# Two objects attract each other gravitationally with a force of 2.5 10-10 N when they are 0.29 m apart.?

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Two objects attract each other gravitationally with a force of 2.5 10-10 N when they are 0.29 m apart. Their total mass is 4.0 kg. Find their individual masses.

• F(g) = gravitational force = GMm / r²

G = gravitational constant = 6.674 x 10 ̄ ¹¹ Nm²kg ̄ ²

=> 2.5 x 10 ̄ ¹⁰ N = 6.674 x 10 ̄ ¹¹[Mm] / (0.29)²

=> Mm = 0.31503 = (m)(4 – m)

=> -m² + 4m – 0.31503 = 0

or m² – 4m + 0.31503 = 0

solving => M = 3.92 kg and m = 0.08 kg (or vice versa)

hope this helps

• Gravitational force Fg between two masses M₁ and M₂ separated by a distance D is given by:

Fg = G / D²

where G is the universal gravitational constant {6.67 * 10^-11 N.m²/kg²}

re-arrange:

M₁ M₂ = Fg D² / G = ( 2.5 10^-10 0.29² / 6.67 * 10^-11 )

M₁ * M₂ = 0.315

M₁ = 0.315 / M₂

Given M₁ + M₂ = 4.0

0.315 / M₂ + M₂ = 4.0 {multiply everything by M₂}

0.315 + M₂² = 4.0M₂

M₂² – 4.0M₂ + 0.315 = 0

a = 1; b = -4.0; c = 0.315

M₂ = [–b ±√(b²–4ac)] / 2a

M₂ = 3.92 kg or 0.08 kg

That’s actually giving you both masses, since

If M₂ = 3.92,

M₁ = (0.315 / 3.92) = 0.08

If M₂ = 0.08,

M₁ = (0.315 / 0.08) = 3.92

• F = m1m2G/d^2

2.510^-10 = m(4-m)6.6710^-11/0.29^2

2.50.29^2 = (4m-m^2)0.667

0.210 -2.668m+0.667m^2 = 0

m = (2.668±√2.668^2-0.2140.667)/1.33 = 3.92 ; 0.08 kg

or

m1m2 = Fd^2/G = 2.510^-10/6.6710^-11*0.29^2 = 0.315

m1+m2 = 4

m1 = 4-m2

4m2-m2^2 = 0.315

m2 = (4±√4^2-1.26)/2 = 3.920 ; 0.080 kg

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