(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).
a)
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(i) Newton's Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the particles and is given by F = G(m1m2)/r^2, where F is the force of attraction between the particles, m1 and m2 are the masses of the particles, r is the distance between them, and G is the universal gravitational constant.
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(ii) Gravitational field is a region in space where a mass experiences a force due to the presence of another mass. The gravitational field strength at a point is the force per unit mass experienced by a small test mass placed at that point. It is given by g = F/m, where F is the force experienced by the test mass and m is the mass of the test mass.
b)
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(i) At the surface of the Earth, the gravitational field strength is equal to the acceleration of free fall (g). Therefore, we have g = GMe/R^2, where Me is the mass of the Earth, R is the radius of the Earth, and G is the universal gravitational constant.
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(ii) The relationship in 8(b)(i) holds under the assumptions that the Earth is a perfect sphere and has a uniform density distribution.
c)
Using the formula F = G(m1m2)/r^2, where m1 is the mass of the star, m2 is the mass of the Earth, r is the distance between them, and G is the universal gravitational constant, we have:
F = (6.67 x 10^-11 N m^2/kg^2) x (2.00 x 10^30 kg) x (5.98 x 10^24 kg) / (1.50 x 10^11 m)^2
F = 3.52 x 10^22 N
d)
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(i) Escape velocity is the minimum velocity required by an object to escape the gravitational field of a planet or a star.
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(ii) The acceleration of free fall (g) depends on the mass and distance of the object experiencing the gravitational force, while the universal gravitational constant (G) is a constant of nature that does not depend on the objects involved. Also, g is a measure of the strength of the gravitational field at a particular point, while G is a measure of the strength of the gravitational force between two objects separated by a distance.
a)
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(i) Newton's Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the particles and is given by F = G(m1m2)/r^2, where F is the force of attraction between the particles, m1 and m2 are the masses of the particles, r is the distance between them, and G is the universal gravitational constant.
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(ii) Gravitational field is a region in space where a mass experiences a force due to the presence of another mass. The gravitational field strength at a point is the force per unit mass experienced by a small test mass placed at that point. It is given by g = F/m, where F is the force experienced by the test mass and m is the mass of the test mass.
b)
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(i) At the surface of the Earth, the gravitational field strength is equal to the acceleration of free fall (g). Therefore, we have g = GMe/R^2, where Me is the mass of the Earth, R is the radius of the Earth, and G is the universal gravitational constant.
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(ii) The relationship in 8(b)(i) holds under the assumptions that the Earth is a perfect sphere and has a uniform density distribution.
c)
Using the formula F = G(m1m2)/r^2, where m1 is the mass of the star, m2 is the mass of the Earth, r is the distance between them, and G is the universal gravitational constant, we have:
F = (6.67 x 10^-11 N m^2/kg^2) x (2.00 x 10^30 kg) x (5.98 x 10^24 kg) / (1.50 x 10^11 m)^2
F = 3.52 x 10^22 N
d)
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(i) Escape velocity is the minimum velocity required by an object to escape the gravitational field of a planet or a star.
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(ii) The acceleration of free fall (g) depends on the mass and distance of the object experiencing the gravitational force, while the universal gravitational constant (G) is a constant of nature that does not depend on the objects involved. Also, g is a measure of the strength of the gravitational field at a particular point, while G is a measure of the strength of the gravitational force between two objects separated by a distance.
