8.3 Challenging Integral Applications

Physics & Engineering Applications: Overview

Hydrostatic Pressure & Force

where is the density, is the depth below the water surface,

and water directly above the plate is

Therefore,

, where the area of the th strip

Pressure is Force per unit area. Thus,

. From here

Water density or

At any point, liquid pressure is same in all directions. Pressure is not constant but increases as depth

increases.

Moments & Center of Mass

Center of Mass: Finding a point on a place of any shape that balances the plate horizontally

, is the total mass of the system,

Sum of individual moments is:

, called the moment of the system about the origin (one

dimensional line, not (x,y) coordinate on 2 dimensional plane).

Moment about the -axis:

, represents tendency to rotate about the -axis

Moment about the -axis:

, represents the tendency to rotate about the -axis

Center of mass coordinate:

Center of mass also called Centroid of plane :

From

,

From

,

Then,

and

Location of the center of mass is independent of density .

If the region lies between two functions and :

and

## Document Summary

Hydrostatic pressure & force where is the density, is the depth below the water surface, and water directly above the plate is. , where the area of the th strip. At any point, liquid pressure is same in all directions. Pressure is not constant but increases as depth increases. Center of mass: finding a point on a place of any shape that balances the plate horizontally. , is the total mass of the system, . Sum of individual moments is: dimensional line, not (x,y) coordinate on 2 dimensional plane). Center of mass coordinate: ( ) . Center of mass also called centroid of plane : , called the moment of the system about the origin (one. , represents tendency to rotate about the -axis. , represents the tendency to rotate about the -axis. From ( ) , ( ) - ( ) , ( Location of the center of mass is independent of density . and .