The precast concrete stairs at the home I recently purchased do not have a footing. Lack of a footing coupled with drainage issues has caused them to sink over time. I'm going to raise them and get them properly supported underneath.
To do that I want to clamp them near their center of gravity (along the length), so that I can jack them up back to where they belong and minimize the risk of breaking the concrete (I'll continuously add temporary support underneath them as I slowly jack them up).
I have dug out the bottom of the steps and determined that they are monolithically poured and solid.
What is the measurement along the length of the steps where the center of gravity will be?
Edit: This is not homework. Here are the stairs in question:
You can see how they are pulling away from the foundation and have rotated forward and dropped. The sidewalk slab in front of it has also sunk and needs to be raised, so it is worse than it seems in this photo.
I attempted to solve this by hand because it seems trivial. I'm glad I asked, because the solution I came up with was wrong. I assumed I could reduce it to working with the ratio of cross sections which are simply three rectangles (since they all have the same width).
If I take the area of each step individually I get:
$$\begin{align} 17.5 \times 18 &= 315 \\ 14 \times 11 &= 154 \\ 47.5 \times 3 &= 142.5 \end{align}$$
I noticed that if I add the bottom two steps together, they are a smaller combined area (and thus volume) than the large step [pretty sure this is where I went wrong ... when you multiply by width this is no longer the case]. So it appeared to me at the time that the COG would be somewhere in the large step. I'll denote the COG as $c$.
So I then used this to give me $c$:
$$18 \times c = 18 \times (17.5 - c) + 154 + 142.5$$
That worked out to 16.98 (17 inches for all practical purposes).
However, after seeing @Chris Johns solution, I realize that what I had in my brain is not correct, and anybody with sinking stairs should ignore everything above.