# How to find frequency of rotational motion without knowing radius?

Apr 2, 2017

$\omega = 10 \frac{r a \mathrm{di} a n s}{s}$

#### Explanation:

I assume that we are referring to a rigid object that rotates at a constant angular frequency $\omega$ around some axis of rotation.

We know the linear speeds ${v}_{1} = 3 \frac{m}{s}$ and ${v}_{2} = 2 \frac{m}{s}$ at two points $1$ and $2$ on the rotating solid, and we know that the radii (distance from the axis of rotation) ${r}_{1}$ and ${r}_{2}$ at these two points are related as ${r}_{2} = {r}_{1} - 10 c m$.

For rigid rotation, it holds for any point on the solid that
$\omega = \frac{v}{r}$, where $v$ is the linear speed at that point and $r$ is the distance from the axis of rotation.

Therefore we know that
$\omega = {v}_{1} / {r}_{1} = {v}_{2} / {r}_{2} = \omega$,
which gives that
${v}_{1} / {r}_{1} = {v}_{2} / \left({r}_{1} - 10 c m\right)$.

Now we can solve for ${r}_{1}$ by multiplying both sides by the denominators
${v}_{1} \left({r}_{1} - 10 c m\right) = {v}_{2} {r}_{1}$,
${r}_{1} = 10 c m {v}_{1} / \left({v}_{1} - {v}_{2}\right) = 30 c m$.

Using our newfound knowledge of the radius ${r}_{1}$, we get that
$\omega = {v}_{1} / {r}_{1} = \frac{3 \frac{m}{s}}{30 c m} r a \mathrm{di} a n s = \frac{3 \frac{m}{s}}{0.3 m} r a \mathrm{di} a n s = 10 \left(\frac{r a \mathrm{di} a n s}{s}\right)$.

Check that you get the same answer when using ${v}_{2}$ and ${r}_{2}$.

Apr 2, 2017

It looks like there is a typo in your question. If what you are really saying is this: ${r}_{2} = {r}_{\textcolor{red}{1}} - 10 c m$ then:
$\omega = \frac{3}{r} _ 1 = \frac{2}{{r}_{1} - 10} \implies {r}_{1} = 30 \text{ cm" implies omega = 10 "rad/s}$