(a) (i) State Newton’s Law of Universal Gravitation. (ii) Define gravitational field. (b) (i) Derive the equation relating the universal gravitational constant, G, and the acceleration of free fall, g, at the surface of the earth from Newton’s law of universal gravitation. (ii) State two assumptions for which the relationship in 8(b)(i) holds.
(c) Calculate the force of attraction between a star of mass 2.00 x 1030 kg and the earth assuming the star is located 1.50 x 108 km from the earth. [Mass of the earth = 5.98 x 1024kg; G = 6.67 x 10-11N m\(^{2}\) kg-2; g = 10 m s\(^{-2}\)
(d) (i) Define escape velocity. (ii) State two differences between the acceleration of free fall (g) and the universal gravitational constant (G).
Explanation
(a) (i) Newton’s law of Universal Gravitation; Any two bodies in the universe attract each other with a force that is (directly) proportional to the product of their masses and inversely proportional to the square of the distance between them. (ii) Definition of gravitational field; A region/space in which gravitational force is felt / experienced.
(b) (i) Derivation of the equation relating G and g at the surface of the earth F = \(\frac{GMm}{R^2}\) mg = \(\frac{GMm}{R^2}\) g = \(\frac{GM}{R^2}\)
(ii) Assumptions for which equation holds; - mass of the earth is concentrated at its centre - earth is spherical - earth is not rotating - possesses uniform density. (c) Calculation of the force of attraction between a star and the earth; F = \(\frac{GM_sM_e}{r_2}\) = 6.67 x 10\(^{11}\) [\(\frac {(2 \times 10^{30}) (5.98 \times 10^{24})} {(1.5 \times 10^{11})^2}\)] = 3.55 x 10\(^{22}\)N
(d) (i) Definition of escape velocity; The minimum uni-directional speed/velocity required by a body to leave the gravitational field/influence of a planet (ii) Differences between acceleration of free fall (g) and universal gravitational constant (G)
gG
Its unit is N m2 kg-2 Its unit is m s-2 / N kg-1
It is constant everywhere It varies from place to place
