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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.

pulley system under equilibrium

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  • $\begingroup$ 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. $\endgroup$
    – AJN
    Commented Nov 13, 2022 at 1:04
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    $\begingroup$ 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. $\endgroup$
    – AJN
    Commented Nov 13, 2022 at 1:07

1 Answer 1

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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 $G_1, R_1, and \ G_2, R_2.$

Then for the system to be in equilibrium we have:

$$M_2\cdot R_2 = M_1\cdot R_1 \rightarrow 20M_1*R_2 =M_1*R_1 $$

$$R_1=20R_2$$

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  • $\begingroup$ Thanks. Was trying to avoid getting into angular momentum and hence introduced gears but it still looks like I am back to square one. Can you please say what would be the result if the two gears had equal radius and thus gear G1 had to rotate x-times more than G2? Would the answer be 20 again? $\endgroup$
    – Ravindra HV
    Commented Nov 13, 2022 at 7:38
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    $\begingroup$ Any other ratio would not be in equilibrium and would create an acceleration of the system. $\endgroup$
    – kamran
    Commented Nov 13, 2022 at 8:56
  • $\begingroup$ Thanks. But that point would not hold if it involved multiple rotations of gear G2 for every turn of G1 … right ? $\endgroup$
    – Ravindra HV
    Commented Nov 13, 2022 at 17:47
  • $\begingroup$ As shown in my answer you can perturb the system and it will start to turn with constant speed. the P2 will move with 1/20 speed of P1 in opposite direction. if no friction. $\endgroup$
    – kamran
    Commented Nov 13, 2022 at 21:54

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