Final answer:
The magnitude of the gravitational force a sphere exerts on another sphere can be calculated using Newton's Law of Universal Gravitation. For the given masses and distance, the magnitude of the force a exerts on b and b exerts on a is 3.708 × 10^-9 N. When the force between the spheres is 3.50 × 10^-9 N, their centers are approximately 1.2 mm apart.
Step-by-step explanation:
The magnitude of the gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (6.674 × 10^-11 N·m²/kg²), m1 and m2 are the masses of the two objects, and r is the distance separating their centers. In this case, we have two spheres with masses 6.5 kg and 8.3 kg, separated by a distance of 0.56 m.
(a) To calculate the magnitude of the gravitational force a exerts on b and b exerts on a, substitute the masses and distance into the formula:
F(a->b) = G * (m(a) * m(b)) / r^2
where m(a) = 6.5 kg, m(b) = 8.3 kg, and r = 0.56 m. Calculate:
F(a->b) = (6.674 × 10^-11 N·m²/kg²) * ((6.5 kg) * (8.3 kg)) / (0.56 m)^2
F(a->b) = 6.674 × 10^-11 N·m²/kg² * 54.45 kg^2 / 0.3136 m²
F(a->b) = 3.708 × 10^-9 N
The magnitude of the gravitational force a exerts on b is 3.708 × 10^-9 N. Similarly, the magnitude of the gravitational force b exerts on a is also 3.708 × 10^-9 N.
(b) To calculate the distance between the centers of the spheres when the force between them is 3.50 × 10^-9 N, rearrange the formula to solve for r:
r = sqrt((G * (m1 * m2)) / F)
Substitute G = 6.674 × 10^-11 N·m²/kg², m1 = 6.5 kg, m2 = 8.3 kg, and F = 3.50 × 10^-9 N:
r = sqrt((6.674 × 10^-11 N·m²/kg² * (6.5 kg * 8.3 kg)) / (3.50 × 10^-9 N))
r = sqrt((6.674 × 10^-11 N·m²/kg² * 54 kg²) / (3.50 × 10^-9 N))
r = sqrt(1.3676 × 10^-6 m²)
r ≈ 0.0012 m ≈ 1.2 mm
Therefore, when the force between the spheres is 3.50 × 10^-9 N, their centers are approximately 1.2 mm apart.