Question

can i get some help with this question please

find the center of mass of a triangle with points (5,8) (9,11) and (5,11)

Answer 1

Is this triangle is of uniform density, or there is a mass at each of the vertices?

Answer 2

uniform density

Answer 3

Join any vertex to the mid-point of the opposite side. The centre of mass is located at 2/3 of the distance from the vertex.

Answer 4

would that be all i need to do for this problem

Answer 5

Yes, but you will do it using analytic geometry.

Answer 6

im confused because that's not how I learned it in class

Answer 7

If you go through all the motions, you will end up with the result equivalent to:
G((x1+x2+x3)/3, (y1+y2+y3)/3)

where the points P1,P2,P3 have coordinates P1(x1,y1),P2(x2,y2),P3(x3,y3),
and G is the centre of gravity.

Answer 8

How did you learn it in class?

Answer 9

for these type of problems my professor wanted us to use x bar and y bar

Answer 10

Can you tell me more?

Answer 11

Do you use calculus?

Answer 12

yes. this is a cal 2 problem

Answer 13

OK, there are different ways to do the same thing. You really are learning Calculus, so you need to do integration.
What I gave you were known results.

Answer 14

Are you familiar with
xBar = integral(x dA) / integral (dA)
and
yBar = integral(y dA) / integral(dA)
?

Answer 15

xbar = 1/A integral xf(x) dx

Answer 16

y bar = 1/A integral 1/2 [f(x)]^2 dx

Answer 17

So your f(x) represents the height of the figure at x, right?

Answer 18

These formulas reduce the double integral (dA) to a simple integral.

Answer 19

i believe so

Answer 20

can you first find the height of the figure in terms of x.
We will call the three points A, B and C respectively.

Answer 21

Have you started by drawing the triangle on paper (or calculator)?
A is the bottom vertex, B is the right vertex, and C is the top-left vertex.

Answer 22

yes i have

Answer 23

We will proceed as follows:
1. We note that the triangle has two sides parallel to the axes, AC // y-axis,
BC // to x-axis.
2. We will find the function f(x) which represents the y-dimension of the triangle for x between 5 and 9
3. We will then proceed to do the integrations you suggested.
Is this ok with you?

Answer 24

yes, this is okay with me.

Answer 25

Can you now start by finding f(x) representing the y-dimension of the triangle.
We know that 5<= x <= 9.
and f(5)=11-8=3, f(9)=0
It should be a linear function.
Confirm the above information on your figure.

Answer 26

We know that the slope of f(x) is -3/4, since f(5)=3, f(9)=0.
So f(x)=-(3/4)x+27/4
Do you agree?

Answer 27

Now you can proceed with the integrations used in your class to find xBar and yBar.

Answer 28

Except that the integral for yBar assumes that the base of the triangle is along the x-axis.
You can still use the same formula, but the resulting yBar will be measured downwards from the side CB.

You can use my formulae to check your results, namely
xBar = (x1+x2+x3)/3
yBar = (y1+y2+y3)/3

Good luck.