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This idea is not limited just to two point masses. In general, if n masses, m 1 , m 2 ,… , m n , are placed on a number line at points x 1 , x 2 ,… , x n , respectively, then the center of mass of the system is given by

x = i = 1 n m i x i i = 1 n m i .

Center of mass of objects on a line

Let m 1 , m 2 ,… , m n be point masses placed on a number line at points x 1 , x 2 ,… , x n , respectively, and let m = i = 1 n m i denote the total mass of the system. Then, the moment of the system with respect to the origin is given by

M = i = 1 n m i x i

and the center of mass of the system is given by

x = M m .

We apply this theorem in the following example.

Finding the center of mass of objects along a line

Suppose four point masses are placed on a number line as follows:

m 1 = 30 kg, placed at x 1 = −2 m m 2 = 5 kg, placed at x 2 = 3 m m 3 = 10 kg, placed at x 3 = 6 m m 4 = 15 kg, placed at x 4 = −3 m .

Find the moment of the system with respect to the origin and find the center of mass of the system.

First, we need to calculate the moment of the system:

M = i = 1 4 m i x i = −60 + 15 + 60 45 = −30.

Now, to find the center of mass, we need the total mass of the system:

m = i = 1 4 m i = 30 + 5 + 10 + 15 = 60 kg .

Then we have

x = M m = −30 60 = 1 2 .

The center of mass is located 1/2 m to the left of the origin.

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Suppose four point masses are placed on a number line as follows:

m 1 = 12 kg, placed at x 1 = −4 m m 2 = 12 kg, placed at x 2 = 4 m m 3 = 30 kg, placed at x 3 = 2 m m 4 = 6 kg, placed at x 4 = −6 m .

Find the moment of the system with respect to the origin and find the center of mass of the system.

M = 24 , x = 2 5 m

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We can generalize this concept to find the center of mass of a system of point masses in a plane. Let m 1 be a point mass located at point ( x 1 , y 1 ) in the plane. Then the moment M x of the mass with respect to the x -axis is given by M x = m 1 y 1 . Similarly, the moment M y with respect to the y -axis is given by M y = m 1 x 1 . Notice that the x -coordinate of the point is used to calculate the moment with respect to the y -axis, and vice versa. The reason is that the x -coordinate gives the distance from the point mass to the y -axis, and the y -coordinate gives the distance to the x -axis (see the following figure).

This figure has the x and y axes labeled. There is a point in the first quadrant at (xsub1, ysub1). This point is labeled msub1.
Point mass m 1 is located at point ( x 1 , y 1 ) in the plane.

If we have several point masses in the xy -plane, we can use the moments with respect to the x - and y -axes to calculate the x - and y -coordinates of the center of mass of the system.

Center of mass of objects in a plane

Let m 1 , m 2 ,… , m n be point masses located in the xy -plane at points ( x 1 , y 1 ) , ( x 2 , y 2 ) ,… , ( x n , y n ) , respectively, and let m = i = 1 n m i denote the total mass of the system. Then the moments M x and M y of the system with respect to the x - and y -axes, respectively, are given by

M x = i = 1 n m i y i and M y = i = 1 n m i x i .

Also, the coordinates of the center of mass ( x , y ) of the system are

x = M y m and y = M x m .

The next example demonstrates how to apply this theorem.

Finding the center of mass of objects in a plane

Suppose three point masses are placed in the xy -plane as follows (assume coordinates are given in meters):

m 1 = 2 kg, placed at ( −1 , 3 ) , m 2 = 6 kg, placed at ( 1 , 1 ) , m 3 = 4 kg, placed at ( 2 , −2 ) .

Find the center of mass of the system.

First we calculate the total mass of the system:

m = i = 1 3 m i = 2 + 6 + 4 = 12 kg .

Next we find the moments with respect to the x - and y -axes:

M y = i = 1 3 m i x i = −2 + 6 + 8 = 12 , M x = i = 1 3 m i y i = 6 + 6 8 = 4.

Then we have

x = M y m = 12 12 = 1 and y = M x m = 4 12 = 1 3 .

The center of mass of the system is ( 1 , 1 / 3 ) , in meters.

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Questions & Answers

can you give me a problem for function. a trigonometric one
geovanni Reply
state and prove L hospital rule
Krishna Reply
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
Jawad
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach $60 billion?
Nguyen Reply
what is the derivative of x= Arc sin (x)^1/2
morfling Reply
y^2 = arcsin(x)
Pitior
x = sin (y^2)
Pitior
differentiate implicitly
Pitior
then solve for dy/dx
Pitior
thank you it was very helpful
morfling
questions solve y=sin x
Obi Reply
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
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)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
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.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
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
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
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
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
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)
Antonio
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
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
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
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
follow algebraic method. look under factoring numerator from Khan academy
moe
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
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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