Answer :
To solve this problem, we need to calculate the gravitational forces exerted by two larger masses on a smaller mass that is placed midway between them. We'll use Newton's law of universal gravitation for this purpose, which is stated as:
[tex]\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force between two masses,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/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.
Let's break down the steps:
1. Determine the Position of the Smaller Mass:
- The smaller mass, [tex]\( 64 \, \text{kg} \)[/tex], is placed exactly midway between the two larger masses, [tex]\( 215 \, \text{kg} \)[/tex] and [tex]\( 579 \, \text{kg} \)[/tex].
- The total distance separating the two larger masses is [tex]\( 0.493 \, \text{m} \)[/tex], so each half-distance (from the smaller mass to either of the larger masses) is [tex]\( \frac{0.493}{2} = 0.2465 \, \text{m} \)[/tex].
2. Calculate the Gravitational Force from the 215 kg Mass:
- Using the formula [tex]\( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)[/tex], calculate the force from the 215 kg mass on the 64 kg mass:
[tex]\[
F_{1\text{-on-}3} = \frac{6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 215 \, \text{kg} \times 64 \, \text{kg}}{(0.2465 \, \text{m})^2}
\][/tex]
- This yields a force of approximately [tex]\( 1.5109 \times 10^{-5} \, \text{N} \)[/tex].
3. Calculate the Gravitational Force from the 579 kg Mass:
- Similarly, calculate the force from the 579 kg mass on the 64 kg mass:
[tex]\[
F_{2\text{-on-}3} = \frac{6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 579 \, \text{kg} \times 64 \, \text{kg}}{(0.2465 \, \text{m})^2}
\][/tex]
- This yields a force of approximately [tex]\( 4.0689 \times 10^{-5} \, \text{N} \)[/tex].
4. Calculate the Net Gravitational Force:
- The two forces act in opposite directions since they pull the smaller mass towards each of the larger masses. To find the net force, subtract the smaller force from the larger force:
[tex]\[
F_{\text{net}} = F_{2\text{-on-}3} - F_{1\text{-on-}3} = 4.0689 \times 10^{-5} \, \text{N} - 1.5109 \times 10^{-5} \, \text{N} = 2.5580 \times 10^{-5} \, \text{N}
\][/tex]
Thus, the magnitude of the net gravitational force exerted by the two larger masses on the 64 kg mass is approximately [tex]\( 2.5580 \times 10^{-5} \, \text{N} \)[/tex].
[tex]\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force between two masses,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/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.
Let's break down the steps:
1. Determine the Position of the Smaller Mass:
- The smaller mass, [tex]\( 64 \, \text{kg} \)[/tex], is placed exactly midway between the two larger masses, [tex]\( 215 \, \text{kg} \)[/tex] and [tex]\( 579 \, \text{kg} \)[/tex].
- The total distance separating the two larger masses is [tex]\( 0.493 \, \text{m} \)[/tex], so each half-distance (from the smaller mass to either of the larger masses) is [tex]\( \frac{0.493}{2} = 0.2465 \, \text{m} \)[/tex].
2. Calculate the Gravitational Force from the 215 kg Mass:
- Using the formula [tex]\( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)[/tex], calculate the force from the 215 kg mass on the 64 kg mass:
[tex]\[
F_{1\text{-on-}3} = \frac{6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 215 \, \text{kg} \times 64 \, \text{kg}}{(0.2465 \, \text{m})^2}
\][/tex]
- This yields a force of approximately [tex]\( 1.5109 \times 10^{-5} \, \text{N} \)[/tex].
3. Calculate the Gravitational Force from the 579 kg Mass:
- Similarly, calculate the force from the 579 kg mass on the 64 kg mass:
[tex]\[
F_{2\text{-on-}3} = \frac{6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 579 \, \text{kg} \times 64 \, \text{kg}}{(0.2465 \, \text{m})^2}
\][/tex]
- This yields a force of approximately [tex]\( 4.0689 \times 10^{-5} \, \text{N} \)[/tex].
4. Calculate the Net Gravitational Force:
- The two forces act in opposite directions since they pull the smaller mass towards each of the larger masses. To find the net force, subtract the smaller force from the larger force:
[tex]\[
F_{\text{net}} = F_{2\text{-on-}3} - F_{1\text{-on-}3} = 4.0689 \times 10^{-5} \, \text{N} - 1.5109 \times 10^{-5} \, \text{N} = 2.5580 \times 10^{-5} \, \text{N}
\][/tex]
Thus, the magnitude of the net gravitational force exerted by the two larger masses on the 64 kg mass is approximately [tex]\( 2.5580 \times 10^{-5} \, \text{N} \)[/tex].
