Chapter 6: Applications of Double Integration
Section 6.5: First Moments
Example 6.5.8
Calculate the coordinates of the center of mass of the lamina whose shape is that of an isosceles right triangle with legs of length 1, and whose density is twice the square of the distance from the vertex of the right angle. Hint: Put the right angle at the origin.
Solution
Mathematical Solution
Figure 6.5.8(a) shows the lamina in red, the center of mass (green dot), and the density as a surface in blue. The relevant calculations are
=
Figure 6.5.8(a) CM, lamina, and
Maple Solution - Interactive
Obtain the equation of the hypotenuse
Tools≻Load Package: Student Precalculus
Loading Student:-Precalculus
Context Panel: Student Precalculus≻ Lines And Segments≻Line≻Equation
Table 6.5.8(a) contains a solution via the task template that implements the CenterOfMass command from the Student MultivariateCalculus package. The given density is .
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Center of Mass≻Cartesian 2-D
Center of Mass for Planar Region in Cartesian Coordinates
Density:
Region:
MomentsMass:
Inert Integral -
Explicit values for and
Plot:
Table 6.5.8(a) Solution by Task Template
A solution from first principles is detailed in Table 6.5.8(b).
Define the density function
Context Panel: Assign Name
Obtain , the total mass of the lamina
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻
Obtain , the total moments about the -axis
Obtain
Table 6.5.8(b) Calculation of the center of mass from first principles
Maple Solution - Coded
Initialize
Define the density function .
Obtain the equation of the hypotenuse of the right triangle
Obtain the total mass of the lamina
Display the unevaluated integral with the Int command, and evaluate the integral with the value command.
Obtain the first moments and
Obtain the coordinates of the center of mass
Implement the relevant arithmetic.
In Table 6.5.8(c), the coordinates of the center of mass are calculated using the CenterOfMass command from the Student MultivariateCalculus package.
Obtain the inert integrals defining the center of mass
Obtain a graph of the region , the density , and the center of mass
Table 6.5.8(c) Coordinates of the center of mass calculated with the CenterOfMass command
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