# Kinematics Mechanics Find the length of the belt and the speed of the rack. [closed] 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.

## closed as off-topic by colormegone, Claude Leibovici, choco_addicted, user91500, WatsonJun 24 '16 at 7:43

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## 1 Answer

Not exactly clear the direction of rotation for shaft A. It looks to me that the arrow on the figure shows that the belt is moving downwards. For axes A to F I am going to be an observer looking from the right. Than A is rotating counterclockwise. I will call this direction +. B will rotate in the same direction, and you already found out the rotation speed is 40 rpm. Axis C will rotate in the - direction, and the angular velocity is 3/2*40=60rpm. You don't care about the diameter.

Axis D (the one between gears 2 and 3) will rotate in the + direction, and then axis F will rotate in the negative direction. The beveled gear will then rotate shaft G counterclockwise, so the rack will move to the left.

Now we are left to decide what is the angular velocities of the rest of the gears/shafts. We now that gear 2 will rotate at 60 rpm. The linear velocities of the three axes (C, D, F) will be the same. We do not know exactly what they are, but we assume that the circular pitch is the same. Now we get $N_C D_2 = N_D D_D = N_F D_3$. We ignore the middle axis, since that is just use to make shafts C and F rotate in the same direction. We do not know What $D_2$ and $D_3$ are, but we can multiply both sides with $\pi$. This will give the circumference, which is just the pitch multiplied by the number of teeth. Since the pitch is the same on both sides, we can divide by it, and we obtain $N_C 26 = N_F 76$, so $N_F= 60*26/76$rpm$\approx20$rpm. By applying the same reasoning, shaft G rotates 10 times slower than shaft F. Now you know the pitch and the number of teeth for gear 4, so you can calculate the radius, and then the linear velocity at the edge.