Answer :
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].
