/angular momentum. Anyway, my answer remains more or less valid: Find the moments of inertia in the system of the three princple axes to get the tensor of inertia Î, then L=Î*omega, and... <look above>
 

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