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

`2.79 * 10^5 \ \text{kg}`

Step-by-step explanation:

Newton's Law of Universal Gravitation:

• 
  `$F= G(m_1 m_2)/(r^2)`
• F = force of gravity (N)
• G = gravitational constant
  `(6.67 * 10^-^1^1 \ N(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 * 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 * 10^-^6 $\ N = 6.67 * 10^-^1^1 \ N (m^2)/(kg^2) * (m_1 m_2)/((2.0 * 10^3 \ m)^2)`

Remove the units for better readability.

• 
  `2.59 * 10^-^6=6.67 * 10^-^1^1 (m_1m_2)/((2.0 * 10^3)^2)`

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

• 
  `(2.59 * 10^-^6)/(6.67 * 10^-^1^1) =(m_1m_2)/((2.0 * 10^3)^2)`

Distribute the power of 2 inside the parentheses.

• 
  `(2.59 * 10^-^6)/(6.67 * 10^-^1^1) =(m_1m_2)/(2.0 * 10^6)`

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

• 
  `3.88305847 * 10^4 = (m_1m_2)/(2.0 * 10^6)`

Multiply both sides of the equation by r.

• 
  `7.76611694 * 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 * 10^1^0= m^2`

Square root both sides of the equation.

• 
  `m=√(7.76611694 * 10^1^0)`

• 
  `m=2.78677536 * 10^5`
• 
  `m=2.79 * 10^5`

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