How Do You Prove the Angular Velocity in Rigid Body Motion?

In summary, the problem involves a thin, homogeneous plate with principal moments of inertia I1, I2, and I3. The origins of two coordinate systems coincide at the center of mass of the plate. At time t=0, the plate is rotating with angular velocity Q about an inclined axis, perpendicular to the x2-axis. The angular velocity about the x2-axis at time t is given by w2(t)=Qcosa*tanh(Qtsina). The problem involves calculating the kinetic energy and square of the angular momentum in terms of the initial conditions, and using the given solution to show that a complete solution for w1 and w3 can be derived, which satisfy the Euler equations.
  • #1
ohsoconfused
1
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Homework Statement


Consider a thin homogeneous plate with principal momenta of inertia
I1 along axis x1,
I2>I1 along x2,
I3 = I1 + I2 along x3

Let the origins of the x and x' systems coincide at the center of mass of the plate. At time t=0, the plate is set rotatint in a force-free manner with angular velocity Q about an axis inclined at an angle of a from the plane of the plate and perpendicular to the x2-axis. If I1/I2 = cos2a, show at time t the angular velocity about the x2-axis is:

w2(t) = Qcosa*tanh(Qtsina).

The Attempt at a Solution


I know that the kinetic energy and the square of the angular momentum are constant, but I'm not positive how to calculate them with the initial conditions given. Past that, the Euler equations simplify somewhat...

(for simplicity's sake, let [x] = the first time derivative of x)

[w2] = w3w1
[w1] = -w2w3
[w3] = w1w2*(I1-I2)/(I1+I2) = w1w2*(cos2a-1)/(cos2q+1)

I'm having trouble seeing how such a bizarre function can even arise, but I think my first problem is calculating KE and L squared in terms of the initial conditions, which I'm blank on.

Anybody help? :frown:
 
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  • #2
ohsoconfused said:

Homework Statement


Consider a thin homogeneous plate with principal momenta of inertia
I1 along axis x1,
I2>I1 along x2,
I3 = I1 + I2 along x3

Let the origins of the x and x' systems coincide at the center of mass of the plate. At time t=0, the plate is set rotatint in a force-free manner with angular velocity Q about an axis inclined at an angle of a from the plane of the plate and perpendicular to the x2-axis. If I1/I2 = cos2a, show at time t the angular velocity about the x2-axis is:

w2(t) = Qcosa*tanh(Qtsina).

The Attempt at a Solution


I know that the kinetic energy and the square of the angular momentum are constant, but I'm not positive how to calculate them with the initial conditions given.

The things I put in red give you a complete geometrical description of the rotation vector at t=0, expressed in the frame (x1,x2,x3) fixed to the rotating body (which is exactly what you need for Euler's equations).
Draw it or something: you'll find the initial values for the 3 components (for instance, you already know that the second component is 0, given that the rotation vector is perpendicular to it). Once you know those, you can fill it in in the expressions for E and M^2.

Past that, the Euler equations simplify somewhat...

(for simplicity's sake, let [x] = the first time derivative of x)

[w2] = w3w1
[w1] = -w2w3
[w3] = w1w2*(I1-I2)/(I1+I2) = w1w2*(cos2a-1)/(cos2q+1)

I'm having trouble seeing how such a bizarre function can even arise, but I think my first problem is calculating KE and L squared in terms of the initial conditions, which I'm blank on.

Suggestion:
If I understand the problem statement well, they GIVE you the solution and you simply have to show that it is correct - you don't need to derive it. So use the solution they give you, to show that you obtain a complete solution for w1 and w3 (derivable from the conservation of M2 and E and given w2), which satisfy the Euler equation.
 

Related to How Do You Prove the Angular Velocity in Rigid Body Motion?

1. What is rigid body motion?

Rigid body motion is the movement of an object as a whole without any deformation or change in shape. This means that all points on the object move the same distance and in the same direction.

2. What are the three types of rigid body motion?

The three types of rigid body motion are translation, rotation, and a combination of translation and rotation. Translation is when an object moves in a straight line, rotation is when the object rotates around a fixed axis, and the combination of translation and rotation is when an object moves in a curved path while also rotating.

3. How is rigid body motion proven?

Rigid body motion can be proven using mathematical principles such as Newton's laws of motion and the concept of center of mass. By applying these principles and using equations, it can be shown that the motion of a rigid body follows certain rules and can be predicted and analyzed.

4. What are some real-world examples of rigid body motion?

Examples of rigid body motion can be seen in everyday life, such as a car moving in a straight line, a pendulum swinging back and forth, or a spinning top rotating around its center. Rigid body motion is also evident in larger scale motions, such as the rotation of the Earth around its axis.

5. How is rigid body motion important in engineering and design?

Rigid body motion is crucial in engineering and design as it helps to understand and predict the behavior of objects and structures. It is used in fields such as mechanical engineering, aerospace engineering, and robotics to design and analyze the motion of machines and structures. By understanding rigid body motion, engineers can ensure that their designs are stable and efficient.

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