Question

How to find frequency of rotational motion without knowing radius?

`v_1` =3m/s, `v_2` =2m/s, `r_2` =r-10cm

Answer 1

`omega = 10 (radians)/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 m/s` and `v_2=2 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 - 10cm`.

For rigid rotation, it holds for any point on the solid that
`omega = 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/(r_1 - 10 cm)`.

Now we can solve for `r_1` by multiplying both sides by the denominators
`v_1(r_1 - 10 cm)= v_2r_1`,
`r_1 = 10 cm v_1/(v_1-v_2) = 30 cm`.

Using our newfound knowledge of the radius `r_1`, we get that
`omega = v_1/r_1 = (3 m/s)/(30 cm) radians= (3 m/s)/(0.3 m) radians = 10 ((radians)/s)`.

Check that you get the same answer when using `v_2` and `r_2`.

Answer 2

Depends what you're asking.

Explanation:

It looks like there is a typo in your question. If what you are really saying is this: `r_2= r_color(red)(1) -10cm` then:

`omega = 3/r_1 = 2/(r_1 - 10) implies r_1 = 30 " cm" implies omega = 10 "rad/s"`