1. Understand the Concept

* Newton's Law of Universal Gravitation: The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The equation is:

F = G * (m1 * m2) / r²

where:

* F is the force of gravity

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

* m1 and m2 are the masses of the objects

* r is the distance between their centers

2. Set up the Equations

* We know:

* F = 2.5 x 10^-10 N

* r = 0.29 m

* m1 + m2 = 4.0 kg (total mass)

* We need to find m1 and m2.

3. Solve for the Masses

* Substitute the known values into the gravitational force equation:

2.5 x 10^-10 N = (6.674 x 10^-11 N m²/kg²) * (m1 * m2) / (0.29 m)²

* Simplify the equation:

(2.5 x 10^-10 N) * (0.29 m)² / (6.674 x 10^-11 N m²/kg²) = m1 * m2

0.315 = m1 * m2

* Solve for one mass in terms of the other:

m1 = 0.315 / m2

* Substitute this expression for m1 into the total mass equation:

0.315 / m2 + m2 = 4.0 kg

* Multiply both sides by m2:

0.315 + m2² = 4.0 m2

* Rearrange into a quadratic equation:

m2² - 4.0 m2 + 0.315 = 0

* Solve the quadratic equation using the quadratic formula:

m2 = [4.0 ± √(4.0² - 4 * 1 * 0.315)] / (2 * 1)

m2 ≈ 3.96 kg or m2 ≈ 0.08 kg

* Find m1 using either of the solutions for m2:

If m2 ≈ 3.96 kg, then m1 ≈ 0.04 kg

If m2 ≈ 0.08 kg, then m1 ≈ 3.92 kg

Therefore, the individual masses are approximately:

* m1 ≈ 0.04 kg

* m2 ≈ 3.96 kg

or

* m1 ≈ 3.92 kg

* m2 ≈ 0.08 kg