Wheels and Crazy Spinning Things!!!

dazedandconfused

For the illustration of the following model, go to http://www.corniceventures.com/images/wheels.JPG

If the 2 wheels ("B1" and "B2") are the same diameter (40 inches) and are rolling forward at 4 miles per hour...Would belts "D1" and "D2" rotate hubs "C1" and "C2" respectively at the same rate (rpm's)?

Here are all the parameters:

"B1" and "B2" Diameter >> 40 inches
"A" Diameter >> 12 inches
"C1" and "C2" Diameter >> 1 inch

Here is what my calculations told me:

"B2" Circumference >> ~125.66 inches
"A" Circumference >> ~37.70 inches
"C1" and "C2" Circumference >> ~3.14 inches
"B2" RPMs at 4 MPH >> ~33.61 rpm
"A" RPMs at 4 MPH >> ~112.05 rpm
"C1" and "C2" RPMs at 4 MPH >> ~1344.54 rpm

To me it seems that belt "D2" would rotate hub "C2" more quickly than belt "D1" would rotate hub "C1" given the same forward speed, but maybe I should trust my calculations! Any help would be appreciated, thanks in advance!

jamesrc

Both big wheels spin with ω = 4mph/40in. (convert that into rpm)

Assuming the belts don't slip:
ω_A*r_A = ω_C1*r_C1

and

ω_B2*r_B2 = ω_C2*r_C2

Note that ω_A = ω_B2 = ω

Rearrange to find one hub speed in terms of the other:

[latex] \omega_{C1} = \frac{r_A}{r_{C1}}\,\frac{r_{C2}}{r_{B2}}\omega_{C2} [/latex]

and since the hubs (C1 & C2) are the same size:

[latex] \omega_{C1} = \frac{r_A}{r_{B2}}\omega_{C2} [/latex]

So the hubs spin at different rates; since the radius (diameter) of B2 is greater than A, Hub C2 will spin faster than hub C1. So, your intuition was good, but there was some error in your calculations (I can't say where you went wrong without seeing your work). You should follow this line of thought out yourself to understand the idea and to double-check the algebra.