A110-kg object and a 410-kg object are separated by 3.80 m.

(a) find the magnitude of the net gravitational force exerted by these objects on a 41.0-kg object placed midway between them. n
(b) at what position (other than an infinitely remote one) can the 41.0-kg object be placed so as to experience a net force of zero from the other two objects?

a)   Fₙ = 2,273 10⁻⁷ N   and   b)    x₃ = 1,297 m

Explanation:

This problem can be solved using the law of universal gravitation and Newton's second law for the equilibrium case. The Universal Gravitation Equation is

    F = G m₁ m₂ / r₁₂²

a) we write Newton's second law

       Σ F = F₁₃ - F₃₂

Body 1 has mass of m₁ = 110 kg and we will place our reference system, body 2 has a mass of m₂ = 410 kg and is in the position x₂ = 3.80 m

Body 3 has a mass of m₃ = 41.0 kg and is in the middle of the other two bodies

      x₃ = (x₂-x₁) / 2

     x₃ = 3.80 / 2 = 1.9 m

     Fₙ = -G m₁ m₃ / x₃² + G m₃ m₂ / x₃²

     Fₙ = G m₃ / x₃² (-m₁ + m₂)

Calculate

     Fₙ = 6.67 10⁻¹¹ 41.0 / 1.9² (- 110 + 410)

     Fₙ = 2,273 10⁻⁷ N

Directed to the right

b) find the point where the force is zero

The distance is

     x₁₃ = x₃ - 0

    x₃₂ = x₂ -x₃= 3.8 -x₃

We write the park equation net force be zero

     0 = - F₁₃ + F₃₂

     F₁₃ = F₃₂

     G m₁ m₃ / x₁₃² = G m₃ m₂ / x₃₂²

     m₁ / x₁₃² = m₂ / x₃₂²

Let's look for the relationship between distances, substituting

     m₁ / x₃² = m₂ / (3.8 - x₃)²

     (3.8 - x₃) = x₃ √ (m₂ / m₁)

     x₃ + x₃ √ (m₂ / m₁) = 3.8

     x₃ (1 + √ m₂ / m₁) = 3.8

     x₃ = 3.8 / (1 + √ (m₂ / m₁))

     x₃ = 3.8 / (1 + √ (410/110))

     x₃ = 1,297 m

When body 3 is in this position the net force on it is zero

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