To determine the center of gravity (CG) of the shaded area in this figure, we will treat the shaded area as a composite shape and find the CG using the principles of centroid calculation for composite areas.

Step-by-Step Breakdown:

1. Understanding the Composite Shape:

The shaded area consists of:

• A triangle $OBD$
• A rectangle $CDAB$
• The total width of the base is $40 \, \text{cm}$.
• The height of point $B$ is $30 \, \text{cm}$, and point $C$ is $26 \, \text{cm}$.

2. Coordinates of the Centroid for Each Shape:

To calculate the CG of the entire composite area, we need to break it down into simpler shapes, find their individual centroids, and then combine them.

• Triangle $OBD$:

  • The base $OB$ is 20 cm (height), and the base is 40 cm (length).
  • The centroid of a triangle is located at a distance of $\frac{1}{3}$ from the base and the height.
  • Therefore, the centroid for triangle $OBD$ is located at $\left(\frac{40}{3}, \frac{30}{3}\right) = \left(13.33 \, \text{cm}, 10 \, \text{cm} \right)$.

• Rectangle $CDAB$:

  • The centroid of the rectangle is located at its center, which is at $\left( \frac{40+40}{2}, \frac{20+26}{2} \right)$ or $(40, 23 \, \text{cm})$.

3. Finding the Area of Each Shape:

• Area of Triangle $OBD$: $A_{\text{triangle}} = \frac{1}{2} \times 40 \, \text{cm} \times 30 \, \text{cm} = 600 \, \text{cm}^2$
• Area of Rectangle $CDAB$: $A_{\text{rectangle}} = 40 \, \text{cm} \times 26 \, \text{cm} = 1040 \, \text{cm}^2$

4. Calculating the Centroid of the Shaded Area:

The centroid of the composite area can be found by: $x_{\text{CG}} = \frac{\sum (A_i \times x_i)}{\sum A_i}, \quad y_{\text{CG}} = \frac{\sum (A_i \times y_i)}{\sum A_i}$

Using the areas and centroids of each part:

• For the x-coordinate: $x_{\text{CG}} = \frac{(600 \times 13.33) + (1040 \times 40)}{600 + 1040}$ $x_{\text{CG}} = \frac{(7998) + (41600)}{1640} = \frac{49598}{1640} \approx 30.24 \, \text{cm}$

• For the y-coordinate: $y_{\text{CG}} = \frac{(600 \times 10) + (1040 \times 23)}{600 + 1040}$ $y_{\text{CG}} = \frac{(6000) + (23920)}{1640} = \frac{29920}{1640} \approx 18.24 \, \text{cm}$

Conclusion:

The center of gravity (CG) of the shaded area is located at approximately $(30.24 \, \text{cm}, 18.24 \, \text{cm})$.

Would you like further details on any of the steps?

Here are 5 related questions:

1. How do we calculate the centroid of a more complex shape?
2. What are the real-life applications of finding the centroid of a shape?
3. How does symmetry affect the position of the centroid?
4. How would the CG change if part of the shaded area was removed?
5. How does the principle of superposition apply when calculating centroids?

Tip: Always remember that for composite shapes, you break them into simpler shapes, calculate individual centroids and areas, and then find the overall centroid using a weighted average.