Throughout the FTC community there is a major gap in understanding how the different lifts work, cascade and continuous. The main consensus that other teams have found is that one must have one motor on a continuous lift and two motors on a cascade. This is not very informative so we decided to investigate further into this problem. Also, if we are able to create the forces acting on the lift we could find out the most optimal setup for the lift regarding the gear ratio and motor.

We first began by drawing what the lift mechanism look like and how we understood it to work. Once this was done we began to describe the forces first. We assumed that there is no friction and it is just half of the system in order to make the math a little bit simpler.

$$F_2 = g(m + \frac{M}{2})$$
$$F_1 = 2F_2 + mg = (3m+M)g$$
$$F_0 = 2F_0 + mg = (7m+2M)g$$
We found that each pulley had to withstand a force that is twice the amount that is being applied by each string for each stage. After we found the forces we were able to find the distance that each stage travelled and found that the first stage travelled the same amount as the pulled string but each distance is doubled as the lift ascends.

Once we found the forces we were able to create a general formula for the energies and we set the two formulas equal and they canceled out.

We were able to make a general formula for the forces that the lift was experiencing and model the distance that each slide travelled. We then checked the work by creating a general formula for the energies and setting them equal to each other. We could then use this information to find the best gear ratio for a certain amount of load so we can have the most efficient lift possible.

$$E = Fd = mad$$
$$E_t = g(m+\frac{M}{2}) 4x = gx_0(2M + 7m)$$
$$F_t = gx_0(2M+7m)$$

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