# Question regarding calculation of angular velocity connected to coordinate transformation

So i need to calculate the angular velocity of the dark disk, $$w^*_3$$. In the master solution they state it is : $$w^*_3 = w_3 + \overline{w_3}$$ with :

$$w_3 = \frac{|\dot{r}|}{4a}$$

$$|\dot{r}| = v_a + \overline{w_3}\cdot a$$

$$|\dot{r}| = v_b - \overline{w_3}\cdot 2a$$

$$\Rightarrow|\dot{r}| = 2a\cdot (w_S+w_R)$$

$$\overline{w_3} = \frac{1}{a}(|r|-v_1)=\frac{1}{2a}(v_b-|r|)=\frac{1}{3a}(v_b-v_a)$$

$$\Rightarrow \overline{w_3} = 2w_r-w_s$$

$$\Rightarrow w_3^* = \frac{5}{2}w_r-\frac{1}{2}w_s$$

So, the question I have is that if I understood correctly, $$w^*_3$$ is the angular velocity of the disk itself (in its coordinate system) if you look at it from the disks center. So shouldn't be $$\overline{w_3}$$ already be my $$w^*_3$$? Because it is the angular velocity with which my disk is actually spinning?(in it's coordinate system)

• No. For example if $w_s = 2w_r$ the disk will move in a circular path, but it will not be rotating about its own center. And if $w_s = 5w_r$ it will rotate about its own center but remain in a fixed position in space. – alephzero Aug 11 '19 at 11:58

## 1 Answer

$$\omega ^*_3$$ is the rotation rate of the disk in the frame pictured (equal to $$2 * pi *$$ frequency point $$P$$ will appear in the 9 o'clock position). It is the sum of the orbit rate of $$\omega _3$$'s disc center about the system center in the frame pictured ($$\omega_3$$), plus the rotation rate about it's own axis in a reference frame where this axis is fixed ($$\bar\omega_3$$). The first term considers the contact points with $$A$$ and $$B$$ fixed and the system rotating with disk $$\omega_3$$, so one turn of the center is one rotation of the disk, and point P makes one revolution. The second term considers $$\omega_3$$'s axis fixed and the contact points free to run.

If we take $$\omega_S$$ as given in each case, we get two different (linear) relations for $$\omega_R$$ and $$\omega_3$$. So $$\omega_R = C_1*\omega_S$$ and $$\bar\omega_R= C_2*\omega_S$$ This lets us set up a general solution $$\omega^*_3 = f(\omega_S,\omega_R)$$, where $$C_1$$ and $$C_2$$ are solved using $$\omega_S$$ and then they are used to compute $$\omega^*_3$$