# Center of gravity/moment of inertia

I am doing a mini hovercraft project. Regarding my design, I plan to place the fan (LIFT) in the middle of a rectangular styrofoam board (15 inches by 8inches). The position of my fan would be (x=7.5, y=4in); this would also be my center of gravity. My origin (coordinate) is at the corner of the styrofoam board. My thrust fan will be placed closer to the origin.

I would like to ask if I do need to include the mass of the styrofoam board (where I will placed all my components)? If yes, what would be its position from the origin.

Another thing, concerning the moment of inertial, I know to get it for a rectangular plane (uniform volume and mass), it is I=1/12 (length^2+width^2) . But with the components to be placed with different masses and volumes, how will I get the moment of inertia? I decided my center of gravity is at the middle of the styrofoam.

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Assuming your question concerns including the mass of the styrofoam when working out the moment of inertia, then yes, you should include it. You can consider the entire mass of the board to be located at its centre of mass, which is at the centre of the board since it is uniform.

The key thing to remember for your second question is that moments of inertia are additive about a common axis. That is to say, if you have a set of objects each with a different moment of inertia about some axis, then the total moment of inertia about that same axis is the sum of each individual moment.

, then the total moment of inertia about that same axis is the sum of each individual moment.
I need more explanation on this part

For example, I would attached only a thrust fan, position is (x=7.5 ,y=2 in), at the back of the rectangular styrofoam board. Lets assume the fan is a rectangular plane as well.
I wanted to get the moment of inertia. So, how will I compute it?

So itf the formula is : total moment of inertia = moment of board on its own axis+ moment of fan on its own axis

My attempt:

total moment of inertia= 1/12 (length of board^2+width of board^2) + 1/12 (length of fan^2+width of fan^2)

I am not sure so the position of the fan relative to the center of mass of the sytro board would not matter for moment of inertia

total moment of inertia = moment of board on its own axis+ moment of fan on its own axis
Not on its own axis. Moments of inertia are only additive about a common axis. So you can't just add the moments of inertia of the two objects about their own axes, you have to pick an axis (wherever you want) and then add the moments of inertia of both objects around that axis.