Kinematics Mechanics Find the length of the belt and the speed of the rack

Question

For A. I know that the formula for belt is simply $L=\frac{Pi(D_a+D_b)}{2}+2C+\frac{(D_b-D_a)^2}{4C}$

Which gives me $L= 116.99"$ since C is equal to $50$

For B. However I'm stuck and can't get the speed of the rack, I think it moves to right based on my inspection.

So I got stuck after solving rpm of Pulley B which is $40$rpm from from velocity ratio of $3:1$ from $(N_aD_a)=(N_bD_b)$ now i know that shaft C is turned 3 times for every 2 times of B. My problem is I need to get the diameter of gear B to relate it to Gear C that drives the same as gear 2.

Please include your formulas and show your solution.

Answer 1

You don't need the diameters. The text says "C turns 3 times for every 2 turns of B." Thus knowing the rotation speed of B you can calculate the rotation speed of C. The rotation speed $f$ is the amount of revolutions $N$ per time interval $\Delta t$:

$$f=\frac{N}{\Delta t}$$

The ratio between two rotation speeds of two connected gears 1 and 2 is the same as the ratio between the amount of revolutions:

$$\frac{f_1}{f_2} = \frac{N_1}{\Delta t} \cdot \frac{\Delta t}{N_2} = \frac{N_1}{N_2}$$