Answer :
To calculate the gravitational force between two masses, we use the formula:
[tex]\[ F = G \times \frac{m_1 \times m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force.
- [tex]\( G \)[/tex] is the gravitational constant, approximately [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex].
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects.
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\( m_1 = 92.0 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 0.894 \, \text{kg} \)[/tex]
- [tex]\( r = 99.3 \, \text{m} \)[/tex]
We need to find the gravitational force [tex]\( F \)[/tex].
1. Substitute the values into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \times \frac{92.0 \times 0.894}{99.3^2} \][/tex]
2. Calculate the numerator:
[tex]\[ 92.0 \times 0.894 = 82.248 \][/tex]
3. Calculate the square of the distance:
[tex]\[ 99.3^2 = 9850.49 \][/tex]
4. Substitute these values into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \times \frac{82.248}{9850.49} \][/tex]
5. Calculate the division:
[tex]\[ \frac{82.248}{9850.49} \approx 0.00834951 \][/tex]
6. Multiply by the gravitational constant:
[tex]\[ F = 6.67430 \times 10^{-11} \times 0.00834951 \][/tex]
7. Calculate the final gravitational force:
[tex]\[ F \approx 5.5671455110243 \times 10^{-13} \, \text{N} \][/tex]
So, the gravitational force between the two masses is:
[tex]\[ \overrightarrow{F} = 5.5671455110243 \times 10^{-13} \, \text{N} \][/tex]
Expressed in a way that matches the requested format:
[tex]\[ \overrightarrow{F} = 0.055671455110243 \times 10^{-11} \, \text{N} \][/tex]
This representation shows that the scientific notation component for the gravitational force is:
[tex]\[ \overrightarrow{F} = 0.055671455110243 \times 10^{-11} \, \text{N} \][/tex]
This aligns with presenting the force in scientific notation, and the power of ten is [tex]\(-11\)[/tex].
[tex]\[ F = G \times \frac{m_1 \times m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force.
- [tex]\( G \)[/tex] is the gravitational constant, approximately [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex].
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects.
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\( m_1 = 92.0 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 0.894 \, \text{kg} \)[/tex]
- [tex]\( r = 99.3 \, \text{m} \)[/tex]
We need to find the gravitational force [tex]\( F \)[/tex].
1. Substitute the values into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \times \frac{92.0 \times 0.894}{99.3^2} \][/tex]
2. Calculate the numerator:
[tex]\[ 92.0 \times 0.894 = 82.248 \][/tex]
3. Calculate the square of the distance:
[tex]\[ 99.3^2 = 9850.49 \][/tex]
4. Substitute these values into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \times \frac{82.248}{9850.49} \][/tex]
5. Calculate the division:
[tex]\[ \frac{82.248}{9850.49} \approx 0.00834951 \][/tex]
6. Multiply by the gravitational constant:
[tex]\[ F = 6.67430 \times 10^{-11} \times 0.00834951 \][/tex]
7. Calculate the final gravitational force:
[tex]\[ F \approx 5.5671455110243 \times 10^{-13} \, \text{N} \][/tex]
So, the gravitational force between the two masses is:
[tex]\[ \overrightarrow{F} = 5.5671455110243 \times 10^{-13} \, \text{N} \][/tex]
Expressed in a way that matches the requested format:
[tex]\[ \overrightarrow{F} = 0.055671455110243 \times 10^{-11} \, \text{N} \][/tex]
This representation shows that the scientific notation component for the gravitational force is:
[tex]\[ \overrightarrow{F} = 0.055671455110243 \times 10^{-11} \, \text{N} \][/tex]
This aligns with presenting the force in scientific notation, and the power of ten is [tex]\(-11\)[/tex].
