The problem presented asks to calculate the center of mass (centroid) of composite figures made from line segments and sections.

Problem 1: Composite Line Sections

The diagram consists of three line segments:

• $l_1 = 20$
• $l_2 = 20$
• $l_3 = 10$

Additionally, the first line segment $l_1$ makes an angle of $60^\circ$ with the horizontal.

To find the coordinates of the centroid $(x_g, y_g)$, the formulas used are:

Centroid in the x-direction:

$x_g = \frac{\sum (x_i \cdot l_i)}{\sum l_i}$

Centroid in the y-direction:

$y_g = \frac{\sum (y_i \cdot l_i)}{\sum l_i}$

Where $x_i$ and $y_i$ are the distances from the origin to the centroids of each line segment, and $l_i$ are the lengths.

• For $l_1$, since it forms an angle of $60^\circ$, the centroid in the x-direction will be influenced by $\cos(60^\circ)$ and in the y-direction by $\sin(60^\circ)$.
• For $l_2$ and $l_3$, they are aligned along the x and y directions.

Problem 2: T-Shaped Object

The second diagram consists of two rectangular sections:

• A vertical rectangle with dimensions $20 \times 10$
• A horizontal rectangle with dimensions $80 \times 10$

To find the center of mass (centroid) for this composite area:

1. Split the figure into two rectangles.
2. Calculate the individual centroids of each rectangle.
3. Apply the weighted average formula for the centroid in both the x and y directions.

Would you like to go into more detail for these calculations? Or should I proceed with solving one of these examples?

Related Questions:

1. What are the typical steps to find the centroid of a composite object?
2. How does the angle affect the centroid calculation in Problem 1?
3. What is the role of weighted averages in centroid calculations?
4. Could we calculate the moment of inertia once the centroid is known?
5. How do centroids help in structural engineering design?

Tip: Always break complex shapes into simpler ones to ease centroid calculations.