Center of mass - applications of integrals, Mathematics

Assignment Help:

Center of Mass - Applications of integrals

In this part we are going to find out the center of mass or centroid of a thin plate along with uniform density ρ. The center of mass or also centroid of a region is the point where the region will be perfectly balanced horizontally whether suspended from that point.

Thus, let's assume that the plate is the region bounded by the two curves f(x) and g(x) on the interval [a,b]. Thus, we wish to find the center of mass of the region below.

1517_Center of Mass - Applications of integrals.png

We'll first require the mass of this plate. The mass is,

M = ρ (area of plate)

= ρ ∫ba f (x) - g (x) dx

Next we'll require the moments of the region. There are two types of moments, represent by Mx and My. The moments calculate the tendency of the region to rotate about the x and y-axis correspondingly. The moments are illustrated by,

Equations of Moments

Mx= ρ ∫ba ½([f (x)]2 - [g (x)]2) dx

My = ρ ∫ba x (f (x) - g (x)) dx

After that the coordinates of the center of mass, (?x,? y) are as follow:

Center of Mass Coordinates

X‾ = My / M = ∫ba x(f (x) - g(x)) dx / ∫ba f (x) - g (x)) dx

= 1/A ∫ba x (f (x) - g (x)) dx

y‾ = Mx / M = ∫ba 1/2 ([f (x)]2 - [g(x)]2) dx / ∫ba f (x) - g (x)) dx

= 1/A ∫ba ½ ([f (x)]2 - [g (x)]2) dx

Where

A = ∫ba f (x) - g (x) dx


Related Discussions:- Center of mass - applications of integrals

Implementation of kruskal algorithm, You are required to implement Kruskal'...

You are required to implement Kruskal's algorithm for finding a Minimum Spanning Tree of Graph.  This will require implementing : A Graph Data Type (including a display meth

the system by graphing, Suppose you are in the market for a new home and a...

Suppose you are in the market for a new home and are interested in a new housing community under construction in a another city. a) The sales representative later shows that there

Find the area of shaded region, Find the area of shaded region, if the side...

Find the area of shaded region, if the side of square is 28cm and radius of the sector is ½ the length of side of square.

Write 3.5 × 104 in decimal notation, Write 3.5 × 104 in decimal notation? ...

Write 3.5 × 104 in decimal notation? Move the decimal point 4 places to the right to get 35,000.

probability that the card is a 8 or an ace, A standard deck of cards conta...

A standard deck of cards contains 52 cards. One card is selected at random. Determine a)    The probability that the card is a 8 or an Ace? b)    The probability that the card is

Defining real numbers, The numbers used to measure quantities such as lengt...

The numbers used to measure quantities such as length, area, volume, body temperature, GNP, growth rate etc. are called real numbers. Another definition of real numbers us

Find probabilities for the standard normal distribution, Q. Find Probabilit...

Q. Find Probabilities for the Standard Normal Distribution? Ans. Suppose the history teacher decides to distribute the final grades of his class with a normal distribution

Mathematical science, state tha different types of models used in operation...

state tha different types of models used in operations research.

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd