The rotating wheels problem?

Question

As shown in figure a smaller wheel of radius $r_1$ and bigger wheel of radius $r_2$ are joined by a string.
When both of them are rotating such that the string does not slip over them then it is obvious that smaller wheel will rotate faster than bigger wheel.
If the bigger wheel is rotating with an angular velocity $\omega$ then what will be the angular velocity of smaller wheel in ideal situation?
Also, can we construct an infinite wheel system as shown below,
It is obvious that angular velocity of each wheel is greater than the previous then it will become very fast as we continue to make it large then what prevents us from creating such a system of wheels.

Answer 1

$\omega=\frac vr$

$\implies w\propto\frac1r$

So as the radius decreases by a factor of $a$, the angular velocity increases by $a$

so

$\omega_2 = a\cdot\omega_1$

Answer 2

The linear speed of the belt is constant, so that

$$\omega_1r_1=\omega_2r_2.$$

This gives you the angular speed ratio for a pair of wheels.

If you concatenate many wheels, the ratio will indeed become huge, and so will the torque required to set the system in motion, to fight inertia, or simply friction. If you manage to produce enough torque, the first belts will slip or break.

Answer 3

For each set of two wheels, assuming massless wheels,

$$ \omega_1r_1 = \omega_2r_2 $$

Supposing identical sets and concatenating them

$$ \omega_1 r_1 = \omega_2 r_2\\ \omega_2 r_1 = \omega_3 r_2\\ \omega_3 r_1 = \omega_4 r_2\\ \cdots $$

then

$$ \omega_n = \left(\frac{r_1}{r_2}\right)^{n-1}\omega_1 $$

Assuming no friction or losses of any kind, we have that the setup is a transformer in the sense that the input energy is equal to the output energy

$$ \tau_n\omega_n = \tau_1\omega_1 $$

with $\tau_1,\tau_n$ the input and output torques respectively. We are considering a finite $n$ In the case of $n\to \infty$ the input should be locked without movement due to the impossibility of infinite energy generation from a finite effort.