# Finding the moment of inertia and center of gravity of a curved shape [closed]

I have to complete an assignment in which I need to calculate the moment applied to a curved shape and I can't seem to figure out how to do just that.

The image above is the experiment we had to do. The top bar (in grey) has a counterweight on the far right side, which we had to adjust before starting manipulations to make sure that the orange part was balanced (when not balanced, the bottom part of the shape hits the bottom of the container).

We then had to add water up to around 150mm and start putting weights on the left side (Wc) until it was balanced again. Then, we noted the height of the water and the weight that we put on the left side (in grams). We then removed a little bit of water and repeated that 5 times with the water above the flat part, and 5 times with the water not completely submerging the flat part.

It is mentioned in the problem that the sum of the moments is equal to 0 and that the only forces that have an impact on the moment are Wc and R.

Now, what we have to do is calculate the moment caused by "R" using two methods and compare the theory and the experiment.

The experiment part works (I think). All we had to do was calculate M = Wc * Lc, where Lc is $0.275m$ as shown on the figure. So, for instance, an height of 156mm and a mass of $450g$ is $0.45 * 9.81 * 0.275$ = $1.214.$

The part where I'm stuck is the theoretical part... We have to calculate the moment like this:

$M = R * yc$

Where R = rho * g * center of gravity * A (let's assume rho is 1000 kg/m3 at 10 degrees celsius, center of gravity is "y bar" and A is area). yc is shown on the figure and is the height at which R is applied.

I have tried several ways to solve this, but when I try with my sample values ($h = 156mm$ and $mc = 450g$), my values are way off (between $24.3%$ to $74%$ off).

My first guess was to try R as $\sqrt{(1000 * 9.81 * (h/2) * h * 0.074)² + (1000 * 9.81 * volume above the curve)²}$ and $y$c as $(h/2) + 2/3 * h$ since the surface can be projected as a rectangle, but that is obviously wrong.

I have tried several other ways to solve this, but I guess I must looking over some easy stuff and I am completely lost now and have no idea how to continue. All other ways to solve this have given me similar or worse results.

I also know that I have to show the details of how I solved this for one case with the water above the flat part and one with the water not completely submerging it, so I'm also guessing I'm missing something crucial as I don't know what the difference would be in the formulas.

To summarize:

• Is the center of mass really $\frac{h}{2}$ even with the water above the flat part
• Is the area $h * 0.074?$
• How would I go about calculating $yc$ and other values that I haven't calculated properly from what I have written in the question?
• Hi, welcome to Physics SE. When typing your next questions use MathJax if you don't know the syntax here is a tutorial meta.math.stackexchange.com/questions/5020/… – Sumant Mar 3 '17 at 2:23
• Unclear what you are asking. Where in the post is moment of inertia mentioned? Are you confusing area moment with MMOI? Remember MMOI is used when you have angular momentum. In this setup you are talking about balance of moments (torques) which oddly has nothing to do with mass moment of inertia. – John Alexiou Mar 3 '17 at 5:57