Determining the balancing force for a system of unequal masses for given scenario (angular-momentum, moment-of-inertia, thought-experiment)

Have already asked this on physics-site but has been closed. As such trying here!

This question is basically a thought experiment.

Let's say there are two pulley's P1 and P2 having equal radius R and negligible mass MP, arranged side-by-side (parallel axis of rotation).

They are connected by a gears G1 and G2 with an unknown ratio. The presence of gears effectively results in pulley P1 having a higher RPM than P2.

If the mass M2 is 20 times the mass of M1, then what is the ratio of the gears that would result in the system being in equilibrium?

If additional values would help, then we can have the pulley radius R to be 5m and Mass M1 to be 5kg (and therefore we have M2 to be 100kg).

To elaborate further, I have tried to map a system involving rotation to a system that is more linear in nature. Specifically in a rotational system, I would have to consider moment-of-inertia. However the equivalent of moment-of-inertia is not as clear in a scenario involving linear motion.

Below is an illustration with pulley's (yellow), gears (orange) and the masses(green) for clarity.

• Since you want to calculate the condition for equilibrium, the linear and angular velocity and acceleration can be assumed as zero. In such case, moment of inertia wouldn't appear in the equations.
– AJN
Nov 13, 2022 at 1:04
• Since the radius of the pulleys are the same, I wouldn't be surprised if the gear ratio turned out to be in the same proportion as the mass ratio.
– AJN
Nov 13, 2022 at 1:07

Let's assume the size of the two pulleys is the same ( if not then we have to multiply the answer by their ratio) and call the radius of $$G_1, R_1, and \ G_2, R_2.$$
$$M_2\cdot R_2 = M_1\cdot R_1 \rightarrow 20M_1*R_2 =M_1*R_1$$
$$R_1=20R_2$$