Question

Two objects attract each other gravitationally with a force of 2.5×10−10 n when they are 0.25 m apart. their total mass is 4.40 kg . find their individual masses.

Answer 1

Let the two masses be m₁ and m₂ kg.

Because their combined mass is 4.40 kg, therefore
m₁ + m₂ = 4.4                (1)

The force of attraction between the masses, when separated by d = 0.25 m, is
F = 2.5 x 10⁻¹⁰ N.
According to Newton's Law of gravitational attraction,
$F = \frac{G m_{1} m_{2}}{d^{2}}$
where
G = 6.673 x 10⁻¹¹ (N-m²)/kg²

Therefore
$\frac{6.673 \times 10^{-11} m_{1}m_{2}}{0.25^{2}} =2.5 \times 10^{-10}$
m₁m₂ = 0.2342            (2)
That is,
m₂ = 0.2342/m₁           3)

Substitute (3) into (1).
m₁ + 0.2342/m₁ = 4.4
m₁² + 0.2342 = 4.4m₁
m₁² - 4.4m₁ + 0.2342 = 0

Solve with the quadratic formula.
m₁ = 0.5[4.4 +/- √(18.423 )]
      = 4.346 or 0.0539 kg

When m₁ = 4.346, then m₂ = 0.2342/4.346 = 0.0539 
When m₁ = 0.0539, then m₂ = 0.2342/0.0539 = 4.346

Answer: 4.346 kg and 0.054 kg