MAT136H1 Lecture Notes - Centroid

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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  

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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 .

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