The gravitational force between two objects masses 10 ^ 24 kg and 10 ^ 27 kg is 6.67N. Calculate the distance…

Analysis

To solve this problem, we use Newton’s Law of Universal Gravitation, which states that the force ($F$) between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Concept

The formula is:

Where:

• $F$ = Gravitational force ($6.67 \, \text{N}$)
• $G$ = Gravitational constant ($6.67 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$)
• $m_1$ = Mass 1 ($10^{24} \, \text{kg}$)
• $m_2$ = Mass 2 ($10^{27} \, \text{kg}$)
• $r$ = Distance between the objects (what we need to find)

Step-by-Step Solution

1. Rearrange the formula to solve for $r$: $$r^2 = \frac{G \cdot m_1 \cdot m_2}{F}$$

2. Substitute the given values into the equation: $$r^2 = \frac{(6.67 \times 10^{-11}) \times (10^{24}) \times (10^{27})}{6.67}$$

3. Simplify the calculation: • The $6.67$ in the numerator and denominator cancel each other out: $$r^2 = 10^{-11} \times 10^{24} \times 10^{27}$$ • Use the law of indices ($a^m \times a^n = a^{m+n}$): $$r^2 = 10^{(-11 + 24 + 27)}$$ $$r^2 = 10^{40}$$

4. Find the square root to get $r$: $$r = \sqrt{10^{40}}$$ $$r = 10^{20} \, \text{m}$$

This can be written as $1.0 \times 10^{20} \, \text{m}$.

Conclusion

Comparing our result to the given options, we find that it matches option A.

Exam Tip: When dealing with powers of 10 in physics calculations, always handle the coefficients (the numbers in front) separately from the exponents. This prevents errors when moving decimals. Also, remember that $\sqrt{10^{40}}$ is simply $10^{40/2} = 10^{20}$.