Gear Rotation puzzle is confusing 50:10 GR but 6 rotations

[sshanker]

Homework Statement

A gear A has 50 teeth and another B has 10 teeth, how many times does the small gear rotate around the big one? I thought 5 but its 6! Note: The gear is curved like 360 degrees.

Relevant Equations

Please help!

A gear A has 50 teeth and another B has 10 teeth, how many times does the small gear rotate around the big one? I thought 5 but its 6! Note: The gear is curved like 360 degrees.

 

[Orodruin]

How many times would it rotate going around if they had the same number of teeth?

 

[sshanker]

How many times would it rotate going around if they had the same number of teeth?

Once, Ofcourse!

 

[sshanker]

Once, Ofcourse!

So, Once 5+1? Hmm..kinda makes sense!

 

[Orodruin]

Once, Ofcourse!

Will it?

How many times will it rotate if it has 200000000000000 teeth and goes around a smaller gear with 10 teeth?

 

[DaveC426913]

Once, Ofcourse!

Try it.
The penny on the left has gone halfway around the penny on the right.
What angle of rotation through its own rotational axis did it experience?

 

[phyzguy]

This is similar to the problem of understanding the length of the sidereal day, which is how long the Earth takes to rotate once on its axis. There are 365.25 solar days in a year, meaning that we see the sun rise and set 365.25 times. But in the same time, the Earth rotates 366.25 times relative to the distant stars. So the time it takes the Earth to rotate once on its axis, instead of being 24 hours, is actually 24 hours * (365.25/366.25) = 23.934 hours = 23 hours 56 minutes.

 

[RPinPA]

Once, Ofcourse!

Do you know about the rotation of the moon? It keeps one face pointed toward the earth. But it is actually rotating. How many rotations does it do in one orbit?

Similar question: Suppose you and I are holding opposite ends of a rope and are circling around each other. Suppose we start out with me facing north as I face you (you are facing south). We go around 180 degrees, half an orbit. I'm now facing south as you face north. We've both rotated. We complete the circle and I am once again facing north and you facing south. Did we each rotate while we orbited? How much?

Also I strongly recommend you do the experimental test which @DaveC426913 is suggesting, with two coins of equal size. Then you'll see experimentally what people are hinting at.

 

[sshanker]

Do you know about the rotation of the moon? It keeps one face pointed toward the earth. But it is actually rotating. How many rotations does it do in one orbit?

Similar question: Suppose you and I are holding opposite ends of a rope and are circling around each other. Suppose we start out with me facing north as I face you (you are facing south). We go around 180 degrees, half an orbit. I'm now facing south as you face north. We've both rotated. We complete the circle and I am once again facing north and you facing south. Did we each rotate while we orbited? How much?

Also I strongly recommend you do the experimental test which @DaveC426913 is suggesting, with two coins of equal size. Then you'll see experimentally what people are hinting at.

Okay, I just tried the experiment with 2 one cent coins and found that thought there were no teeth to simulate it perfectly, the driver coin took almost exactly 2 rotations...

 

[sshanker]

Try it.
The penny on the left has gone halfway around the penny on the right.
What angle of rotation through its own rotational axis did it experience?

View attachment 253114

Think, I see the point...Thank you!

 

[sshanker]

Will it?

How many times will it rotate if it has 200000000000000 teeth and goes around a smaller gear with 10 teeth?

The larger gear would have passed 10 of its 200000000000000 teeth in one rotation I would assume!

 

[DaveC426913]

Think, I see the point...Thank you!

So, do you have an answer to Orodruin's post #2? And can you then extrapolate a more general rule that applies to 50:10 - or any other combination of gears?

 

[sshanker]

So, do you have an answer to Orodruin's post #2? And can you then extrapolate a more general rule that applies to 50:10 - or any other combination of gears?

To Orodruin's question, It's tricky, but my understanding as of now is that, any larger gear driving a smaller one,say 20 teeth gear driving a 10 teeth one would rotate exactly once, so a 40 teeth gear would rotate like 1/2 a rotation? and so on? If I'm right?

 

[DaveC426913]

To Orodruin's question, It's tricky, but my understanding as of now is that, any larger gear driving a smaller one,say 20 teeth gear driving a 10 teeth one would rotate exactly once, so a 40 teeth gear would rotate like 1/2 a rotation? and so on? If I'm right?

Well, that's only the one specious example - gears that have a 2:1 ratio - so it doesn't really help answer either post #2 or your OP question.

BTW, I think it might help to make the distinction between - and use the terms - driver gear and driven gear. That will help clarify the question as well as the answer.

Also, note the fact that the OP question, as stated, isn't asking about gear ratios of two rotating gears so much as it is asking about one gear rotating around a stationary gear.

 

[sshanker]

Well, that's only the one specious example - gears that have a 2:1 ratio - so it doesn't really help answer either post #2 or your OP question.

BTW, I think it might help to make the distinction between - and use the terms - driver gear and driven gear. That will help clarify the question as well as the answer.

Is there a physics concept that in central to this phenomenon? I'll try looking that up and see if that helps me in my understanding?

 

[Orodruin]

Look at it like this. If you have a gear with 10 teeth and there is a long row of teeth on the flat ground, how much would the gear rotate if it advanced 10 teeth? Note that the situation with another gear is different from this situation because in order to go one lap around, the gear that is moving must also rotate a bit extra (how much) to compensate for the fact that the other gear is also curved.

 

[Tom.G]

This problem taken to the extreme is the basis for a Planetary Gear set. Here is a 14 minute video that describes more than you want to know about them. Some of the visuals may help you to sort things out though.


Hope it helps!
Tom

 

[Cutter Ketch]

Suppose the little gear didn’t rotate at all relative to the big gear. Suppose it kept the same tooth pointed toward the big gear during the whole orbit. In going around the big gear the little gear has to rotate once relative to the room in order to not rotate at all relative to the big gear. This is the extra rotation. The number of rotations it takes to keep the gears meshed are relative to this “not rolling” condition so you always get one more rotation than required by the gear count.

 

[DaveC426913]

The penny on the left has gone halfway around the penny on the right.

I presume your answer was intended to address my post #6 about pennies, not sshanker's opening post #1.

Yes, the penny on the left has gone halfway around the penny on the right. That was a given.

The question was: what angle of rotation has the left penny undergone? (i.e. it starts with its face oriented upright - with text legible - but what orientation is its face when it ends? What angle has it rotated though? 90? 180? 360?) (Hint: the diagram in post 6 shows you the answer.)

 

[Andigles]

One thing to notice about this exercise it’s that the gear needs to be pointing “upward”, not from the big gear reference, but from the initial position reference. That’s how I could get it.

 

[etotheipi]

Old thread, but it is a fun little problem. When the tooth of the smaller gear that was initially in contact with the larger gear is next in contact with the larger gear (i.e. the centre of the smaller gear has traversed an angle $\theta = \frac{2\pi r}{R}$), the smaller gear has rotated about its centre by a little more than $2\pi$, specifically by an angle of $2\pi + \frac{2\pi r}{R} = 2\pi (1+ \frac{r}{R})$. The smaller gear returns to its initial position after $\frac{R}{r}$ of these motions, by which point it will have rotated about its centre by $2\pi (\frac{R}{r} + 1)$, i.e. it has completed $\frac{R}{r} + 1$ revolutions about its centre.

Alternatively, consider that the centre of the smaller gear has orbital angular velocity $\vec{\Omega}=\Omega \hat{z}$, whilst its spin angular velocity is $\vec{\omega} = \omega \hat{z}$. The no slip condition ensures that $$\vec{\Omega} \times (R+r)\hat{r} + \vec{\omega} \times (-r\hat{r}) = \vec{0} \implies \Omega (R + r) = r\omega$$That is to say that the spin angular velocity differs from the orbital angular velocity by a factor$$\omega = \frac{R+r}{r} \Omega$$For instance, if both gears have the same radius, then $\omega = 2\Omega$ and the smaller gear will complete two revolutions about its centre in the time its centre takes to complete one revolution about the larger gear!