Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve the problem of determining the gravitational force between a [tex]\( 45 \, \text{kg} \)[/tex] person and the Earth, with a given distance from the Earth's center, we will use Newton's law of universal gravitation. The formula for gravitational force is:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]),
- [tex]\( m_1 \)[/tex] is the mass of the person ([tex]\( 45 \, \text{kg} \)[/tex]),
- [tex]\( m_2 \)[/tex] is the mass of the Earth ([tex]\( 5.98 \times 10^{24} \, \text{kg} \)[/tex]),
- [tex]\( r \)[/tex] is the distance between the person and the center of the Earth ([tex]\( 6.38 \times 10^6 \, \text{m} \)[/tex]).
Let's break down the steps:
1. Identify the given values:
[tex]\[ G = 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \][/tex]
[tex]\[ m_1 = 45 \, \text{kg} \][/tex]
[tex]\[ m_2 = 5.98 \times 10^{24} \, \text{kg} \][/tex]
[tex]\[ r = 6.38 \times 10^6 \, \text{m} \][/tex]
2. Substitute these values into the gravitational force formula:
[tex]\[ F = \left( 6.67430 \times 10^{-11} \right) \frac{45 \times 5.98 \times 10^{24}}{\left( 6.38 \times 10^6 \right)^2} \][/tex]
3. Calculate the expression within the numerator:
[tex]\[ 45 \times 5.98 \times 10^{24} = 269.1 \times 10^{24} \, \text{kg} \][/tex]
4. Calculate the expression within the denominator:
[tex]\[ \left( 6.38 \times 10^6 \right)^2 = 40.7044 \times 10^{12} \, \text{m}^2 \][/tex]
5. Simplify the fraction:
[tex]\[ \frac{269.1 \times 10^{24}}{40.7044 \times 10^{12}} = \frac{269.1}{40.7044} \times 10^{12} = 6.61 \times 10^{12} \, \text{kg} \, \text{m}^{-2} \][/tex]
6. Multiply this result by the gravitational constant:
[tex]\[ F = \left( 6.67430 \times 10^{-11} \right) \times \left( 6.61 \times 10^{12} \right) \][/tex]
7. Perform the final multiplication:
[tex]\[ F \approx 441.24323906015076 \, \text{N} \][/tex]
Thus, the gravitational force [tex]\( F \)[/tex] is approximately [tex]\( 441 \, \text{N} \)[/tex]. Among the provided options, the closest value to our calculated result is:
[tex]\[ \text{C. } 441 \, \text{N} \][/tex]
Therefore, the correct answer is [tex]\( \text{C. } 441 \, \text{N} \)[/tex].
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]),
- [tex]\( m_1 \)[/tex] is the mass of the person ([tex]\( 45 \, \text{kg} \)[/tex]),
- [tex]\( m_2 \)[/tex] is the mass of the Earth ([tex]\( 5.98 \times 10^{24} \, \text{kg} \)[/tex]),
- [tex]\( r \)[/tex] is the distance between the person and the center of the Earth ([tex]\( 6.38 \times 10^6 \, \text{m} \)[/tex]).
Let's break down the steps:
1. Identify the given values:
[tex]\[ G = 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \][/tex]
[tex]\[ m_1 = 45 \, \text{kg} \][/tex]
[tex]\[ m_2 = 5.98 \times 10^{24} \, \text{kg} \][/tex]
[tex]\[ r = 6.38 \times 10^6 \, \text{m} \][/tex]
2. Substitute these values into the gravitational force formula:
[tex]\[ F = \left( 6.67430 \times 10^{-11} \right) \frac{45 \times 5.98 \times 10^{24}}{\left( 6.38 \times 10^6 \right)^2} \][/tex]
3. Calculate the expression within the numerator:
[tex]\[ 45 \times 5.98 \times 10^{24} = 269.1 \times 10^{24} \, \text{kg} \][/tex]
4. Calculate the expression within the denominator:
[tex]\[ \left( 6.38 \times 10^6 \right)^2 = 40.7044 \times 10^{12} \, \text{m}^2 \][/tex]
5. Simplify the fraction:
[tex]\[ \frac{269.1 \times 10^{24}}{40.7044 \times 10^{12}} = \frac{269.1}{40.7044} \times 10^{12} = 6.61 \times 10^{12} \, \text{kg} \, \text{m}^{-2} \][/tex]
6. Multiply this result by the gravitational constant:
[tex]\[ F = \left( 6.67430 \times 10^{-11} \right) \times \left( 6.61 \times 10^{12} \right) \][/tex]
7. Perform the final multiplication:
[tex]\[ F \approx 441.24323906015076 \, \text{N} \][/tex]
Thus, the gravitational force [tex]\( F \)[/tex] is approximately [tex]\( 441 \, \text{N} \)[/tex]. Among the provided options, the closest value to our calculated result is:
[tex]\[ \text{C. } 441 \, \text{N} \][/tex]
Therefore, the correct answer is [tex]\( \text{C. } 441 \, \text{N} \)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
