Respuesta :
Answer:
The individual masses of the two objects are 0,06 kg and 3,94 kg.
Explanation:
To solve this we need to use Newton's law of universal gravitation that states that the force of gravity between two massive bodies 1 and 2, is proportional to their masses M₁ and M₂ and inversely proportional to the square of the distance d that separates them:
[tex]F=\frac{G M_{1}M_{2}}{d^{2} }[/tex]
with [tex]G=6,67 x 10^{-11} \frac{Nm^{2} }{kg^{2} }[/tex] being the universal gravitational constant.
We are told:
- the value of F when it says that the objects attract each other gravitationally with a force of [tex]2.5x10^{-10}N[/tex],
- the value of d when it says that they are 0.25 m apart.
When it says that their total mass is 4.00 kg it means that:
[tex]M_{1} +M_{2} =4.00\ kg[/tex]
[tex]M_{2} =4.00\ kg-M_{1}[/tex]
replacing all of this in the original equation [tex]F=\frac{G M_{1}M_{2}}{d^{2} }[/tex] we get
[tex]2.5x10^{-10}N=6.67x10^{-11}\frac{Nm^{2}}{kg^{2}} \frac{M_{1}(4.00kg-M_{1})}{(0.25)^{2}m^{2}}[/tex]
[tex]2.5x10^{-10}N=\frac{6.67x10^{-11}Nm^{2}}{(0.25)^{2}m^{2}kg^{2}}(M_{1}4.00kg-M_{1} ^{2})[/tex]
[tex]2.5x10^{-10}N=\frac{6.67x10^{-11}N}{(0.25)^{2}kg^{2}}(M_{1}4.00kg-M_{1} ^{2})[/tex]
[tex]\frac{2.5x10^{-10} }{6.67x10^{-11}}(0.25)^{2}kg^{2}=4.00kg\ M_{1}-M_{1} ^{2}[/tex]
if we put all terms in one side we get
[tex]M_{1} ^{2} -4.00kg\ M_{1} +\frac{2.5x10^{-10}}{6.67x10^{-11}}(0.25)^{2}kg^{2} =0[/tex]
[tex]M_{1} ^{2} -4.00kg\ M_{1} +0.23\ kg^{2} =0[/tex]
we have to solve a quadratic equation [tex]ax^{2} +bx+c=0[/tex] where a=1, b=-4.00 and c=0.23. Here x=M₁.
We replace these values in the quadratic formula for the roots:
[tex]x=\frac{-b+-\sqrt{b^{2}-4.a.c}}{2a}[/tex]
so we get
[tex]x=\frac{4.00+-\sqrt{16-4.1.0.23}}{2}[/tex]
[tex]x=\frac{4.00+-\sqrt{16-0.92}}{2}[/tex]
[tex]x=\frac{4.00+-\sqrt{15.08}}{2}[/tex]
[tex]x=\frac{4.00+-3.88}{2}[/tex]
the possible roots are
[tex]x_{1} =\frac{4.00+3.88}{2}=\frac{7.88}{2}=3.94[/tex]
[tex]x_{2} =\frac{4.00-3.88}{2}=\frac{0.12}{2}=0.06[/tex]
given that M₁+M₂=4.00 kg the possible values of M₂ are then
[tex]M_{2}=4.00kg-3.94kg=0.06kg[/tex]
[tex]M_{2}=4.00kg-0.06kg=3.94kg[/tex]
Either one of the individual masses M₁ and M₂ could be 0.06 kg or 3.94 kg.
