Monday, November 26, 2018

Calculus of a Single Variable, Chapter 7, 7.6, Section 7.6, Problem 26

First lets find the bounds of integration. When looking at the graph the furthest that the lamina is bounded on the y-axis is where the curves interest. Lets find those points.
y+2=y^2
0=(y+1)(y-2)
Therefore the y bounds are y=-1 and y=2. Then we will integrate between the furthest right curve (x=y) and the furthest left curve.
The center of Mass is:
(x_(cm),y_(cm))=(M_y/M, M_x/M)
Where the moments of mass are defined as:
M_x=int int_A rho(x,y)*y dy dx
M_y=int int_A rho(x,y)*x dy dx
The total mass is defined as:
M=int int_A rho(x,y)dy dx
First, lets find the total mass.
M=int^2_-1 [int^(x=y+2)_(x=y^2) rho dx] dy
M=rho int^2_-1 [(y+2)-(y^2)] dy
M=rho [(1/2)y^2+2y-(1/3)y^3]|^2_-1
M=9/2 rho
Now lets find the x moment of mass.
M=int^2_-1 y*[int^(x=y+2)_(x=y^2) rho dx] dy
M=rho int^2_-1 y*[(y+2)-(y^2)] dy
M=rho int^2_-1 (y^2+2y-y^3) dy
M=rho ((1/3)y^3+y^2-(1/4)y^4)|^2_-1
M_x=9/4 rho
Now the y moment of mass.
M=int^2_-1 [int^(x=y+2)_(x=y^2) rho x dx] dy
M=rho/2 int^2_-1 [x^2|^(y+2)_(y^2)] dy
M=rho/2 int^2_-1 (y^2+4y+4-y^4) dy
M=rho/2((1/3)y^3+2y^2+4y-(1/5)y^5)|^2_-1
M_y=36/5 rho
Therefore the center of mass is:
(x_(cm),y_(cm))=(M_y/M, M_x/M)=((36/5 rho)/(9/2 rho),(9/4 rho)/(9/2 rho))=(8/5,1/2)
The moments of inerita or the second moments of the lamina are:
I_x=int int_A rho(x,y)*y^2 dy dx
I_y=int int_A rho(x,y)*x^2 dy dx
I won't solve these integrals step by step since they are very similar to the others, but you will find that:
I_x=63/20 rho
I_y=423/28 rho

No comments:

Post a Comment

Why is the fact that the Americans are helping the Russians important?

In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...