Calculating Masses Using Gravitational Force Equations

[ELLE_AW]

Homework Statement

Two celestial bodies form an isolated system in a remote region in space. When they are separated by a distance of 5000m, and they attract each other with a force of 2.67 N. If the total mass of the system is 2.5 x 10^9, then what is the mass of each celestial body?

Relevant Equations

F=Gm1m2/r^2

F=Gm1m2/r^2
2.67 = (6.67x10^-11)(m1xm2)/25000000
M1xM2 = 1 x 10^18
M2 = 1x10^18/M1 (Equation 1)

From the question stem, we know M1 + M2 = 2.5x10^9 (Equation 2)

So, substituting Equation 1 into Equation 2 we get:
1x10^18/m1 + m1 = 2.5 x 10^9

I'M STUCK FROM HERE ONWARDS... in the solutions manual they show that the m1 gets squared and you end up with a quadratic equation that you have to solve to get the two values for the masses. But, where does the squaring come in? I don't get it.

NEVERMIND, I got it! Just rusty algebra.. haha!

 

[gneill]

Either there is something missing from the problem statement or you need to make an assumption about the masses involved, such as the two being equal, if you want to arrive at a single numerical answer for the masses. Otherwise there would be an infinite number of combinations of mass values that would satisfy the given constraints.

If the solution involves m1 being squared then I strongly suspect that they expect you to assume that m1 = m2.

Can you provide any other information from the original problem statement as it was presented to you?

 

[jbriggs444]

Either there is something missing from the problem statement or you need to make an assumption about the masses involved, such as the two being equal, if you want to arrive at a single numerical answer for the masses. Otherwise there would be an infinite number of combinations of mass values that would satisfy the given constraints.

It seems quote solvable. You have two equations and two unknowns.
$$m_1+m_2=m_{tot}$$ $$\frac{Gm_1m_2}{r^2}=F$$
$F$, $m_{tot}$ and $r$ are all given. So you solve the first equation for $m_1$ in terms of $m_2$ and the second equation turns into a quadratic in $m_1$.

By inspection, the two solutions (if any) will be mirror images of one another.

 

[gneill]

By inspection, the two solutions (if any) will be mirror images of one another.

And will there be specific values for either, or will there be an expression relating the two values?

 

[jbriggs444]

And will there be specific values for either, or will there be an expression relating the two values?

Specific values for both. Once you solve the quadratic for $m_1$, you can easily find $m_2$.

 

[gneill]

I concede your point. There are indeed a pair of mass values that either body can have that will resolve the problem. (I think I may have been dozing while I posted before. My apologies to @ELLE_AW). Cheers, jbriggs.