# Euler's Equations, a freely rotating lamina

• Oijl
In summary, using Euler's equations and the results of Problems 10.23 and 10.30, it can be shown that the component of angular velocity in the plane of the lamina has a constant magnitude. This is demonstrated by showing that the time derivative of w1^2 + w2^2 is equal to zero, indicating that the angular velocities in the x and y directions are always opposite and cancel each other out. This holds true even if the lamina is not rotating about a principal axis.
Oijl

## Homework Statement

Consider a lamina rotating freely (no torques) about a point O of the lamina. Use Euler's equations to show that the component of $$\omega$$ in the plane of the lamina has constant magnitude.

[Hint: Use the reults of Problems 10.23 and 10.30. According to Problem 10.30, if you choose the direction e3 normal to the plane of the lamina, e3 points along a principal axis. Then what you have to prove is that the time derivative of $$\omega$$$$_{1}$$$$^{2}$$ + $$\omega$$$$_{2}$$$$^{2}$$ is zero.]

## Homework Equations

The result of 10.23 is that Izz = Ixx + Iyy
The result of 10.30 is that for a lamina rotating about a point O in the body, the axis through O and perpendicular to the plane is a principal axis.

Euler's equations, as given in my book, are
y1W1 - (y2 - y1)w2w3 = N1
y2W2 - (y3 - y1)w3w1 = N2
y3W3 - (y1 - y2)w1w2 = N3
where y is lambda, the eigenvalue, w is omega, W is omega dot, and N is the torque.

## The Attempt at a Solution

If the lamina is rotating freely, the N1=N2=N3=0.
If I choose e3 to be normal to the plane of the lamina, then e3 points along a principal axis, and that means that

w3 = constant,
so
W3 = 0.

Those last two statements I'm not sure about.

However, I still know that I would like to prove that the time derivative of w1^2 + w2^2 = 0. That is,

2w1W1 + 2w2W2 = 0

And solving Euler's equations for W1 and W2, with N1,2,3 = 0, you have

W1 = ((y2-y3)/y1)w2w3
W2 = ((y3-y1)/y2)w3w1

Plugging these into 2w1W1 + 2w2W2 = 0 and doing algebra, I found that, if 2w1W1 + 2w2W2 = 0 is true, then

y2^2 - y2y3 + y1y3 - y1^2 = 0.

Which doesn't seem to necessarily be true.

Help? Thank you!

I don't believe that you can assume that w3 is constant. The problem does not state that the lamina is rotating about a principal axis. Instead, use the result you quoted ($\lambda_{3}$=$\lambda_{1}$+$\lambda_{2}$) and use Euler's equations to show that

$\omega_{1}$$\omega_{1}^{.}$+$\omega_{2}$$\omega_{2}^{.}$=0

Also, if the lamina WERE rotating about its principal axis that is perpendicular to the surface then the angular velocity would always point along this direction in the absence of torques and$\omega_{1}$ and $\omega_{2}$ would be zero.

Last edited:
Oops. Those dots were supposed to be above omega one and omega two. I'm sure you get the idea though.

## 1. What are Euler's equations?

Euler's equations are a set of three differential equations that describe the motion of a rigid body that is freely rotating around a fixed point. They were first derived by Leonhard Euler in the 18th century and are widely used in physics and engineering.

## 2. What is a freely rotating lamina?

A freely rotating lamina is a flat, rigid object that is able to rotate freely around a fixed point without any external forces acting on it. It is often represented as a thin sheet or disc and is used as a simplified model for more complex rotating objects.

## 3. How are Euler's equations derived?

Euler's equations are derived using principles of classical mechanics, specifically the laws of motion and conservation of angular momentum. They can also be derived from the Euler-Lagrange equations, which are a formulation of the principle of least action.

## 4. What are the applications of Euler's equations?

Euler's equations have numerous applications in physics and engineering. They are used to study the motion of objects such as gyroscopes, satellites, and spinning tops. They are also used in the design and analysis of aircraft, spacecraft, and other rotating machinery.

## 5. Are there any limitations to Euler's equations?

While Euler's equations are a powerful tool for studying the motion of rigid bodies, they do have some limitations. They assume that the object is perfectly rigid and that there are no external forces acting on it. In reality, most objects are not perfectly rigid and are subject to external forces, so these assumptions may not always hold true.

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