- #1

Leo Liu

- 344

- 145

- Homework Statement:
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- Relevant Equations:
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The "egg" initially spun around axis 1 with at ##\omega_s##. After being disturbed, it has started to possesses angular velocities along 2 and 3. The question is to find the rotational speed of ##\vec \omega=\vec\omega_1+\vec\omega_2+\vec\omega_3## to a fixed observer.

It is calculated that ##\omega_2\text{ & }\omega_3## fluctuates at a frequency of ##\lambda=\left|\frac{I_1-I_{\perp}}{I_{\perp}}\right|\omega_s##, where ##I_{\perp}## represents the inertia around 2 or 3. The equations of angular speed 2,3 are:

$$\begin{cases}\omega_2=\omega_\perp \cos\lambda t\\

\omega_3=\omega_\perp\sin\lambda t\end{cases}$$

And they are shown by the diagram below.

Then the author makes the following argument:

I am totally bewildered by this derivation. While it is dimensionally correct, how can one add angular frequency to angular velocity, and why does the sum give you the answer? Could someone explain the reasoning behind this?As the drawing shows, ω2 and ω3 are the components of a vector ω⊥ that rotates in the 2–3 plane at rate γ. An observer fixed to the body would see ω⊥ rotate relative to the body about the 1 axis at angular frequency γ. Since the 1, 2, 3 axes are fixed to the body and the body is rotating about the 1 axis at rate ωs, the rotational speed of ω⊥ relative to an observer fixed in space is $$\lambda+\omega_s=\frac{I_1\omega_s}{I_\perp}$$

(γ=##\lambda##)