Answer :
To find the gravitational force between a baseball and a bowling ball when their centers are 0.5 meters apart, we use Newton's law of universal gravitation. The formula for the gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [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 first object (the baseball), [tex]\(0.145 \text{ kg} \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the second object (the bowling ball), [tex]\(6.8 \text{ kg} \)[/tex],
- [tex]\( r \)[/tex] is the distance between the centers of the two objects, [tex]\(0.5 \text{ m} \)[/tex].
Using these values, we substitute into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \frac{(0.145 \times 6.8)}{(0.5)^2} \][/tex]
First, calculate the numerator:
[tex]\[ 0.145 \times 6.8 = 0.986 \][/tex]
Next, calculate the square of the distance:
[tex]\[ 0.5^2 = 0.25 \][/tex]
Then, divide the product of the masses by the square of the distance:
[tex]\[ \frac{0.986}{0.25} = 3.944 \][/tex]
Now, multiply by the gravitational constant:
[tex]\[ F = 6.67430 \times 10^{-11} \times 3.944 = 2.63234392 \times 10^{-10} \text{ N} \][/tex]
Therefore, the gravitational force between the baseball and the bowling ball is approximately:
[tex]\[ 2.63234392 \times 10^{-10} \text{ N} \][/tex]
Matching this result with the given choices, the closest answer is:
[tex]\[ 2.6 \times 10^{-10} \text{ N} \][/tex]
Thus, the correct answer is:
[tex]\[ 2.6 \times 10^{-10} \text{ N} \][/tex]
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [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 first object (the baseball), [tex]\(0.145 \text{ kg} \)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the second object (the bowling ball), [tex]\(6.8 \text{ kg} \)[/tex],
- [tex]\( r \)[/tex] is the distance between the centers of the two objects, [tex]\(0.5 \text{ m} \)[/tex].
Using these values, we substitute into the formula:
[tex]\[ F = 6.67430 \times 10^{-11} \frac{(0.145 \times 6.8)}{(0.5)^2} \][/tex]
First, calculate the numerator:
[tex]\[ 0.145 \times 6.8 = 0.986 \][/tex]
Next, calculate the square of the distance:
[tex]\[ 0.5^2 = 0.25 \][/tex]
Then, divide the product of the masses by the square of the distance:
[tex]\[ \frac{0.986}{0.25} = 3.944 \][/tex]
Now, multiply by the gravitational constant:
[tex]\[ F = 6.67430 \times 10^{-11} \times 3.944 = 2.63234392 \times 10^{-10} \text{ N} \][/tex]
Therefore, the gravitational force between the baseball and the bowling ball is approximately:
[tex]\[ 2.63234392 \times 10^{-10} \text{ N} \][/tex]
Matching this result with the given choices, the closest answer is:
[tex]\[ 2.6 \times 10^{-10} \text{ N} \][/tex]
Thus, the correct answer is:
[tex]\[ 2.6 \times 10^{-10} \text{ N} \][/tex]
