How to find moments and center of mass for a given shape using integration?

In summary, the conversation was about calculating the moments Mx and My and the center of mass of a lamina with density ρ = 4 and a given shape, which is a function of f(x) = 1/3x. The formula for finding Mx, My, and (x,y) was provided in a website and the person asking for help was hoping for a proof as their book did not provide one. They eventually found the answer by looking up the formula.
  • #1
tnutty
326
1
Not really a homework problem but related. In one of my homework problem, it says ,
Calculate the moments Mx and My and the center of mass of a lamina with density ρ = 4 and the given shape.
Mx = 2
My = 12
(x, y) = (2,1/3 )

I got the answer. YOUR WELCOME.


But, I only got it because I looked up the formula to find Mx,My and (x,y).
the formula is given in this site near the intro http://www.math.utep.edu/Faculty/javila/Calculus%20II/Math%201312%20problems%20for%20section%207.6.pdf

But I was hoping one of you could prove this, since my book didn't.
 
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  • #2
What is the given shape?
 
  • #3
it is a function of f(x) = 1/3x.
 

Related to How to find moments and center of mass for a given shape using integration?

What is the purpose of using integration in scientific applications?

Integration is used in scientific applications to find the area under a curve, which can represent physical quantities such as displacement, velocity, acceleration, and force. This allows scientists to analyze and understand complex systems and phenomena.

What are the different methods of integration used in scientific applications?

The most commonly used methods of integration in scientific applications are the Riemann sum, trapezoidal rule, and Simpson's rule. These methods approximate the area under a curve by dividing it into smaller, simpler shapes and summing their individual areas.

How is integration used in physics and engineering?

In physics and engineering, integration is used to calculate work, power, and energy. It is also used in the study of motion, where the area under a velocity-time graph represents displacement, and the area under an acceleration-time graph represents change in velocity.

What are some real-world examples of using integration in scientific applications?

Integration is used in real-world examples such as calculating the amount of medicine in a patient's bloodstream over time, determining the amount of energy produced by a power plant, and analyzing stock market trends.

What are the limitations of integration in scientific applications?

One major limitation of integration is that it can only provide an approximate value for the area under a curve, as it relies on dividing the curve into smaller, simpler shapes. Additionally, integration may not be possible for all functions, and some functions may require complex techniques to integrate accurately.

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