Answer:
[tex]2.79 \times 10^5 \ \text{kg}[/tex]
Explanation:
Newton's Law of Universal Gravitation:
- [tex]$F= G\frac{m_1 m_2}{r^2}[/tex]
- F = force of gravity (N)
- G = gravitational constant [tex](6.67 \times 10^-^1^1 \ N\frac{m^2}{kg^2})[/tex]
- [tex]m_1[/tex] = mass of Object 1 (kg)
- [tex]m_2[/tex] = mass of Object 2 (kg)
- r = distance between the center of mass (m)
Let's convert our given information to scientific notation:
- [tex]2000 \ m \rightarrow 2.0 \times 10^3 \ m[/tex]
Now using the gravitational force and the distance between centers of mass that are given, we can plug these into Newton's law:
- [tex]2.59 \times 10^-^6 $\ N = 6.67 \times 10^-^1^1 \ N \frac{m^2}{kg^2} \times \frac{m_1 m_2}{(2.0 \times 10^3 \ m)^2}[/tex]
Remove the units for better readability.
- [tex]2.59 \times 10^-^6=6.67 \times 10^-^1^1 \frac{m_1m_2}{(2.0 \times 10^3)^2}[/tex]
Divide both sides of the equation by the gravitational constant G.
- [tex]\frac{2.59 \times 10^-^6}{6.67 \times 10^-^1^1} =\frac{m_1m_2}{(2.0 \times 10^3)^2}[/tex]
Distribute the power of 2 inside the parentheses.
- [tex]\frac{2.59 \times 10^-^6}{6.67 \times 10^-^1^1} =\frac{m_1m_2}{2.0 \times 10^6}[/tex]
If we evaluate the left side of the equation, we get:
- [tex]3.88305847 \times 10^4 = \frac{m_1m_2}{2.0 \times 10^6}[/tex]
Multiply both sides of the equation by r.
- [tex]7.76611694 \times 10^1^0= m_1m_2[/tex]
In order to find the mass of one asteroid, we can use the fact that both asteroids have the same mass, therefore, we can rewrite [tex]m_1m_2[/tex] as [tex]m^2[/tex].
- [tex]7.76611694 \times 10^1^0= m^2[/tex]
Square root both sides of the equation.
- [tex]m=\sqrt{7.76611694 \times 10^1^0}[/tex]
- [tex]m=2.78677536 \times 10^5[/tex]
- [tex]m=2.79 \times 10^5[/tex]
Since m is in units of kg, we can state that the mass of each asteroid is 2.79 * 10⁵ kg.
