angular speed question (sprocket assembly)

[messa]

Hello, I need help here! This is my problem:

The sprocket assembly for a bicycle is shown in the figure. If the sprocket of a radius r1 rotates through an angle of theta1 radians, find the corresponding angle of rotation for the sprocket of radius r2.

The figure shown is a picture of two wheel sprockets. The first one r1 is bigger than the second one r2. There are no numbers or anything so I don't know how to complete the problem. I would assume that I would have to use the equation for angular speed which is (2pi)(rotations per time unit). But I have no numbers, so how do I go about this??

 

[mark07]

Re: angular speed question

If the sprocket of a radius r1 rotates through an angle of theta1 radians, ...

then find the linear distance traveled: \(\displaystyle \L r_1 \theta_1\). In other words, you can think of it as the distance traveled by a bug sitting on the first sprocket.

Let's say the second one rotates through an angle of \(\displaystyle \L \theta_2\). Then the linear distance traveled is \(\displaystyle \L r_2 \theta_2\).

Since the sprockets are attached, they travel the same linear distance. Then,

\(\displaystyle \L r_1 \theta_1 = r_2 \theta_2\)

which gives

\(\displaystyle \L \theta_2 = \frac{r_1}{r_2} \theta_1\)

 

[messa]

okay, that makes sense but when you divide by r2 why isn't the theta1 divided as well: theta2=r1theta1/r2 ??

Is there a formula for this, or is it something you just have to work through?

 

[mark07]

okay, that makes sense but when you divide by r2 why isn't the theta1 divided as well: theta2=r1theta1/r2 ??

I hate to do this but I will answer your question with a question...

Isn't 5 times (2/3) same as 2 times (5/3) which is also ( (5*2) / 3 )??

Is there a formula for this, or is it something you just have to work through?

I can't really say anything without knowing how your textbook covers it. I simply used basic algebra/geometry approach without any physics in it (angular velocity etc). See the following tutorial,

http://www.mathwords.com/a/arc_circle.htm

 

[messa]

yeah, thanks. It's much easier to see with real numbers. That was probably a silly question!!
Thank you