Question

The gravitational force between two asteroids is 2.59 × 10 (exponent)-6 N. The centers of mass are 2000 meters away and their masses are equal. What is the mass of each asteroid?​

Answer 1

Answer:

$2.79 \times 10^5 \ \text{kg}$

Explanation:

Newton's Law of Universal Gravitation:

• $F= G\frac{m_1 m_2}{r^2}$
• F = force of gravity (N)
• G = gravitational constant $(6.67 \times 10^-^1^1 \ N\frac{m^2}{kg^2})$
• $m_1$ = mass of Object 1 (kg)
• $m_2$ = mass of Object 2 (kg)
• r = distance between the center of mass (m)

Let's convert our given information to scientific notation:

• $2000 \ m \rightarrow 2.0 \times 10^3 \ m$

Now using the gravitational force and the distance between centers of mass that are given, we can plug these into Newton's law:

• $2.59 \times 10^-^6 \ N = 6.67 \times 10^-^1^1 \ N \frac{m^2}{kg^2} \times \frac{m_1 m_2}{(2.0 \times 10^3 \ m)^2}$

Remove the units for better readability.

• $2.59 \times 10^-^6=6.67 \times 10^-^1^1 \frac{m_1m_2}{(2.0 \times 10^3)^2}$

Divide both sides of the equation by the gravitational constant G.

• $\frac{2.59 \times 10^-^6}{6.67 \times 10^-^1^1} =\frac{m_1m_2}{(2.0 \times 10^3)^2}$

Distribute the power of 2 inside the parentheses.

• $\frac{2.59 \times 10^-^6}{6.67 \times 10^-^1^1} =\frac{m_1m_2}{2.0 \times 10^6}$

If we evaluate the left side of the equation, we get:

• $3.88305847 \times 10^4 = \frac{m_1m_2}{2.0 \times 10^6}$

Multiply both sides of the equation by r.

• $7.76611694 \times 10^1^0= m_1m_2$

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 $m_1m_2$ as $m^2$.

• $7.76611694 \times 10^1^0= m^2$

Square root both sides of the equation.

• $m=\sqrt{7.76611694 \times 10^1^0}$

• $m=2.78677536 \times 10^5$
• $m=2.79 \times 10^5$

Since m is in units of kg, we can state that the mass of each asteroid is 2.79 * 10⁵ kg.