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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 function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
But using L'Hopitals rule is x=1 is defined
Could U explain better boss?
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
can we apply l hospitals rule for function of two variables
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
I agree with Andrew
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
i mean can we replace the roles of x and y and call x=2 as function
if x =y and x = 800 what is y
Joys Reply
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The problem is the question, is not a problem where it is, but what it is
I think you should first know the basics man: PS
Yes, what factorization is
Antonio bro is x=2 a function?
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
Why y, if domain its usually defined as x, bro, so you creates confusion
Its f(x) =y=2 for every x
Yes but he said could you put x = 2 as a function you put y = 2 as a function
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
The said x=2 and that 2 is y
that 2 is not y, y is a variable 2 is a constant
So 2 is defined as f(x) =2
No y its constant =2
what variable does that function define
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Sorry x=2
And you are right, but os not a function of x, its a function of y
As function of x is meaningless, is not a finction
yeah you mean what I said in my first post, smh
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
OK you can call this "function" on a set {2}, but its a single value function, a constant
well as long as you got there eventually
2x^3+6xy-4y^2)^2 solve this
follow algebraic method. look under factoring numerator from Khan academy
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
It's an integral easy
V=1/3 h π (R^2+r2+ r*R(
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
is x=2 a function?
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
don ger it
what is a limit?
it is the value the function approaches as the input approaches that value.
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
is x=2 a function?
Practice Key Terms 6

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