Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang Lecture 7 Gravity and satellites Newton's Law of Universal Gravitation Gravitation is a force of attraction that acts between any two masses. The gravitation force acting between two masses, m1 and m: is a force of attraction that acts from the centre of each mass acts equally on each mass (Newton's third law) is weaker if the masses are further apart. Gravitation is inversely proportional to the squared distance between the centres of the masses depends directly on the mass of each body involved Gm1m Therefore, the universal gravitational force is given by: F = where F is the gravitational force acting on each body (N) m1 and m are the masses of the objects (kg) is the distance between the centres of the objects (m) The universal gravitational constant G is equal to 6.67 10 Nm kg 11. The gravitational force that acts on a 1000 kg space probe on the surface of Mars is 3 4.43 10 N. The radius of Mars is 3400 km. Without using the mass of Mars, determine the gravitational force that acts on the space probe when it is: a. 3400 km above the surface of Mars. b. 6800 km above the surface of Mars. Gravitational field strength A gravitational field is a region in which any object will experience a gravitational force. The magnitude of a gravitational field is known as the gravitational field strength, g. This is defined as the gravitational force that acts on each kilogram of an object in the field and is given by: g = where G is the universal gravitational constant 6.67 10 Nm kg 11 M is the mass of the central planet (kg) is the distance from the centre of the central planet (m) g is the gravitational field strength (N/kg) NOTE that the gravitation field strength does not depend on the mass of the object in the gravitational field. The gravitational force acting on an object in a gravitational field is also called the weight, W, of the object m and is given by: W = mg = Physics 1
Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang a. Calculate the gravitational field strength g at the surface of the Earth (mass of Earth is radius of Earth is 6 6.4 10 m ). 4 6.0 10 kg, b. Calculate the gravitational field strength g at the surface of the Moon (mass of Moon is 7.34 10 kg, radius of Moon is 6 1.74 10 m ). c. Determine the weight of an astronaut, whose total mass is 100 kg, at each of these locations. The planet Alpha, whose mass is M, has one moon Beta of mass 0.01M. The mean distance between the centres of Alpha and Beta is. Physics
Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang a. If an asteroid is at point X, exactly halfway between the centres of Alpha and Beta, calculate the value of the ratio: force exerted on asteroid by Alpha force exerted on asteroid by Beta b. Point Y represents the distance from planet Alpha where the magnitude of the net gravitational force is zero. What is this distance in terms of? A broader concept of free falling As long as an object is accelerating towards the centre of the central planet with an acceleration equal to the gravitational field strength at that location, the motion of the object could be considered as free falling. A person during free falling would experience apparent weightlessness (a = g and N = 0) Motion of the planets A satellite is an object that is in a stable orbit around a more massive central planet. The only force acting on a satellite is the gravitational attraction between it and the central planet. Satellites are in continuous free-fall (uniform circular motion) with the centripetal acceleration being equal to the gravitational field strength at the location of their orbit. F For the satellite orbiting around the central planet, the gravitational force exerted by the central planet on the satellite provides the centripetal force. F m = = ma v a = 4π T a = a = g g = G is the universal gravitational constant 6.67 10 Nm kg 11 M is the mass of the central planet (kg) is the distance from the centre of the central planet (m) g is the gravitational field strength (N/kg) Physics 3
Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang a = g a = g v = v = v = (orbital speed) 4π T = 3 4π = T 3 T = 4π Kepler's third law Assume that somewhere in space there is a small spherical planet with a radius of 30 km. By some chance a person living on this planet visits Earth. He finds that he weighs the same on Earth as he did on his home planet, even though Earth is so much larger. Earth has a radius of 6 6.37 10 m and a mass of 4 5.98 10 kg. The acceleration due to gravity (g), or the gravitational field, at the surface of Earth, is approximately 10 N/kg. The universal gravitational constant, G, is 6.67 10 Nm kg 11. What is the value of the gravitational field on the surface of the visitor's planet? What is the mass of the visitor's planet? Explain your answer by showing clear working. The visitor's home planet is in orbit around its own small star at a radius of orbit of has a mass of 5 5.7 10 kg. What would be the period of the orbit of the visitor's planet? Show working. 9 1.0 10 m. The star Physics 4
Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang Nick is a geostationary satellite. Its period of orbit is 4 hours so that it revolves at the same rate at which the Earth turns. Given that the mass of the Earth is a. its orbital radius 4 6.0 10 kg and the mass of Nick is 100 kg, calculate: b. the gravitational field strength at this radius c. Nick's orbital speed d. its acceleration. Ceres, the first asteroid to be discovered, was found in 1801. Ceres has a mass of of 385 km. a. What is the gravitational field strength at the surface of Ceres? 0 7.0 10 kg and a radius b. Determine the speed required by a satellite in order to remain in orbit 10 km above the surface of Ceres. Physics 5
Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang The dwarf planet Pluto was discovered in 1930, and was thought to be the outermost member of our solar 1 system. It can be considered to orbit the Sun in a circle of radius 6 billion kilometres ( 6.0 10 m). In 003 a new dwarf planet, Eris, was discovered. It has approximately the same mass as Pluto, but the average 1 radius of its orbit around the Sun is 10.5 billion kilometres ( 10.5 10 m). Which of the choices (A-D) below gives the best estimate of the ratio A. 0.33 B. 0.57 C. 1.75 D. 3.06 gravitational attraction of the Sun on Eris gravitational attraction of the Sun on Pluto? The period of Pluto around the Sun is 48 Earth-years. How many Earth-years does Eris take to orbit the Sun? The planet Mars has a mass of approximately half that of Earth. 3 6.4 10 kg, which is approximately 1 10 that of Earth, and its radius is Which of the following (A-D) gives the best value for the acceleration due to gravity at the surface of Mars? A. 1 m/s B..5 m/s C. 4 m/s D. 5 m/s The Mars probe that was launched in August 005 is now orbiting Mars in an orbit with an average radius of 7 3.00 10 m. What is the period of the orbit in seconds? Physics 6