Hm.
As far as I understand, we have a parabolic plate and we need to find the equation of its parabolic side given the coordinates of its center of mass, right?
Suppose the parabola is y = ax^2 + bx + c.
We see that it passes through (0, 0). So:
y(0) = c = 0
So, we are left with y = ax^2 + bx.
Now, to find a and b we need to consider the center of mass. So, apply the formulas for them with f(x) = ax^2 + bx. Looking at the picture, we get the limits:
0 < y < ax^2 + bx
0 < x < 1
Can you calculate the integrals?
#double integral
rotund shadow
sleek lake
thank you for taking your time to give me an answer, I will now be able to calculate the integrals
rotund shadow
Great! You're welcome 😁
sleek lake
Great! You're welcome 😁
I think I have an error 😅, when I calculate m I have this
and by calculating xg I find this
rotund shadow
I think I have an error 😅, when I calculate m I have this
Wait, wait. Why did you take those limits?
I already said that the limis are 0 < y < ax^2 + bx, 0 < x < 1.
sleek lake
leaving a and b unknown ?
rotund shadow
Yeah. We will be able to find them after we find both integrals.
sleek lake
so double integral with one of them has 0 and 1 for limits (x) and the other 0 and ax^2+bx (y)
rotund shadow
Yes.
sleek lake
So m could be that ?
rotund shadow
So m could be that ?
Yes. That is the mass of the plate.
Now, you need to find the moments.
sleek lake
Xg and Yg or Mx and My ?
sleek lake
I'm still searching
rotund shadow
Alright.
sleek lake
I understand it's a small surface element multiplied by lever arm
and with µx/m we will have Xg
rotund shadow
Yeah, we will have M(x)/M = X(g) and M(y)/M = X(y). That will be the system of equations.
So, did you calculate M(x)?
sleek lake
First, calculate M(x) and M(y).
I really don't know how to calculate these static moments with the data I have
rotund shadow
Well, for M you integrated just dxdy.
For M(x) you do the same, but integrate xdxdy. And for M(y) you integrate ydxdy.
sleek lake
Like that ?
rotund shadow
Can't see it very well. It should be (6a^2 + 15ab + 10b^2)/60.
sleek lake
I have a mistake (6a+15ab+10b^2)/60 , you have in parentheses 6a^2 and me 6a
And M(y) it could be that
rotund shadow
No, that's incorrect.
You mixed up the limits. The inner integral is xdy. If you want, you can factor out x from it.
sleek lake
but if I keep the same limits I will find the same thing as for M(x) right?
sleek lake
Fine, let me show.
okay
rotund shadow
This is what we get.
sleek lake
Ohhh okay ! So I had to reverse to calculate Mx in my double integral and then after all these steps you have to solve the system to have a and b, right?
rotund shadow
Yes.
To solve the system, I recommend expressing a or b from the first equation and substituting into the second. That will give you a quadratic equation.
sleek lake
To solve the system, I recommend expressing a or b from the first equation and s...
oh okay
sleek lake
the only way to check is to take for example the first equation and replace b with the number and see if we had the same a, right ?
because a = -1 and b = 2
rotund shadow
because a = -1 and b = 2
Yes, very good!
So, our parabola is y = -x^2 + 2x.
sleek lake
Yes, very good!
So, our parabola is y = -x^2 + 2x.
YES Thank you very much for your explanations and your time !!
umbral glacierBOT
@sleek lake has given 1 rep to @rotund shadow
rotund shadow
YES Thank you very much for your explanations and your time !!
You're welcome!
sleek lake
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