Inequalities in Force of Gravitation between Three Bodies

[jantdroid]

Homework Statement

Given that the force of gravitation between Planet A (the one in the left side of the drawing), F_a=3000/d_a² and the force of gravitation between Planet B and the rocket, F_b= 6000/ db^2. Assuming that the three bodies involved is in stationary. What are the distances (ranges) from Planet A where the gravitational force is above 0.001 Newtons?

Homework Equations

The Attempt at a Solution

First, I solved if the Fa>0.001,

Fa=3000/d_a²
0.001<3000/d_a²
1732.054< d_a²

Therefore: 0 < d_a² < 1732.054

Then I calculated if F_b>0.001
0.001<6000/d_b²
2442.75<d_b²

I subtracted the 2442.75 km from 6 000 000 km because the question asks what distances from planet A where the gravitational force is above 0.001 N.
5997557.25<d_a

0<d_a<5997557.25

Not sure if my answer was correct though, I'm still concerned with the middle body which is the rocket.

 

[Simon Bridge]

First, I solved if the Fa>0.001, ...

That won't work - no.
Do you need the distance along the line joining planets A and B?

Planets A and B are separated by distance D. At a distance x along the line from A to B, the magnitude of the force of gravity towards A is given by: $F= F_A-F_B = \cdots$ ... you finish it.
Note: you'll need to express $d_A$ and $d_B$ in terms of x.

 

[jantdroid]

That won't work - no.
Do you need the distance along the line joining planets A and B?

Planets A and B are separated by distance D. At a distance x along the line from A to B, the magnitude of the force of gravity towards A is given by: $F= F_A-F_B = \cdots$ ... you finish it.
Note: you'll need to express $d_A$ and $d_B$ in terms of x.

I think so, the question kinda confuses me.

"What are the distances (ranges) from Planet A to B where the gravitational force is above 6 000 000 and I'm concerned about the rocket in the middle too.

 

[Simon Bridge]

Which question is confusing you, there are several?
The trick with this question is to work out the equation for the sum of the forces - just do that part.
Don't use any of the given values - use letters for them instead.

 

[jantdroid]

Which question is confusing you, there are several?
The trick with this question is to work out the equation for the sum of the forces - just do that part.
Don't use any of the given values - use letters for them instead.

The one that says find the distance(s) where gravitational force is above 0.001 N

 

[Simon Bridge]

OK - so did you follow the suggestion?

 

[jantdroid]

OK - so did you follow the suggestion?

Yah, the equation is on the 4th degree. Not sure how to proceed with it by picking out the critical points and do some test points. :3

 

[BiGyElLoWhAt]

Yah, the equation is on the 4th degree. Not sure how to proceed with it by picking out the critical points and do some test points. :3

Are you saying the sum of the forces is a degree 4 polynomial? Because if it is, I think you made a mistake somewhere.

 

[jantdroid]

Are you saying the sum of the forces is a degree 4 polynomial? Because if it is, I think you made a mistake somewhere.

Yup, my classmate tried to solved it too still get the 4th degree equation. lol

 

[Simon Bridge]

Please show your working, with the reasoning behind each step ...
I'm getting a 4th order polynomial that can be reduced to a 2nd order one.

 

[jantdroid]

Please show your working, with the reasoning behind each step ...
I'm getting a 4th order polynomial that can be reduced to a 2nd order one.

 

[jantdroid]

Please show your working, with the reasoning behind each step ...
I'm getting a 4th order polynomial that can be reduced to a 2nd order one.

F=FA−FB

Since the force needed is greater than 0.001N

0.001< 3000/da² - 6000/db²

transposing the 0.001 value to the right side of the equation,
0< 3000/da² - 6000/db² - 0.001

Finding the LCD of the solution

0< ( 3000db²-6000da²-0.001da²db² ) db² da²

Given that
db= 6 000 000 - da

then,

db²= (6 000 000)² - 12 000 000 da + da²

substitute all to db and b²

0 < 3000( 3x10¹³ - 12 000 000 da + da²) - 6000 da² - 0.001 (3x10¹³ - 12 000 000 da + da²) da² ) / da² (3.6x10 ¹³ - 12 000 000 da + da² )

 

[Simon Bridge]

Checking - I made a mistake in my quick once over :)

It's easier to handle big number symbolically:

putting x=d_A, (makes it easier to type) then d_B=D-x: D=6,000,000km
putting G=3000N.km² means I can write the net force in the direction of planet A as:

$$F=\frac{G}{x^2}-\frac{2G}{(D-x)^2}$$ ... put H=F/G and multiply through by the common denominator:
$\implies H(D-x)^2x^2 = D^2-2Dx-x^2$
$\implies HD^2x^2-2HDx^3+Hx^4= D^2-2Dx-x^2$
$\implies Hx^4-2HDx^3 +(HD^2+1)x^2 + 2Dx - D^2 = 0$

You should double-check my working...
But you get the idea?

The solution we want lies between the planets, which corresponds to the only positive value of x < D.
$F=0.001\text{N} \implies H=(1/3000000) \text{km}^{-2} = 2/D$
... which simplified the equation a bit.

It is possible to solve a 4th order polynomial analytically, or you may prefer a numerical solution via Newton/Raphson.

 

[jantdroid]

Checking - I made a mistake in my quick once over :)

It's easier to handle big number symbolically:

putting x=d_A, (makes it easier to type) then d_B=D-x: D=6,000,000km
putting G=3000N.km² means I can write the net force in the direction of planet A as:

$$F=\frac{G}{x^2}-\frac{2G}{(D-x)^2}$$ ... put H=F/G and multiply through by the common denominator:
$\implies H(D-x)^2x^2 = D^2-2Dx-x^2$
$\implies HD^2x^2-2HDx^3+Hx^4= D^2-2Dx-x^2$
$\implies Hx^4-2HDx^3 +(HD^2+1)x^2 + 2Dx - D^2 = 0$

You should double-check my working...
But you get the idea?

The solution we want lies between the planets, which corresponds to the only positive value of x < D.
$F=0.001\text{N} \implies H=(1/3000000) \text{km}^{-2} = 2/D$
... which simplified the equation a bit.

It is possible to solve a 4th order polynomial analytically, or you may prefer a numerical solution via Newton/Raphson.

I don't quite get where the "H" came from and what exactly is the answer (distance) from planet a to b

 

[jantdroid]

x:

Checking - I made a mistake in my quick once over :)

It's easier to handle big number symbolically:

putting x=d_A, (makes it easier to type) then d_B=D-x: D=6,000,000km
putting G=3000N.km² means I can write the net force in the direction of planet A as:

$$F=\frac{G}{x^2}-\frac{2G}{(D-x)^2}$$ ... put H=F/G and multiply through by the common denominator:
$\implies H(D-x)^2x^2 = D^2-2Dx-x^2$
$\implies HD^2x^2-2HDx^3+Hx^4= D^2-2Dx-x^2$
$\implies Hx^4-2HDx^3 +(HD^2+1)x^2 + 2Dx - D^2 = 0$

You should double-check my working...
But you get the idea?

The solution we want lies between the planets, which corresponds to the only positive value of x < D.
$F=0.001\text{N} \implies H=(1/3000000) \text{km}^{-2} = 2/D$
... which simplified the equation a bit.

It is possible to solve a 4th order polynomial analytically, or you may prefer a numerical solution via Newton/Raphson.

Here, me trying to apply your method,

Let F> 0.001

0.001 < G (D-x)² - 2G (x²) / (D-x)² (x²)

0.001 < G (D²- 2Dx +x²) - 2Gx / (D²-2Dx+x²) (x²)

0.001 < GD² - 2GDx + Gx² -2Gx²/ X²D² - 2Dx³ + x⁴

0 < GD²- 2GDx - Gx² / x²D²- 2Dx³ + x⁴ - 0.001

Finding the LCD:

0 < ( GD²- 2GDx -Gx² - 0.001 D²x² - 0.002 Dx³ + 0.001 x⁴ / x²D² - 2x³ +x⁴ ) x²D² - 2x³ + x⁴

and now, I am lost. lol

 

[Simon Bridge]

I don't quite get where the "H" came from...

H is what you get by dividing the first equation through by the common factor of G.

...and what exactly is the answer (distance) from planet a to b

The distance from planet A to planet B is given in the problem statement - I gave this distance the label D, so D=6000000km.

The problem want you to find the distance from planet A to the place where the gravity experienced by the ship is above 0.001N strong.

Let F> 0.001

Put F = 0.001N ... in order for the force to be stronger than this, the ship must be closer to planet A.

There are two places between A and B where the force will have this magnitude, but only one where the direction is towards planet A.

0 < ( GD2- 2GDx -Gx2 - 0.001 D2x2 - 0.002 Dx3 + 0.001 x4 / x2D2 - 2x3 +x4 ) x2D2 - 2x3 + x4

... I didn't bother to check the algebra ... I had 0.001/G = H because it's tidier at the outset.
I chose "H" because it's next in the alphabet from F and G.

Now you have to decide how to solve for x.
You can either use the link I gave you, or find x where F=0 as your starting point for a numerical calculation.

 

[jantdroid]

∠⊆⊆⊕⊕→ℝℝβ

H is what you get by dividing the first equation through by the common factor of G.
The distance from planet A to planet B is given in the problem statement - I gave this distance the label D, so D=6000000km.

The problem want you to find the distance from planet A to the place where the gravity experienced by the ship is above 0.001N strong.Put F = 0.001N ... in order for the force to be stronger than this, the ship must be closer to planet A.

There are two places between A and B where the force will have this magnitude, but only one where the direction is towards planet A.

... I didn't bother to check the algebra ... I had 0.001/G = H because it's tidier at the outset.
I chose "H" because it's next in the alphabet from F and G.

Now you have to decide how to solve for x.
You can either use the link I gave you, or find x where F=0 as your starting point for a numerical calculation.

Nice! Thanks for giving me this technique on making equations neat, I'll study how to get that quartic equation right away. I'll get back to you once I've confirmed the answer. Thank you! :D