Lamina

Definition

A lamina is a two-dimensional flat plate or region with mass that is distributed across its area. It is often used in problems involving moments and centers of mass to calculate properties like the centroid.

5 Must Know Facts For Your Next Test

1. The density function $\sigma(x, y)$ describes how mass is distributed over the lamina.
2. The moment about the x-axis for a lamina can be found using $M_x = \iint_R y \sigma(x, y) \, dA$.
3. The moment about the y-axis for a lamina can be found using $M_y = \iint_R x \sigma(x, y) \, dA$.
4. The center of mass (centroid) $(\bar{x}, \bar{y})$ of a lamina can be calculated using $\bar{x} = \frac{M_y}{m}$ and $\bar{y} = \frac{M_x}{m}$ where $m$ is the total mass.
5. For uniform density $\sigma$, the centroid simplifies to geometric centroid calculations.

Review Questions

• How do you calculate the moment about the x-axis for a given lamina?
• What formula would you use to find the center of mass of a lamina?
• Explain how the density function affects the calculation of moments for a lamina.

Related terms

Moment: A measure of the tendency of one or more forces to cause rotation about a point or axis.

Center of Mass: The point in an object where all its mass can be considered to be concentrated.

Density Function: \textit{(denoted by }\sigma(x,y)\textit{)} A function that describes how mass is distributed over an area or volume.