Find the mass and center of mass of the lamina
Answers
Answer:
.The mass of a rectangular lamina with boundaries a≤x≤b a≤x≤b and c≤y≤d c≤y≤d is given by
M = ∫dc ∫ba ρ(x,y) dx dy .
M = ∫cd ∫ab ρ(x,y) dx dy .
The coordinates of the centroid ("center-of-mass" in physical applications) are found from
x¯¯¯ = ∫dc ∫ba x⋅ρ(x,y) dx dyM ,
x¯ = ∫cd ∫ab x⋅ρ(x,y) dx dyM ,
y¯¯¯ = ∫dc ∫ba y⋅ρ(x,y) dx dyM .
y¯ = ∫cd ∫ab y⋅ρ(x,y) dx dyM .
Note that since the density gets larger as we go toward the "upper-right" corner, (7,2) , (7,2) , we should expect the centroid to be "above and to the right" of the center of the rectangle at (72,1) . (72,1) . [Later, I'll mention a handy way to get the mass without integration when the density function is linear in each dimension...]
For the sake of guidance, I get M = 273 , x¯¯¯ = 1127273 , y¯¯¯ = 8753 ⋅ 273 . M = 273 , x¯ = 1127273 , y¯ = 8753 ⋅ 273 . (No one said the centroid coordinates would be pretty -- just rational...)
Answer:
The mass of a rectangular lamina with boundaries a≤x≤b a≤x≤b and c≤y≤d c≤y≤d is given by
M = ∫dc ∫ba ρ(x,y) dx dy .
M = ∫cd ∫ab ρ(x,y) dx dy .
The coordinates of the centroid ("center-of-mass" in physical applications) are found from
x¯¯¯ = ∫dc ∫ba x⋅ρ(x,y) dx dyM ,
x¯ = ∫cd ∫ab x⋅ρ(x,y) dx dyM ,
y¯¯¯ = ∫dc ∫ba y⋅ρ(x,y) dx dyM .
y¯ = ∫cd ∫ab y⋅ρ(x,y) dx dyM .
Note that since the density gets larger as we go toward the "upper-right" corner, (7,2) , (7,2) , we should expect the centroid to be "above and to the right" of the center of the rectangle at (72,1) . (72,1) . [Later, I'll mention a handy way to get the mass without integration when the density function is linear in each dimension...]
For the sake of guidance, I get M = 273 , x¯¯¯ = 1127273 , y¯¯¯ = 8753 ⋅ 273 . M = 273 , x¯ = 1127273 , y¯ = 8753 ⋅ 273 . (No one said the centroid coordinates would be pretty -- just rational...)
EDIT: The "short-cut" I was holding off mentioning is that when linear functions are involved, some calculations can be greatly simplified because of the way some integrals work. The density function in this problem, ρ(x,y)=3x+4y+5 ρ(x,y)=3x+4y+5 , is linear in both the x− x− and y− y− directions. In this situation, the mass of the lamina is just the density at the geometrical center of the region (not the centroid; the two only coincide for uniform density) times the area. For this problem, we thus find
M = A ⋅ ρ(xm,ym) = (b−a) (d−c)⋅ ρ(a+b2,c+d2)
M = A ⋅ ρ(xm,ym) = (b−a) (d−c)⋅ ρ(a+b2,c+d2)
= 7 ⋅ 2 ⋅ ρ(72,1) = 14 ⋅(3⋅72 + 4⋅1 + 5) = 14 ⋅ 19.5 = 273