homework and exercises - Find center of mass and moment applied on beam structure

I have a simple physics problem to solve but it is giving me a slightly difficult time to figure out. The problem:

I have a beam structure with same cross section. It consists of three beam. First in vertical with length 1meter. Second beam joined on top with a angle of 20degrees to the right with lenth 1.1meter and on top of the second beam, a third beam parallel to the ground with 1meter length.

I need to find the center of gravity and how much moment I have on the base of the structure so I would add a mass that would be sufficient to avoid the structure to fall over.

any ideas how?

Answer

I need to find the center of gravity

Center of mass (CoM) is found by adding up all particles $i$:

$$\vec r_{cm}=\frac{\sum m_i \vec r_i}{\sum m_i}$$

$\vec r_{cm}$ is the coordinate vector of the CoM.

Since you have straight beams (assumingly cylinder shaped), start out with symmetry considerations for each of them. The CoM for each beam must be at the exact center (assuming constant density throughout). Put these three CoMs into the formula above to find the combined CoM of the whole structure.

A side note: If the gravitational acceleration $\vec g$ is constant throughout, then the CoM is equal to the center of gravity.

and how much moment i have on the base of the structure so I would add a mass that would be sufficient to avoid the structure to fall over.

Formula for torque about an axis (the tip of the base point):

$$\vec \tau=\vec r_{cm} \times \vec F$$

In your 2D-case simplified to:

$$\tau=r_{cm,\perp} F=r_{cm} F \sin \theta$$

where $r_{cm} \sin \theta$ must be $r_{cm}$'s horizontal component (since gravity $\vec F$ is vertical).

Find this torque of gravity on the structure, and then place an object on the base in whatever manner the problem allows so it gives the opposite torque equal in magnetude.