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Looking at the following question: Question The part of the solution I am interested in is as follows enter image description here

Since the tangential velocities are the same, we could use equation 3 to solve for the angular acceleration of gear B. What I'm wondering is, why are the tangential accelerations of the gears at point P the same? I can't seem to wrap my head around this part since they are both obviously spinning at different rates.

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  • $\begingroup$ I can't seem to wrap my head around this part since they are both obviously spinning at different rates. ... think in terms of number of teeth per second ... are the two rotational rates still different? $\endgroup$
    – jsotola
    Sep 9, 2022 at 23:23
  • $\begingroup$ have you ever picked up two gears in your hands and meshed them? ... it adds a lot of understanding when you do that $\endgroup$
    – jsotola
    Sep 9, 2022 at 23:25

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The velocity at that point has to be the same as the teeth lock the two gears together. One cannot move without the other.

Of course, in the real world gears can move independently either due to them moving apart - shaft play etc or the teeth break off.

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  • $\begingroup$ The trick to thinking about this is to not overthink it. $\endgroup$
    – DKNguyen
    Sep 9, 2022 at 13:59
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Wouldn't it make sense that if the two gears had different tangential accelerations that there would be a conflict of forces? Think of it as two non-toothed disks contacting each other.

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As you said, both te gears rotate at different rates. At the same time it is important to understand the gears rotate with different "ANGULAR RATES" i.e., both of them rotate with different angular speeds. Since it is often confusing to think in terms of angular rates(angular speed) and linear rates(tangentaial speed/acceleration), think about it in a different way,

  1. Consider the point 'P' in the image where both the gear mesh with each other. It is clear that the teeth of the driving gear (Gear A) pushes the teeth of the griven gear (gear B). This means one teeth from the driving gear pushes only one teeth in the driven gear. So you can say that the number of teeth which is moving across the point 'P' per unit time is same for both gears.

  2. If you are aware of 'Universal law of gearing' you'll be knowing that in order for two gears to be in mesh, both the gears must have the same module (same teeth size).

Logically you can deduce that if the number of gears per unit time and the gear size is same for both the gears, the tangential acceleration (hence, tangential velocity) must be same for both the gears at the periphery (hence, at the point of meshing 'P' ).

Hope this helps.

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