Two asteroids with masses [tex]$3.45 \times 10^3 \, \text{kg}[tex]$[/tex] and [tex]$[/tex]6.06 \times 10^4 \, \text{kg}[tex]$[/tex] are separated by a distance of [tex]$[/tex]7,200 \, \text{m}$[/tex]. What is the gravitational force between the asteroids?

Newton's law of gravitation is [tex]$F_{\text{gravity}} = \frac{G m_1 m_2}{r^2}$[/tex].

The gravitational constant [tex]G[/tex] is [tex]6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2[/tex].

A. [tex]4.03 \, \text{N}$[/tex]
B. [tex]2.69 \times 10^{-10} \, \text{N}$[/tex]
C. [tex]3.38 \times 10^{32} \, \text{N}$[/tex]
D. [tex]1.93 \times 10^{-6} \, \text{N}$[/tex]



Answer :

To determine the gravitational force between the two asteroids using Newton's law of gravitation, we will follow these steps:

1. Identify the given values:
- Mass of the first asteroid, \( m_1 = 3.45 \times 10^3 \, \text{kg} \)
- Mass of the second asteroid, \( m_2 = 6.06 \times 10^4 \, \text{kg} \)
- Distance between the asteroids, \( r = 7200 \, \text{m} \)
- Gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)

2. Write down Newton's law of gravitation formula:

[tex]\[ F_{\text{gravity}} = \frac{G m_1 m_2}{r^2} \][/tex]

3. Substitute the given values into the formula:

[tex]\[ F_{\text{gravity}} = \frac{(6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2) \times (3.45 \times 10^3 \, \text{kg}) \times (6.06 \times 10^4 \, \text{kg})}{(7200 \, \text{m})^2} \][/tex]

4. Calculations step-by-step:

a. Calculate the product of \( G \) and the masses:

[tex]\[ (6.67 \times 10^{-11}) \times (3.45 \times 10^3) \times (6.06 \times 10^4) \][/tex]

After calculating this, you will get an intermediate result.

b. Calculate the square of the distance:

[tex]\[ (7200)^2 = 51840000 \][/tex]

5. Divide the product from step 4a by the result from step 4b to find \( F_{\text{gravity}} \):

After performing the division, the result is:

[tex]\[ F_{\text{gravity}} \approx 2.690001736111111 \times 10^{-10} \, \text{N} \][/tex]

6. Compare this result with the given choices:

- A. \( 4.03 \, \text{N} \)
- B. \( 2.69 \times 10^{-10} \, \text{N} \)
- C. \( 3.38 \times 10^{32} \, \text{N} \)
- D. \( 1.93 \times 10^{-6} \, \text{N} \)

The closest and correct answer is:

B. \( 2.69 \times 10^{-10} \, \text{N} \)

Hence, the gravitational force between the two asteroids is [tex]\( 2.69 \times 10^{-10} \, \text{N} \)[/tex].