A planet of mass M has a moon of mass m in a circular orbit of radius R. An object is placed between the planet and the moon on the line joi

Question

A planet of mass M has a moon of mass m in a circular orbit of radius R. An object is placed between the planet and the moon on the line joining the center of the planet to the center of the moon so that the net gravitational force on the object is zero. How far is the object placed from the center of the planet

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Minh Khuê 3 years 2021-09-05T02:03:28+00:00 1 Answers 35 views 0

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  1. ThanhThu
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    2021-09-05T02:05:23+00:00
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    Answer:

    r =\frac{ 1 \pm \sqrt{ \frac{m}{M} } }{1 - \frac{m}{M} }

    Explanation:

    Let’s apply the universal gravitation law to the body (c), we use the indications 1 for the planet and 2 for the moon

              ∑ F = 0

               -F_{1c} + F_{2c} = 0

                 F_{1c} = F_{2c}

    let’s write the force equations

                 G \frac{m_c M}{r^2} = G \frac{m_c m}{(d-r)^2}

    where d is the distance between the planet and the moon.

                  \frac{M}{r^2} = \frac{m}{(d-r)^2}

                 (d-r)² = \frac{m}{M} \ \ r^2  

                  d² – 2rd + r² = \frac{m}{M} \ \ r^2

                  d² – 2rd + r² (1 – \frac{m}{M}) = 0

                  (1 – \frac{m}{M})  r² – 2d r + d² = 0

    we solve the second degree equation

                  r = [2d ± \sqrt{ 4d^2 - 4 ( 1 - \frac{m}{M} ) } ] / 2 (1- \frac{m}{M})

                  r = [2d ±  2d \sqrt{ \frac{m}{M} }] / 2d (1- \frac{m}{M})

                  r =\frac{ 1 \pm \sqrt{ \frac{m}{M} } }{1 - \frac{m}{M} }

    there are two points for which the gravitational force is zero

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