Answer:
M = 3.9406 kg and m = 0.0594 kg
Explanation:
The gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance that separates them. Mathematically it is expressed as follows:
Fg = (G×M×m)/r² Formula (1)
Where:
Fg is the gravitational force (N)
G is the universal gravitation constant, G = 6.67 × 10⁻¹¹ (N×m²)/kg²
M and m are the masses of the bodies that interact (kg).
r is the distance that separates them (m).
Known Data
Fg = 2.5 × 10⁻¹⁰ N
r = 0.25 m
G = 6.67 × 10⁻¹¹ (N×m²)/kg²
Problem development
We propose 2 equations
M + m = 4kg
M = 4 - m equation (1)
We replace in formula (1)
2.5 × 10⁻¹⁰ = (6.67 × 10⁻¹¹ × M × m)/(0.25)²
2.5 × 10⁻¹⁰ × (0.25)² = (6.67 × 10⁻¹¹ × M × m)
(2.5 × 10⁻¹⁰ × (0.25)²)/(6.67 × 10⁻¹¹) = M × m
M × m = 0.234 equation (2)
We replace M = 4 - m in equation (2)
(4 - m) × m = 0.234
4m - m² = 0.234
m² - 4m + 0.234 = 0 (quadratic equation)
We apply the formula for the quadratic equation and obtain 2 values for m that meet the conditions:
m = 3.9406 kg or m = 0.0594 kg
We replace m in equation (1)
M = 4 - 3.9406 = 0.0594 kg or M = 4 - 0.0594 = 3.9406
To meet the condition that M + m must give 4 kg, one mass must be equal 3.9406 and the other must equal 0.0594, then:
M = 3.9406 kg and m = 0.0594 kg
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