Lamina

Definition

A lamina is a two-dimensional flat object with mass that has a density function defined over its surface. It is typically used to calculate moments and centers of mass in planar regions.

5 Must Know Facts For Your Next Test

1. A lamina's density can vary across its surface, often represented by a function $\sigma(x, y)$.
2. The mass of a lamina is calculated using double integrals over the region it occupies: $m = \iint_R \sigma(x, y) \, dA$.
3. The moments about the x-axis and y-axis are given by $M_x = \iint_R y \sigma(x, y) \, dA$ and $M_y = \iint_R x \sigma(x, y) \, dA$, respectively.
4. The center of mass $(\bar{x}, \bar{y})$ for a lamina is found using $\bar{x} = \frac{M_y}{m}$ and $\bar{y} = \frac{M_x}{m}$.
5. A homogeneous lamina has constant density, simplifying the calculations to geometric centroids.

Review Questions

• What is the formula for the mass of a lamina with varying density?
• How do you calculate the moment about the x-axis for a given lamina?
• What are the coordinates of the center of mass for a lamina?

Related terms

Density Function: A function that describes how mass is distributed over an area or volume, often denoted as $\sigma(x,y)$ for two-dimensional objects.

Moment:

The product of the distance from an axis and some quantity (e.g., force or mass); in this context, it refers to moments about the x or y axes.

Center of Mass:

The point at which the total mass of an object can be considered to be concentrated; calculated using moments and total mass.