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1 Gravity Knowledge and understanding When you have finished this chapter, you should be able to: define weight as the force on an object due to a gravitational field explain that a change in gravitational potential energy is related to work done define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field, E p G m m 1 2 r analyse information using the expression F mg to determine the weight force for a body on Earth and for the same body on other planets. \\\\\\\ \\\\\\\ \\\\\\\ \\\\\\\ \\\\ \\\\\\\ \\\\\\ \\\\\\\ \\\ \\\\\\\ \\\\\\\ \\\\ \\\\\\\ \\\\\\\
Module 1 Chapter 1 \\ Gravity 3 Figure 1.1 Gravity is acting on the airborne skier. Figure 1.2 Many forces act on the occupants of a roller-coaster. 1.1 Mass and weight Any object released from the window of a high-rise building will gain speed as it falls. This means two things: first, there must be a force acting on the object; second, the object accelerates. If you drop a stone and a piece of paper from near the ceiling of a room, everyone knows that the stone reaches the floor first. But what happens if you crumple the piece of paper into the shape of a ball? During the first Moon landing, in 1969, the astronauts carried out a simple experiment by dropping a hammer and a feather and observing that both reached the Moon s surface at the same time. The rate at which these objects accelerate towards the surface of the Earth (or the Moon) is called the acceleration due to gravity. On the Moon, there is no atmosphere and so no air resistance to interfere with the motion of the feather (in particular). On the Earth, however, wind resistance is quite significant and so a feather and hammer, when dropped, do not reach the ground at the same time. Gravitation itself is a natural phenomenon where objects, simply because they have mass, attract each other. It is one of the fundamental forces of physics that keeps the Earth and other planets in orbit around the Sun. It keeps the Moon in orbit around the Earth and thus gives force to the tides. It is involved in the formation of stars and the heating of their centres to very high temperatures. Gravity affects objects that have mass. In popular science, mass and weight are often confused. We talk of hopping on the scales to measure our weight in kilograms, when the kilogram is the unit that is associated with mass, not weight. Mass and weight are two distinctly different quantities. The mass of an object is the same regardless of the gravitational field that it experiences. Mass is a scalar quantity. A 10 kg mass on the Earth still has a mass of 10 kg when in the gravitational field of the Moon. The product of its volume and density remains unchanged. Its weight, however, will be different. Weight is a vector quantity. Its weight is a direct result of the force of the gravitational field acting upon it. To find weight, we use the equation: gravity Calculating the weight of objects on other planets, pages 7 8, Practical Physics for Senior Students, HSC The acceleration that a body experiences when placed in a gravitational field. On Earth the accepted value is 9.8 m s 2, although it changes slightly with altitude and latitude mass The amount of matter in an object as defined by the product of its volume and density. It is measured in kilograms (kg) and is a scalar quantity weight The actual force of gravitation acting on an object at or near the surface of the Earth or other astronomical body. It is measured in newtons (N) and is a vector quantity F mg where F weight in newtons (N) m mass in kilograms (kg) g acceleration due to gravity (m s 2 )
4 Nelson Physics Stage 6 HSC Figure 1.3 Gravitation keeps the planets in orbit around the Sun (not to scale). Question 1 \\ WORKED EXAMPLE An object has a mass of 25 kg. a Calculate its weight on Earth given that g 9.8 m s 2. b Calculate its weight on Mars given that the acceleration due to gravity on Mars is 3.7 m s 2. Answer a b F mg 25 9.8 245 N The weight of the object on Earth is 245 N. F mg 25 3.7 92.5 N The weight of the object on Mars is 92.5 N. Determining g using a simple pendulum, pages 2 5, Practical Physics for Senior Students, HSC gravitational field Exists in any region where there is a gravitational effect. Gravitational field strength is a vector 1.2 Gravitational fields The force of gravity is the most universal of all forces. It acts on our bodies to ensure that our feet remain planted on the ground. It keeps satellites (including the Moon) in orbit around the Earth and the planets in orbit around the Sun. A gravitational field is said to exist in any region where there is a gravitational effect, i.e. where there is force on any mass placed in that area. Because force is a vector, gravitational field strength is also a vector. Consider the region near the surface of the Earth. The mass of the Earth may be considered to be concentrated at its centre. A gravitational field exists and its direction is towards the centre of the Earth. We define the strength of the field (from Newton s Second Law) as the gravitational force per unit mass, i.e. g m F The unit of g is newtons per kilogram, which is written as N kg 1.
Module 1 Chapter 1 \\ Gravity 5 On a larger scale, the gravitational field for an isolated point mass would be as shown in Figure 1.5. Note that in each case, the lines, called lines of force, indicate the direction of force on a mass placed at that point. The strength of the field decreases with the square of the distance, as shown graphically in Figure 1.6. For the isolated mass, the magnitude of the gravitational field strength, g, is given by: where g GM r 2 M mass involved (e.g. mass of a planet) r distance from the centre of that mass G a constant The constant G is called the universal gravitational constant. In the SI system, the value of this constant is 6.67 10 11 N m 2 kg 2. This can be used to calculate the gravitational field at any distance r from a mass; so it can be used to calculate the gravity on the surface of the Earth. Question 2 \\ WORKED EXAMPLE Given that the radius at a particular location on the Earth is 6365.0 km, calculate the gravity there. Consider the mass of the Earth to be 5.974 10 24 kg. Earth Figure 1.4 The gravitational fi eld at the surface of the Earth is downwards and perpendicular to the surface. m W Earth Figure 1.5 The gravitational fi eld for an isolated point mass m. g g = GM r 2 Answer g GM r 2 (6.67 10 11 ) (5.974 10 24 ) (6.365 10 6 ) 2 9.84 N kg 1 Earth s surface distance from centre of the Earth Figure 1.6 Graph of gravitational fi eld strength against distance. The gravitational field strength at a distance from the centre of the Earth is g GM 2, which means that the gravitational field strength is inversely r proportional to the square of the distance from the centre of the Earth, i.e. g 1 r. 2 Let us consider what happens to the gravitational field strength when the distance from the centre of the Earth is doubled. Let the radius of the Earth be R, then twice the distance from the centre of the Earth is 2R. Hence: g 1 r becomes 2 g 1 (2R) 2 g 1 4R 2 So doubling the radius has led to a reduction in gravitational field strength to one-quarter that on the Earth s surface. This means that the weight of an object will be one-quarter of what it is at the Earth s surface, even though its mass will not have changed.
6 Nelson Physics Stage 6 HSC \\ WORKED EXAMPLE Question 3 What is the gravitational fi eld strength at a distance of 2000 km above the Earth s surface? The mass of the Earth is 6.0 10 24 kg. The radius of the Earth is 6.4 10 6 m. The distance from the centre of the Earth is thus (6.4 10 6 2.0 10 6 ) m. The physics of the gravity of different planets, page 6, Practical Physics for Senior Students, HSC Answer From g GM r 2 g 6.67 10 11 6.0 10 24 (6.4 10 6 2.0 10 6 ) 2 5.7 N kg 1 Note that this means that the acceleration of a body such as a satellite at this height is 5.7 m s 2. Variations in gravity \ \\ DID YOU KNOW? Although the accepted value for gravity on the Earth s surface is 9.8 N kg 1, the actual value for g will depend on both geographical location and altitude. Minor variations in the acceleration due to gravity occur because: the Earth is not a perfect sphere (it is fl attened at the poles) the value for g at the poles is greater than at the Equator as the poles are closer to the centre of the Earth of variations in the thickness of the Earth s crust and the minerals found there the density of mineral deposits and positioning of tectonic plate boundaries affects the local value for g the spin of the Earth reduces the value of g this effect is greatest at the Equator and least at the poles and is due to a centrifugal effect that reduces the value of g of the altitude above the Earth s surface: at higher altitudes, gravity is less. The result of these effects is that the gravitational fi eld at the poles is 9.832 m s 2, at the Equator is 9.789 m s 2 and on Mount Everest is 9.77 m s 2. Figure 1.7 Mt Everest at higher altitudes, gravity is less.
Module 1 Chapter 1 \\ Gravity 7 Problem set 1A For the problems in this section assume: g (on the Earth s surface) 9.8 m s 2 g m (gravity on the Moon) 1.6 m s 2 G (the universal constant of gravitation) 6.67 10 11 N m 2 kg 2 M E (the mass of the Earth) 6.0 10 24 kg m (the mass of the Moon) 7.34 10 22 kg r (the radius of the Moon s orbit around the Earth) 3.84 10 8 m r E (the radius of the Earth) 6.38 10 6 m r m (the radius of the Moon) 1.74 10 6 m M S (the mass of the Sun) 2.0 10 30 kg Question 1 An astronaut of mass 60 kg travels from the Earth to the Moon. a b Calculate his weight on the Earth. Calculate his weight on the Moon. Question 2 Explain why mass and weight are often confused with each other. Question 3 a b What is the weight of a 1.0 kg mass on the surface of the Earth where the gravitational field strength is 9.8 N kg 1? How far from the centre of the Earth is the mass when its weight is 4.9 N? Question 4 Find the gravitational field strength at a point whose distance from the Earth s surface is equal to the radius of the Earth. Gravitational potential energy, kinetic energy and conservation of energy Question 5 In terms of G, M E and r E, find the gravitational field strength, g, at the surface of the Earth. Question 6 a b Calculate the gravitational field strength on the surface of the Moon due to the Moon s gravitational field. Calculate the acceleration due to the Moon s gravity on the surface of the Moon. Question 7 At what distance from the Earth would a spacecraft experience zero net gravitational force due to the opposing pulls of the Earth and the Moon? 1.3 Gravitational potential energy Potential energy is the energy that a body has due to its position or configuration. Gravitational potential energy (E p ) is the energy that a body has due to its position within a gravitational field. At the Earth s surface, the gravitational potential energy is equal to the work done in moving a mass to a distance h above the ground. From Preliminary potential energy The energy that a body has due to its position or configuration; stored energy gravitational potential energy (E p ) The energy of a mass due to its position within a gravitational field
8 Nelson Physics Stage 6 HSC Physics, you will recall that work is the product of force and distance, so mathematically: W Fs (mg) h mgh where m mass (kg) g force of gravity (m s 2 ) h distance above Earth s surface (m) E p r 0 r, distance from Earth s centre E p 0 E Gm M E r Figure 1.8 Gravitational potential energy in the Earth s fi eld. Note that in this case the mass of the planet (Earth) M E. So in this case, E p mgh. In this unit we are dealing with the gravitational potential energy associated with systems of large masses, where the distances between the masses are also very large. When dealing with small masses near the surface of the Earth, it is satisfactory to think of objects at the Earth s surface as having E p 0. For larger distances, we need a different approach. Work is done in moving an object away from the source of a gravitational field. The further the object is from the source of the field, the less work has to be done to it to move it away from the field as the gravitational field itself becomes less strong. To describe a universal relationship, gravitational potential energy is defined as being zero at a distance of infinity from the gravitational source. As a consequence of this, work is always done to move an object against the field so that it gains potential energy, E p. So the potential energy at point x is less than at point infinity because work has to be done to move it to point infinity (where E p 0). This means that E p at point x is negative! Mathematically, we find that gravitational potential energy is as follows: where E p Gm m 1 2 r E p gravitational potential energy (J) m 1 mass of planet (kg) m 2 mass of object (kg) r distance separating objects (m) \\ WORKED EXAMPLE Question 4 Calculate the energy required to move a satellite of mass 3500 kg from the Earth s surface to a position beyond the Earth s gravitational fi eld (mass of the Earth 6.0 10 24 kg, radius of Earth 6.4 10 6 m). Answer First, fi nd the gravitational energy of the satellite on the Earth s surface. E p Gm m 1 2 r 6.67 10 11 6.0 10 24 3500 6.4 10 6 2.2 10 11 J Since the gravitational energy at a point outside the Earth s gravitational fi eld is zero, the energy needed to go from the Earth s surface to beyond that point (infi nity) is 2.2 10 11 J.
Module 1 Chapter 1 \\ Gravity 9 Summary of gravity Mass is the amount of matter of which an object is composed and is a scalar quantity. It is the product of density and volume and is measured in kilograms (kg). Weight is the force on an object due to a gravitational field. It is a vector quantity. The acceleration due to gravity at the Earth s surface is taken to be 9.8 m s 2. Weight is found by the formula F mg. Weight is measured in newtons (N). The universal law of gravitation for an isolated mass can be used to find its gravitational field: g GM r. 2 The universal gravitational constant, G, is 6.67 10 11 N m 2 kg 2. Gravitational potential energy is the energy of a mass due to its position within a gravitational field. The gravitation potential energy of a mass at a particular point within a gravitational field is equivalent to the work done in moving the mass from that point to a point of infinite distance: E p Gm m 1 2 r. Review questions For the questions in this section assume: g (on the Earth s surface) 9.8 m s 2 g m (gravity on the Moon) 1.6 m s 2 g M (gravity on Mars) 3.7 m s 2 G (the universal constant of gravitation) 6.67 10 11 N m 2 kg 2 M E (the mass of the Earth) 6.0 10 24 kg m (the mass of the Moon) 7.34 10 22 kg r (the radius of the Moon s orbit around the Earth) 3.84 10 8 m r E (the radius of the Earth) 6.38 10 6 m r m (the radius of the Moon) 1.74 10 6 m M S (the mass of the Sun) 2.0 10 30 kg Question 1 Calculate the weight of a 625 kg lunar module on the Moon. Question 2 A vehicle has a weight of 2127.5 N on Mars. a Calculate its mass. b Explain why its mass is the same both on Earth and on Mars. Question 3 The gravitational field at the Earth s surface has an average accepted value of 9.8 m s 2 or N kg 1. Show that these two sets of units are equivalent. Question 4 The acceleration due to gravity on Earth is 9.8 m s 2 and on Mercury is 3.8 m s 2. If a person has a weight of 450 N on Earth, what would be the person s weight on the surface of Mercury? Question 5 The weight of an astronaut on the Moon is one-sixth of his weight on Earth. Calculate the acceleration due to gravity on the Moon.
10 Nelson Physics Stage 6 HSC Question 6 a Calculate the gravitational field on Venus, given that it has a mass of 4.9 10 24 kg and a radius of 6052 km. b Find the weight of an 80 kg person on the surface of Venus. Question 7 A satellite is in orbit at a distance of 350 km above the Earth s surface. Calculate the gravitational field strength at this distance. Question 8 The gravitational field strength experienced by a satellite orbiting Earth is 4.5 N kg 1. Calculate how high above the Earth s surface the satellite is in orbit. Question 9 A satellite of mass 200 kg is launched into a uniform circular orbit of radius 6.5 10 6 m around the Earth. a Calculate the magnitude of the gravitational potential energy, E p, of the satellite. b From this circular orbit, the satellite can escape the Earth s gravitational field when its kinetic energy is equal to the magnitude of the gravitational potential energy. Use this relationship to calculate the escape velocity of the satellite. Question 10 Determine the energy required to move a rocket of mass 25000 kg from the Earth s surface to a position beyond the Earth s gravitational field. Question 11 Calculate the amount of work done in moving a mass of 3000 kg from the surface of the Earth to a height 32 km above the Earth s surface. Question 12 A satellite is gradually increasing the size of its orbit around the Earth. Explain whether this is an increase or decrease in potential energy. Question 13 To place a satellite of mass 10000 kg in orbit around the Earth, 5.73 10 11 J of work is done. Calculate how far from the Earth s surface the satellite is placed in its orbital path.