Understanding the Concepts

* Gravitational Force: The force of attraction between any two objects with mass. It depends on the masses of the objects and the distance between their centers.

* Newton's Law of Universal Gravitation: This law describes the gravitational force:

* F = G * (m1 * m2) / r²

* Where:

* F is the gravitational force

* G is the gravitational constant (approximately 6.674 x 10⁻¹¹ N m²/kg²)

* m1 and m2 are the masses of the two objects

* r is the distance between the centers of the two objects

Setting up the Problem

1. Earth's Gravity: We need to find the force of gravity Earth exerts on a particle. Let's assume the particle has a mass of 1 kg (we can choose any mass for this example).

* Earth's mass (M) = 5.972 x 10²⁴ kg

* Earth's radius (R) = 6.371 x 10⁶ m

* Force of gravity (Fg) = G * (M * 1 kg) / R²

* Fg ≈ 9.8 N (approximately the acceleration due to gravity at the Earth's surface)

2. The Small Ball:

* Mass of the ball (m) = 100 kg

* We want to find the distance (r) where the ball's gravitational pull on the 1 kg particle equals 9.8 N.

Solving for Distance

1. Equate the Forces: We want the force from the ball (Fb) to be equal to the force from Earth (Fg):

* Fb = Fg

* G * (m * 1 kg) / r² = 9.8 N

2. Solve for r:

* r² = (G * m * 1 kg) / 9.8 N

* r = √((G * m * 1 kg) / 9.8 N)

* Substitute the values of G, m, and the force (9.8 N):

* r ≈ √((6.674 x 10⁻¹¹ N m²/kg² * 100 kg * 1 kg) / 9.8 N)

* r ≈ 8.2 x 10⁻⁵ m

Answer:

The particle would have to be placed approximately 8.2 x 10⁻⁵ meters (or 0.082 millimeters) away from the center of the 100 kg ball to experience the same gravitational force as it does from Earth.

Important Note: This is a theoretical calculation. In reality, it's practically impossible to create such a precise scenario, as other gravitational influences (like nearby objects) would interfere.