Composite Laminas - Rectangle with Circle Removed

Basic Concepts

A lamina is a flat or thin object with uniform thickness. It’s often assumed to be perfectly flat and rigid in physics problems.
A composite lamina is a shape made up by adding or subtracting basic shapes like rectangles, circles, triangles, etc.
In this case, the composite lamina is a rectangle with a circle removed from it.

The centre of mass of a composite lamina is the point at which the entire weight of the object can be assumed to act.
When a rectangle has a circle removed from it, we find the centre of mass by considering the original rectangular lamina, the circular lamina removed, and then subtracting one from the other.

For a rectangle, the centre of mass lies at the intersection of diagonals. For a circle, it lies at the centre of the circle.
To calculate the centre of mass of the composite shape, we first find the separate mass moments for the rectangle and the circle and then subtract, using the formula: [(mass of rectangle x position of its centre) - (mass of circle x position of its centre)] / (mass of rectangle - mass of circle).
The mass moment of a lamina about a certain axis is the product of its mass and the perpendicular distance of its centre of mass from the axis.

Since we’re assuming that the lamina has uniform thickness and consistent material, it thus has a uniform density.
This means that if a shape within the lamina is scaled up or down, its mass will increase or decrease proportionally to the area.

The principles about composite laminas apply to both theoretical problems and real-world applications like engineering and design.
For example, when designing an object with a hole in it (like a window frame), the overall balance and stability can be affected by the position and size of the hole, which change the centre of mass and hence the object’s dynamics.
Calculating the centre of mass can help to determine how the composite lamina will behave under forces, such as gravity or pressure, and how it can be balanced, pivoted, or rotated.

In further implications, the effects of external forces might also be accounted, such as gravitational pull, buoyancy, magnetic fields, etc.
Moreover, the non-uniform density might also be considered, which would require integration to calculate the mass and centre of mass more accurately.