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Sagot :
To determine the gravitational force between two spherical objects in contact with each other, we can use Newton's law of universal gravitation. This law states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) (measured from their centers) is given by the formula:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- \( G \) is the gravitational constant with a value of \( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \),
- \( m_1 \) and \( m_2 \) are the masses of the spheres,
- \( r \) is the separation distance between the centers of the two spheres.
Here's how we would solve this step-by-step:
1. Identify the masses and radii:
- Mass of the first sphere (\( m_1 \)): \( 200 \, \text{kg} \)
- Radius of the first sphere: \( 4 \, \text{m} \)
- Mass of the second sphere (\( m_2 \)): \( 400 \, \text{kg} \)
- Diameter of the second sphere: \( 12 \, \text{m} \)
- Radius of the second sphere (half of the diameter): \( 6 \, \text{m} \)
2. Determine the distance between the centers of the two spheres:
Since the spheres are in contact, the distance between their centers \( r \) is the sum of their radii.
[tex]\[ r = \text{radius}_1 + \text{radius}_2 = 4 \, \text{m} + 6 \, \text{m} = 10 \, \text{m} \][/tex]
3. Plug the values into the gravitational force formula:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
Substitute the known values:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \cdot \frac{200 \, \text{kg} \cdot 400 \, \text{kg}}{(10 \, \text{m})^2} \][/tex]
4. Simplify the expression:
First, calculate the product of the masses:
[tex]\[ 200 \, \text{kg} \cdot 400 \, \text{kg} = 80000 \, \text{kg}^2 \][/tex]
Then, calculate the square of the distance:
[tex]\[ (10 \, \text{m})^2 = 100 \, \text{m}^2 \][/tex]
Substitute these into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \cdot \frac{80000 \, \text{kg}^2}{100 \, \text{m}^2} \][/tex]
5. Evaluate the fractions:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{80000 \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}}{100} \][/tex]
6. Finish the calculations:
[tex]\[ F = 6.67430 \times 10^{-11} \times 800 \][/tex]
[tex]\[ F = 5.3394399999999997 \times 10^{-8} \, \text{N} \][/tex]
So, the gravitational force between the two spherical bodies in contact with each other is approximately:
[tex]\[ 5.33944 \times 10^{-8} \, \text{Newtons} \][/tex]
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- \( G \) is the gravitational constant with a value of \( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \),
- \( m_1 \) and \( m_2 \) are the masses of the spheres,
- \( r \) is the separation distance between the centers of the two spheres.
Here's how we would solve this step-by-step:
1. Identify the masses and radii:
- Mass of the first sphere (\( m_1 \)): \( 200 \, \text{kg} \)
- Radius of the first sphere: \( 4 \, \text{m} \)
- Mass of the second sphere (\( m_2 \)): \( 400 \, \text{kg} \)
- Diameter of the second sphere: \( 12 \, \text{m} \)
- Radius of the second sphere (half of the diameter): \( 6 \, \text{m} \)
2. Determine the distance between the centers of the two spheres:
Since the spheres are in contact, the distance between their centers \( r \) is the sum of their radii.
[tex]\[ r = \text{radius}_1 + \text{radius}_2 = 4 \, \text{m} + 6 \, \text{m} = 10 \, \text{m} \][/tex]
3. Plug the values into the gravitational force formula:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
Substitute the known values:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \cdot \frac{200 \, \text{kg} \cdot 400 \, \text{kg}}{(10 \, \text{m})^2} \][/tex]
4. Simplify the expression:
First, calculate the product of the masses:
[tex]\[ 200 \, \text{kg} \cdot 400 \, \text{kg} = 80000 \, \text{kg}^2 \][/tex]
Then, calculate the square of the distance:
[tex]\[ (10 \, \text{m})^2 = 100 \, \text{m}^2 \][/tex]
Substitute these into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \cdot \frac{80000 \, \text{kg}^2}{100 \, \text{m}^2} \][/tex]
5. Evaluate the fractions:
[tex]\[ F = 6.67430 \times 10^{-11} \, \frac{80000 \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}}{100} \][/tex]
6. Finish the calculations:
[tex]\[ F = 6.67430 \times 10^{-11} \times 800 \][/tex]
[tex]\[ F = 5.3394399999999997 \times 10^{-8} \, \text{N} \][/tex]
So, the gravitational force between the two spherical bodies in contact with each other is approximately:
[tex]\[ 5.33944 \times 10^{-8} \, \text{Newtons} \][/tex]
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