At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the force of gravity acting on the two spheres given the specified conditions, we can use Newton's Universal Law of Gravitation. Here is a detailed, step-by-step solution for calculating the gravitational force:
1. Identify the given variables:
- Mass of the first object (\( m_1 \)) = 1000 kg
- Location of the first object = 1 m
- Mass of the second object (\( m_2 \)) = 1000 kg
- Location of the second object = 9 m
2. Calculate the distance (\( r \)) between the two masses:
- Since the positions of mass \( m_1 \) and mass \( m_2 \) are 1 m and 9 m respectively along the same axis, the distance between them is:
[tex]\[ r = 9 \, \text{m} - 1 \, \text{m} = 8 \, \text{m} \][/tex]
3. Recall the gravitational constant (\( G \)):
- The universal gravitational constant, \( G \), is \( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \).
4. Apply Newton's Law of Gravitation:
- The formula for gravitational force \( F \) is given by:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
5. Substitute the known values into the formula:
- \( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
- \( m_1 = 1000 \, \text{kg} \)
- \( m_2 = 1000 \, \text{kg} \)
- \( r = 8 \, \text{m} \)
[tex]\[ F = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \cdot \frac{1000 \, \text{kg} \cdot 1000 \, \text{kg}}{(8 \, \text{m})^2} \][/tex]
6. Calculate the square of the distance:
-
[tex]\[ (8 \, \text{m})^2 = 64 \, \text{m}^2 \][/tex]
7. Complete the calculation:
-
[tex]\[ F = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \cdot \frac{1000000 \, \text{kg}^2}{64 \, \text{m}^2} \][/tex]
Simplifying this:
[tex]\[ F = \frac{6.67430 \times 10^{-11} \times 1000000}{64} \, \text{N} \][/tex]
[tex]\[ F = \frac{6.67430 \times 10^{-5}}{64} \, \text{N} \][/tex]
Dividing the numbers:
[tex]\[ F = 1.042859375 \times 10^{-6} \, \text{N} \][/tex]
8. Express the result in scientific notation:
- The gravitational force \( F \) acting between the two spheres is:
[tex]\[ F = 1.043 \times 10^{-6} \, \text{N} \][/tex]
Therefore, the force of gravity acting on the spheres, under the given conditions, is [tex]\( 1.043 \times 10^{-6} \, \text{N} \)[/tex].
1. Identify the given variables:
- Mass of the first object (\( m_1 \)) = 1000 kg
- Location of the first object = 1 m
- Mass of the second object (\( m_2 \)) = 1000 kg
- Location of the second object = 9 m
2. Calculate the distance (\( r \)) between the two masses:
- Since the positions of mass \( m_1 \) and mass \( m_2 \) are 1 m and 9 m respectively along the same axis, the distance between them is:
[tex]\[ r = 9 \, \text{m} - 1 \, \text{m} = 8 \, \text{m} \][/tex]
3. Recall the gravitational constant (\( G \)):
- The universal gravitational constant, \( G \), is \( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \).
4. Apply Newton's Law of Gravitation:
- The formula for gravitational force \( F \) is given by:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
5. Substitute the known values into the formula:
- \( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
- \( m_1 = 1000 \, \text{kg} \)
- \( m_2 = 1000 \, \text{kg} \)
- \( r = 8 \, \text{m} \)
[tex]\[ F = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \cdot \frac{1000 \, \text{kg} \cdot 1000 \, \text{kg}}{(8 \, \text{m})^2} \][/tex]
6. Calculate the square of the distance:
-
[tex]\[ (8 \, \text{m})^2 = 64 \, \text{m}^2 \][/tex]
7. Complete the calculation:
-
[tex]\[ F = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \cdot \frac{1000000 \, \text{kg}^2}{64 \, \text{m}^2} \][/tex]
Simplifying this:
[tex]\[ F = \frac{6.67430 \times 10^{-11} \times 1000000}{64} \, \text{N} \][/tex]
[tex]\[ F = \frac{6.67430 \times 10^{-5}}{64} \, \text{N} \][/tex]
Dividing the numbers:
[tex]\[ F = 1.042859375 \times 10^{-6} \, \text{N} \][/tex]
8. Express the result in scientific notation:
- The gravitational force \( F \) acting between the two spheres is:
[tex]\[ F = 1.043 \times 10^{-6} \, \text{N} \][/tex]
Therefore, the force of gravity acting on the spheres, under the given conditions, is [tex]\( 1.043 \times 10^{-6} \, \text{N} \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
