Discover a wealth of information and get your questions answered on IDNLearn.com. Get prompt and accurate answers to your questions from our community of knowledgeable experts.
Sagot :
Certainly! Let's solve this problem step-by-step.
We are asked to find the distance between two men, each with a mass of 90 kg, given the gravitational force between them is [tex]\(8.64 \times 10^{-8}\)[/tex] N. The gravitational constant [tex]\(G\)[/tex] is [tex]\(6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We start by recalling the formula for the gravitational force between two masses:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\(F\)[/tex] is the gravitational force,
- [tex]\(G\)[/tex] is the gravitational constant,
- [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are the masses,
- [tex]\(r\)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\(m_1 = 90 \text{ kg}\)[/tex],
- [tex]\(m_2 = 90 \text{ kg}\)[/tex],
- [tex]\(F = 8.64 \times 10^{-8} \text{ N}\)[/tex],
- [tex]\(G = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We need to solve for [tex]\(r\)[/tex]. Rearranging the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r^2 = G \cdot \frac{m_1 \cdot m_2}{F} \][/tex]
Substituting the given values into the equation:
[tex]\[ r^2 = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot \frac{90 \text{ kg} \cdot 90 \text{ kg}}{8.64 \times 10^{-8} \text{ N}} \][/tex]
First, let's compute the numerator and the denominator separately:
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot 8100 \text{ kg}^2 \][/tex]
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \cdot 8100 \text{ m}^2 \][/tex]
[tex]\[ \text{Numerator} = 5.4027 \times 10^{-7} \text{ m}^2 \][/tex]
[tex]\[ \text{Denominator} = 8.64 \times 10^{-8} \text{ N} \][/tex]
Now, divide the two results to find [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = \frac{5.4027 \times 10^{-7} \text{ m}^2}{8.64 \times 10^{-8} \text{ N}} \][/tex]
[tex]\[ r^2 \approx 6.25 \text{ m}^2 \][/tex]
Finally, take the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{6.25 \text{ m}^2} \][/tex]
[tex]\[ r \approx 2.50 \text{ m} \][/tex]
Thus, the distance between the two men is approximately 2.5 meters.
The correct answer is:
D. 2.5 m
We are asked to find the distance between two men, each with a mass of 90 kg, given the gravitational force between them is [tex]\(8.64 \times 10^{-8}\)[/tex] N. The gravitational constant [tex]\(G\)[/tex] is [tex]\(6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We start by recalling the formula for the gravitational force between two masses:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\(F\)[/tex] is the gravitational force,
- [tex]\(G\)[/tex] is the gravitational constant,
- [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are the masses,
- [tex]\(r\)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\(m_1 = 90 \text{ kg}\)[/tex],
- [tex]\(m_2 = 90 \text{ kg}\)[/tex],
- [tex]\(F = 8.64 \times 10^{-8} \text{ N}\)[/tex],
- [tex]\(G = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right)\)[/tex].
We need to solve for [tex]\(r\)[/tex]. Rearranging the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r^2 = G \cdot \frac{m_1 \cdot m_2}{F} \][/tex]
Substituting the given values into the equation:
[tex]\[ r^2 = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot \frac{90 \text{ kg} \cdot 90 \text{ kg}}{8.64 \times 10^{-8} \text{ N}} \][/tex]
First, let's compute the numerator and the denominator separately:
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \text{ N} \cdot \left( \frac{\text{m}^2}{\text{kg}^2} \right) \cdot 8100 \text{ kg}^2 \][/tex]
[tex]\[ \text{Numerator} = 6.67 \times 10^{-11} \cdot 8100 \text{ m}^2 \][/tex]
[tex]\[ \text{Numerator} = 5.4027 \times 10^{-7} \text{ m}^2 \][/tex]
[tex]\[ \text{Denominator} = 8.64 \times 10^{-8} \text{ N} \][/tex]
Now, divide the two results to find [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = \frac{5.4027 \times 10^{-7} \text{ m}^2}{8.64 \times 10^{-8} \text{ N}} \][/tex]
[tex]\[ r^2 \approx 6.25 \text{ m}^2 \][/tex]
Finally, take the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{6.25 \text{ m}^2} \][/tex]
[tex]\[ r \approx 2.50 \text{ m} \][/tex]
Thus, the distance between the two men is approximately 2.5 meters.
The correct answer is:
D. 2.5 m
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
