# 6.6 Moments and centers of mass  (Page 2/14)

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

$\stackrel{–}{x}=\frac{\sum _{i=1}^{n}{m}_{i}{x}_{i}}{\sum _{i=1}^{n}{m}_{i}}.$

## Center of mass of objects on a line

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

$M=\sum _{i=1}^{n}{m}_{i}{x}_{i}$

and the center of mass of the system is given by

$\stackrel{–}{x}=\frac{M}{m}.$

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:

$\begin{array}{cccc}{m}_{1}=30\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{1}=-2\phantom{\rule{0.2em}{0ex}}\text{m}\hfill & & & {m}_{2}=5\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{2}=3\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ {m}_{3}=10\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{3}=6\phantom{\rule{0.2em}{0ex}}\text{m}\hfill & & & {m}_{4}=15\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{4}=-3\phantom{\rule{0.2em}{0ex}}\text{m}.\hfill \end{array}$

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:

$\begin{array}{cc}\hfill M& =\sum _{i=1}^{4}{m}_{i}{x}_{i}\hfill \\ & =-60+15+60-45=-30.\hfill \end{array}$

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

$\begin{array}{cc}\hfill m& =\sum _{i=1}^{4}{m}_{i}\hfill \\ & =30+5+10+15=60\phantom{\rule{0.2em}{0ex}}\text{kg}\text{.}\hfill \end{array}$

Then we have

$\stackrel{–}{x}=\frac{M}{m}=\frac{-30}{60}=-\frac{1}{2}.$

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

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

$\begin{array}{cccc}{m}_{1}=12\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{1}=-4\phantom{\rule{0.2em}{0ex}}\text{m}\hfill & & & {m}_{2}=12\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{2}=4\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ {m}_{3}=30\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{3}=2\phantom{\rule{0.2em}{0ex}}\text{m}\hfill & & & {m}_{4}=6\phantom{\rule{0.2em}{0ex}}\text{kg,}\phantom{\rule{0.2em}{0ex}}\text{placed at}\phantom{\rule{0.2em}{0ex}}{x}_{4}=-6\phantom{\rule{0.2em}{0ex}}\text{m}.\hfill \end{array}$

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

$M=24,\stackrel{–}{x}=\frac{2}{5}\phantom{\rule{0.2em}{0ex}}\text{m}$

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 $\left({x}_{1},{y}_{1}\right)$ 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).

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 ${m}_{1},{m}_{2}\text{,…},{m}_{n}$ be point masses located in the xy -plane at points $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\text{,…},\left({x}_{n},{y}_{n}\right),$ respectively, and let $m=\sum _{i=1}^{n}{m}_{i}$ 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

${M}_{x}=\sum _{i=1}^{n}{m}_{i}{y}_{i}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{M}_{y}=\sum _{i=1}^{n}{m}_{i}{x}_{i}.$

Also, the coordinates of the center of mass $\left(\stackrel{–}{x},\stackrel{–}{y}\right)$ of the system are

$\stackrel{–}{x}=\frac{{M}_{y}}{m}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\stackrel{–}{y}=\frac{{M}_{x}}{m}.$

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):

$\begin{array}{c}{m}_{1}=2\phantom{\rule{0.2em}{0ex}}\text{kg, placed at}\phantom{\rule{0.2em}{0ex}}\left(-1,3\right),\hfill \\ {m}_{2}=6\phantom{\rule{0.2em}{0ex}}\text{kg, placed at}\phantom{\rule{0.2em}{0ex}}\left(1,1\right),\hfill \\ {m}_{3}=4\phantom{\rule{0.2em}{0ex}}\text{kg, placed at}\phantom{\rule{0.2em}{0ex}}\left(2,-2\right).\hfill \end{array}$

Find the center of mass of the system.

First we calculate the total mass of the system:

$m=\sum _{i=1}^{3}{m}_{i}=2+6+4=12\phantom{\rule{0.2em}{0ex}}\text{kg}\text{.}$

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

$\begin{array}{}\\ \\ {M}_{y}=\sum _{i=1}^{3}{m}_{i}{x}_{i}=-2+6+8=12,\hfill \\ {M}_{x}=\sum _{i=1}^{3}{m}_{i}{y}_{i}=6+6-8=4.\hfill \end{array}$

Then we have

$\stackrel{–}{x}=\frac{{M}_{y}}{m}=\frac{12}{12}=1\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\stackrel{–}{y}=\frac{{M}_{x}}{m}=\frac{4}{12}=\frac{1}{3}.$

The center of mass of the system is $\left(1,1\text{/}3\right),$ in meters.

can you give me a problem for function. a trigonometric one
state and prove L hospital rule
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
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?
what is the derivative of x= Arc sin (x)^1/2
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
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?
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)
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
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
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
y=800
800
Bg
how do u factor the numerator?
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
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio