Principle Axes and Euler's Equation

In summary, the conversation discusses finding the force on each bearing of a rotating flat rectangular plate with specific dimensions and angular velocity. The solution involves using Euler's equations to relate angular velocity and moment of inertia, and then finding the tensor of inertia and using it to calculate the external torque.
  • #1
Ed Quanta
297
0
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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  • #2
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
 
  • #3
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>
 

1. What are principle axes and why are they important in science?

Principle axes refer to the three orthogonal axes (x, y, and z) that define the orientation of a rigid body in three-dimensional space. These axes are important because they allow us to describe and understand the rotational motion of an object.

2. What is Euler's equation and how is it related to principle axes?

Euler's equation is a mathematical relationship that describes the motion of a rigid body in three-dimensional space. It relates the angular velocity of a body to its moment of inertia and the torques acting on it. It is closely related to principle axes because it is derived from the principle of conservation of angular momentum, which is based on the orientation of the principle axes.

3. How are principle axes and Euler's equation used in engineering and physics?

In engineering and physics, principle axes and Euler's equation are used to analyze and predict the behavior of rotating objects. They are particularly useful in fields such as mechanics, robotics, and aerospace engineering, where understanding rotational motion is essential for designing and controlling complex systems.

4. What is the significance of the moment of inertia in principle axes and Euler's equation?

The moment of inertia is a measure of an object's resistance to rotational motion. It plays a critical role in principle axes and Euler's equation, as it determines the rotational energy and angular momentum of a body. The distribution of mass around the principle axes affects the moment of inertia and, therefore, the rotational behavior of an object.

5. Can principle axes and Euler's equation be applied to non-rigid bodies?

No, principle axes and Euler's equation are specifically designed for rigid bodies, which have a fixed shape and do not deform under external forces. They cannot be applied to non-rigid bodies, such as fluids or gases, which do not have a distinct shape and can change their orientation under applied forces.

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