Question

The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's orbit. Pluto was subsequently discovered near its predicted position. But it now appears that the discovery was fortuitous, since Pluto is small and the irregularities in Neptune's orbit were not well known. Answer the following to illustrate that Pluto has a minor effect on the orbit of Neptune compared with other planets. (a) Calculate the acceleration of gravity at Neptune due to Pluto when they are 3.2 1012 m apart. The mass of Pluto is 1.3 1022 kg. g1 = m/s2 (b) Calculate the acceleration of gravity at Neptune due to Uranus, when they are 3.2 1012 m apart. The mass of Uranus is 8.68 1025 kg. g2 = m/s2 (c) Give the ratio of this acceleration to that due to Pluto. g2 g1 =

Answer 1

Answer:

a) g1 = 8.467 10⁻¹⁴ m / s² , b)  g2 = 5.65 10⁻¹⁰ m / s² , c) g2 / g1 = 6.6 10³

Explanation:

a) for this we will use Newton's second law where force is the force of universal gravitation

         F = m a

        G m M / r² = m a

        a = g = G M / r²

By the effect of Pluto

        g1 = 6.67 10⁻¹¹  1.3 10²² / (3.2 10¹²)²

        g1 = 8.467 10⁻¹⁴ m / s²

b) By the effect of Uranus

        g2 = 6.67 10⁻¹¹ 8.68 10²⁵ / (3.2 10¹²)²

        g2 = 5.65 10⁻¹⁰ m / s²

c) the relationship between these accelerations

        g2 / g1 = 5.65 10⁻¹⁰ / 8.47 10⁻¹⁴

        g2 / g1 = 0.66 10⁴

        g2 / g1 = 6.6 10³