DHS 07 Gravitational Field (Lecture Notes)

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field Page 1 of 20 Guiding Questions • How do two masses interact? Do they need to be in physical contact to do so? • What do field lines represent? Do field lines represent similar things for gravitational fields and electric fields? • How can we understand the motion of planets and satellites? Are we at the centre of the universe? Content • Gravitational field • Gravitational force between point masses • Gravitational field of a point mass • Gravitational field near to the surface of the Earth • Gravitation potential • Circular orbits Learning Outcomes Students should be able to: Gravitational field (a) show an understanding of the concept of a gravitational field as an example of field of force and define gravitational field strength at a point as the gravitational force exerted per unit mass placed at that point (b) recognise the analogy between certain qualitative and quantitative aspects of gravitational and electric fields. [To be taught in the topic of “Electric Field”] Gravitational force between point masses (c) recall and use Newton's law of gravitation in the form 12 2 Gm mF r= Gravitational field of a point mass (d) derive, from Newton's law of gravitation and the definition of gravitational field strength, the equation 2 GMg r= for the gravitational field strength of a point mass (e) recall and apply the equation 2 GMg r = for the gravitational field strength of a point mass to new situations or to solve related problems Gravitational field near the surface of the Earth (f) show an understanding that near the surface of the Earth, gravitational field strength is approximately constant and is equal to the acceleration of free fall Gravitational potential (g) define the gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to that point (h) solve problems using the equation GM rφ = − for the gravitational potential in the field of a point mass Circular orbits (i) analyse circular orbits in inverse square law fields by relating the gravitational force to the centripetal acceleration it causes (j) show an understanding of geostationary orbits and their application 9749 H2 Physics Topic 7 Gravitational Field Year 5 (2024) DUNMAN HIGH SCHOOL

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field Page 2 of 20 0. Field Concepts 1. Introduction Gravitational force is a force that is evident in our everyday lives and plays a crucial role in many processes on Earth. For instance, the ocean tides are caused by the gravitational attraction of both the Moon and Sun on the Earth’s oceans. The falling of objects when released is also caused by the gravitational pull of the Earth on all objects. In terms of planetary motion, gravitational force is responsible for keeping the Earth in its orbit ar ound the Sun, which in turn gives rise to four seasons in some countries, as Earth’s tilted axis always points in the same direction when it orbits the Sun.

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field Page 3 of 20 (c) recall and use Newton's law of gravitation in the form 12 2 Gm mF r = 2. Gravitational force acting between two point masses m1 and m2 with separation r Newton's law of gravitation states that the gravitational force of attraction between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the separation between their centres. This means that if there are two point masses 1m and 2m and they are separated by distance r, the magnitude of the gravitational force attracting them to each other is 12 2 Gm mF r= where G = 6.67 × 10−11 N m 2 kg −2, which is the constant of proportionality known as gravitational constant (provided in Data list). Example 1 (Use of Formula for Newton's law of gravitation) There are (a) a boy and a girl, with masses 65 kg and 45 kg respectively, separated by 1.1 m, and (b) the Earth and its Moon , with masses 6.0 × 10 24 kg and 7.4 × 10 22 kg respectively, separated by 3.8 × 108 m. Determine the gravitational force between masses in (a) and (b). Solution: (a) F = 12 2 Gm m r = 11 2 (6.67 10 )(65)(45) (1.1) −× = 1.6 × 10−7 N (b) F = 12 2 Gm m r = 11 24 22 82 (6.67 10 )(6.0 10 )(7.4 10 ) (3.8 10 ) −××× × = 2.1 × 1020 N F F m1 m2 r Note: (1) Point masses have non-zero mass and no volume. If two objects are placed sufficiently far apart such that their dimensions become negligible compared to the distance separating them, the two objects can be considered point masses. (2) The gravitational forces between two masses are equal and opposite and constitute an action and re action pair of forces (Newton’s 3 rd Law). The forces always act along the line joining the two point masses. To show its attractive nature, it is written (with negative sign) as: 12 2 Gm mFr r= −   The negative sign is ignored when only the magnitude of the force is required. INQUIRY 01 : What is the significance of the answers?

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field Page 4 of 20 (a) show an understanding of the concept of a gravitational field as an example of field of force and define gravitational field strength at a point as the gravitational force exerted per unit mass placed at that point. 3.1. Gravitational field Think about it: How can two objects exert attractive force on each other when they are not in contact with each other? Every object with mass sets up a gravitational field in its surrounding space. When two objects enter each other’s gravitational fields, they will be attracted towards each other. Hence, when an object with mass is placed in a gravitational field (an example of field of force 1), there would be a gravitational force acting on it. INQUIRY 02 : Gravitational field is invisible and is represented by imaginary field lines. How would the Earth’s gravitational field (both over large distances from Earth and near Earth) looks like? (a) In Fig. 1, several small masses are placed far from the Earth. Draw (using pencil) the direction of gravitational forces acting on them by Earth; (b) In Fig. 2, several small masses are placed equidistant from each other near the Earth’s surface. Draw the direction of gravitational forces acting on them by Earth. Fig. 1: • The gravitational field around Earth is non-uniform. • The Earth is usually assumed to be a point mass with all its mass concentrated at its centre. The field lines should be drawn radially pointing towards the centre of Earth. Fig. 2: • The gravitational field near Earth’s surface is uniform. Within a uniform field, the field strength is the same at all points. • The field lines should be drawn parallel to each other and of equal spacing. 1 A field of force is a region of space where there is a force acting on an object placed in that field. An object placed in an ordinary space (like in deep outer space, which is not a field) would not have any force acting on it. surface of the Earth Graphs play an important role in this topic. Let us investigative various key graphs through the “Inquiry” questions. Fig. 2 Fig. 1

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field Page 5 of 20 The direction of a field at a point is along a tangent to the field line at that point, as shown in Fig. 3. The density of the field lines at a point (number of lines per unit area) corresponds to the strength of the field at that point. A denser arrangement of field lines indicates greater gravitati