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Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):

m 1 = 5 kg, placed at ( −2 , −3 ) , m 2 = 3 kg, placed at ( 2 , 3 ) , m 3 = 2 kg, placed at ( −3 , −2 ) .

Find the center of mass of the system.

( −1 , −1 ) m

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Center of mass of thin plates

So far we have looked at systems of point masses on a line and in a plane. Now, instead of having the mass of a system concentrated at discrete points, we want to look at systems in which the mass of the system is distributed continuously across a thin sheet of material. For our purposes, we assume the sheet is thin enough that it can be treated as if it is two-dimensional. Such a sheet is called a lamina    . Next we develop techniques to find the center of mass of a lamina. In this section, we also assume the density of the lamina is constant.

Laminas are often represented by a two-dimensional region in a plane. The geometric center of such a region is called its centroid    . Since we have assumed the density of the lamina is constant, the center of mass of the lamina depends only on the shape of the corresponding region in the plane; it does not depend on the density. In this case, the center of mass of the lamina corresponds to the centroid of the delineated region in the plane. As with systems of point masses, we need to find the total mass of the lamina, as well as the moments of the lamina with respect to the x - and y -axes.

We first consider a lamina in the shape of a rectangle. Recall that the center of mass of a lamina is the point where the lamina balances. For a rectangle, that point is both the horizontal and vertical center of the rectangle. Based on this understanding, it is clear that the center of mass of a rectangular lamina is the point where the diagonals intersect, which is a result of the symmetry principle    , and it is stated here without proof.

The symmetry principle

If a region R is symmetric about a line l , then the centroid of R lies on l .

Let’s turn to more general laminas. Suppose we have a lamina bounded above by the graph of a continuous function f ( x ) , below by the x -axis, and on the left and right by the lines x = a and x = b , respectively, as shown in the following figure.

This image is a graph of y=f(x). It is in the first quadrant. Under the curve is a shaded region labeled “R”. The shaded region is bounded to the left at x=a and to the right at x=b.
A region in the plane representing a lamina.

As with systems of point masses, to find the center of mass of the lamina, we need to find the total mass of the lamina, as well as the moments of the lamina with respect to the x - and y -axes. As we have done many times before, we approximate these quantities by partitioning the interval [ a , b ] and constructing rectangles.

For i = 0 , 1 , 2 ,… , n , let P = { x i } be a regular partition of [ a , b ] . Recall that we can choose any point within the interval [ x i 1 , x i ] as our x i * . In this case, we want x i * to be the x -coordinate of the centroid of our rectangles. Thus, for i = 1 , 2 ,… , n , we select x i * [ x i 1 , x i ] such that x i * is the midpoint of the interval. That is, x i * = ( x i 1 + x i ) / 2 . Now, for i = 1 , 2 ,… , n , construct a rectangle of height f ( x i * ) on [ x i 1 , x i ] . The center of mass of this rectangle is ( x i * , ( f ( x i * ) ) / 2 ) , as shown in the following figure.

This figure is a graph of the curve labeled f(x). It is in the first quadrant. Under the curve and above the x-axis there is a vertical shaded rectangle. the height of the rectangle is labeled f(xsubi). Also, xsubi = f(xsubi/2).
A representative rectangle of the lamina.

Questions & Answers

what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
Practice Key Terms 6

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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