To find the center of mass of the planar lamina shown, we will need to break the figure into simpler shapes (like rectangles), calculate the area and the coordinates of the centroid of each shape, and then use these to determine the overall center of mass of the composite shape.

Here are the steps involved:

1. Divide the shape into simpler components:

The lamina can be broken down into two rectangles:

• Top rectangle: width = $\frac{7}{8}$, height = 6
• Bottom rectangle: width = 2 (from $x = -1$ to $x = 1$), height = $\frac{7}{8}$

2. Find the centroids of each shape:

The centroid $(\bar{x}, \bar{y})$ of a rectangle can be found at the center of the rectangle.

• Top rectangle centroid:

  • $x_1 = 0$ (since it is centered at the origin)
  • $y_1 = 6 + \frac{\frac{7}{8}}{2} = 6.4375$ (since the centroid of the rectangle is at half the height of the rectangle)

• Bottom rectangle centroid:

  • $x_2 = 0$ (since it is also centered at the origin)
  • $y_2 = \frac{\frac{7}{8}}{2} = 0.4375$

3. Find the areas of each shape:

The area $A$ of a rectangle is width $\times$ height.

• Top rectangle area:
  • $A_1 = \frac{7}{8} \times 6 = 5.25$
• Bottom rectangle area:
  • $A_2 = 2 \times \frac{7}{8} = 1.75$

4. Calculate the center of mass:

The center of mass $(\bar{x}, \bar{y})$ of the composite shape is calculated using the weighted average of the centroids:

• For $\bar{x}$:
  $\bar{x} = \frac{x_1 A_1 + x_2 A_2}{A_1 + A_2}$ Since both centroids are centered at $x = 0$, we have: $\bar{x} = 0$

• For $\bar{y}$:
  $\bar{y} = \frac{y_1 A_1 + y_2 A_2}{A_1 + A_2}$ Substituting the values: $\bar{y} = \frac{6.4375 \times 5.25 + 0.4375 \times 1.75}{5.25 + 1.75}$ $\bar{y} = \frac{33.796875 + 0.765625}{7}$ $\bar{y} = \frac{34.5625}{7} \approx 4.9375$

5. Final Answer:

The center of mass is $(0, 4.937)$.

Would you like any further details or have questions about this calculation? Here are some related questions you could explore:

1. How do you find the centroid of more complex shapes?
2. What happens if the density of the material is not uniform?
3. Can you calculate the moment of inertia for this shape?
4. How does the centroid change if we rotate the shape?
5. How can this concept of center of mass be applied to three-dimensional objects?

Tip: Always break down complex shapes into simpler components to make centroid and center of mass calculations easier.