Thin Plates with Constant Density (Calculus II)

In summary, the conversation is about finding the center of mass of a thin plate with constant density over a given region bounded by a parabola and a line. The equations and attempted solution are discussed, with a correction made for an algebra mistake. The question of whether the formulas used are a certified way to find the center of mass is also posed.
  • #1
dm41nes
4
0
Thank you in advance for the help!

Homework Statement



Find the center of mass of a thin plate of constant density (delta) covering the given region.
The region bounded by the parabola y= x - x2 and the line y= -x



Homework Equations



See attachment question 15 p1

The Attempt at a Solution



See attachment question 15 p2
 

Attachments

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  • #2
I like your last solution on attachment 2 except...
Integral from 0 to 2 (x-x^2-x) dx is not Int (2x - x^2) dx
I think you just have an algebra mistake there.
 
  • #3
Thank you, well it was x-x^2-(-x). So, I added the double negative to the other x. Thats how I was able to get 2x.


Are these forumulas a certified way to find the center of mass?
 

1. What are thin plates with constant density?

Thin plates with constant density are flat, two-dimensional objects with a uniform mass distribution throughout. They can be thought of as infinitely thin sheets of material with a specific density.

2. How is calculus used in studying thin plates with constant density?

Calculus is used to model and analyze the behavior of thin plates with constant density. Specifically, techniques such as integration, differentiation, and the application of the fundamental theorem of calculus are used to calculate properties such as mass, center of mass, and moments of inertia.

3. What is the equation for calculating the mass of a thin plate with constant density?

The equation for calculating the mass of a thin plate with constant density is M = D * A, where M is the mass, D is the density, and A is the area of the plate. This equation is derived by integrating the density function over the area of the plate.

4. How is the center of mass of a thin plate with constant density calculated?

The center of mass of a thin plate with constant density is calculated by finding the weighted average of the x and y coordinates of the plate. This can be done using the equations x-bar = (1/M)*∫x*dA and y-bar = (1/M)*∫y*dA, where M is the mass of the plate and x and y are the coordinates of each point on the plate. These integrals can be evaluated using calculus techniques.

5. What is the significance of the moment of inertia for thin plates with constant density?

The moment of inertia for thin plates with constant density is a measure of the resistance of the plate to rotational motion. It is calculated using the equation I = ∫r^2*dA, where r is the distance from the axis of rotation to each point on the plate. This property is important in engineering and physics applications, as it helps determine the stability and motion of objects.

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