   Chapter 14.4, Problem 28E

Chapter
Section
Textbook Problem

Finding the Center of Mass Using TechnologyIn Exercises 25–28, use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. r = 1 + cos θ , ρ = k

To determine

To Calculate: The mass and center of mass of the lamina bounded by the graphs of the given equations.

Explanation

Given:

r=1+cosθ,ρ=k

Calculation:

First, sketch the graph as shown below,

Draw the reference axis in xy coordinates.

Then. plot the provided equations:

r=1+cosθ,0θ2π

 θ 0 π3 π2 π 4π3 2π R 2 1.5 1 0 -1.5 2

Further, mark the points on the plot and connect them.

Shade the region inside the graph as the area of interest is the area inscribed by the equations.

The final plot is shown below as:

Therefore, the mass of the lamina is:

m=02π01+cosθkrdrdθ=02π[kr22]01+cosθdθ=02πk2(1+2cosθ+cos2θ)dθ=k2[θ+2sinθ+1+cos2θ2]02πm=3kπ2

Now, to find the center of the mass calculate the moment of inertia about both the axes.

The moment of inertia about x-axis is:

Mx=02π01+cosθkr2sinθdrdθMx=02π[kr33]01+cosθsinθdθMx=02πk3(1+cosθ)3sinθdθMx=0

The moment of inertia about y-axis is:

My=02π01+cosθkr2

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