Math Help - A Seemingly Simple Question Regarding the ratio of Two Gears

1. A Seemingly Simple Question Regarding the ratio of Two Gears

This is a problem posed by my pre-calc instructor. I feel like this is somehow rudimentary, but I can't find a solution. The problem goes as follows:

Two gears are connected such that rotation of the smaller wheel causes the larger wheel to rotate. Through how many degrees will the larger wheel rotate if the smaller wheel rotates through an angle of 60 degrees.

He does supply a small diagram of a small circle (circle A) with a radius of 2 inches next to a larger circle (B) with a radius of 5 inches.

So far I've worked out the circumferences of both circles, with circle A being approx. 12.6 inches and circle B being approx. 31.4 inches. I'm also able to find the arc length of the 60 degree angle on the smaller circle, since 60 degrees = pi/3 and the arc is s = radius x theta radians, so the arc of the angle on circle A is approx. 2.1 inches.

Ultimately I'm confused in my thinking, because it seems like it'd be a simple ratio problem like between two 60/90/30 triangles of varying size. Problem is, I can't verify that the 60 degree change in the rotation of circle A being equates to 60 degrees of change in circle B. If anyone out there can give me a hand, I'll be infinitely grateful.

2. Originally Posted by Sinetologist
This is a problem posed by my pre-calc instructor. I feel like this is somehow rudimentary, but I can't find a solution. The problem goes as follows:

Two gears are connected such that rotation of the smaller wheel causes the larger wheel to rotate. Through how many degrees will the larger wheel rotate if the smaller wheel rotates through an angle of 60 degrees.

He does supply a small diagram of a small circle (circle A) with a radius of 2 inches next to a larger circle (B) with a radius of 5 inches.

So far I've worked out the circumferences of both circles, with circle A being approx. 12.6 inches and circle B being approx. 31.4 inches. I'm also able to find the arc length of the 60 degree angle on the smaller circle, since 60 degrees = pi/3 and the arc is s = radius x theta radians, so the arc of the angle on circle A is approx. 2.1 inches.

Ultimately I'm confused in my thinking, because it seems like it'd be a simple ratio problem like between two 60/90/30 triangles of varying size. Problem is, I can't verify that the 60 degree change in the rotation of circle A being equates to 60 degrees of change in circle B. If anyone out there can give me a hand, I'll be infinitely grateful.
Ok I think you just confused yourself a bit. I'll just need to make a distinction here from basics to help you get back on track.

Imagine 2 gears (I'm sure you've already done so) with radius R and r respectively. For the gears to work effectively, they must come in contact at the same rate to fit together.

Thus the linear velocity of the two gears must be the same, which implies that in the set amount of time, a single teeth on each gear will travel the same distance. This distance is the arc length of the circle.

So using the formula for arclength, we find that $r\theta = R \phi$. Now for gears of two different radius, clearly the angles must be different. It's simply a matter of substituting an angle to get the other one, given the radius of both.

Hope this clears up your confusion.

3. I think I see what you're saying. By using $r\theta = R \phi$, then then $S = 2.1 inches$. This gives me $2.1/5 = \theta$, giving an output of .42 rads, which converted to degrees comes out to $.42 ( 180 / \pi ) = 24.06 degrees$. This seems like it checks out!

Interestingly enough, I got a somewhat similar figure by dividing the angle of circle A by the ratio of the two circumferences (it ended up looking like 60/ 2.49 rev of A to every 1 rev of B); it's not as accurate compared to the calculations above, but I wasn't sure whether or not to write it off as circumstantial before I got a response on this thread.

I can't thank you enough for setting me straight on this. I had a real hard time on this one, much harder than it should've been. I'll try and be more alert next time. Thanks again!

4. Originally Posted by Sinetologist
I think I see what you're saying. By using $r\theta = R \phi$, then then $S = 2.1 inches$. This gives me $2.1/5 = \theta$, giving an output of .42 rads, which converted to degrees comes out to $.42 ( 180 / \pi ) = 24.06 degrees$. This seems like it checks out!

Interestingly enough, I got a somewhat similar figure by dividing the angle of circle A by the ratio of the two circumferences (it ended up looking like 60/ 2.49 rev of A to every 1 rev of B); it's not as accurate compared to the calculations above, but I wasn't sure whether or not to write it off as circumstantial before I got a response on this thread.
Well the ratio of circumferences is simply the ratio of the arc lengths of the gears with angle $2\pi$

5. Originally Posted by Sinetologist
...(it ended up looking like 60/ 2.49 rev of A to every 1 rev of B); it's not as accurate compared to the calculations above, ....
pi(10) / pi(4) = 10/4 = 5/2 = 2.5