ASRJC Gravitational Field Notes

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-1 Additional Notes Topic 7 Gravitational Field Content • Gravitational Field • Gravitational force between point masses • Gravitational field of a point mass • Gravitational field near to the surface of the Earth • Gravitational potential • Circular orbits Learning Outcomes: Candidates should be able to: (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. (will be discussed in JC2) (c) recall and use Newton's law of gravitation in the form = 12 2 Gm mF r . (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. (f) show an understanding that near the surface of Earth g is approximately constant and equal to the acceleration of free fall. (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=- r for the gravitational potential in the field of a point mass. (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. Nature of Science A good theory should lead to new discoveries. Newton’s theory of gravitation has enabled us to work out problems connected with space travel, leading to new knowledge about the solar system and the discovery of new planets. Relating Science and Society Explore how data from satellites is used in Singapore, making reference to the work of the Centre for Remote Imaging, Sensing and Processing (CRISP) and the Meteorological Service Singapore (MSS). Examples of how the data is used by Singapore include the monitoring of the haze problem and the prediction of the weather. By analysing satellites currently deployed around the globe, compare the functionality and purposes of various types of satellites (e.g. geostationary versus polar and low earth orbit), and weigh the costs and benefits of launching our own satellites. 60 ASMC, Regional Haze Situation. Retrieved from [http://asmc.asean.org/home/]. 61 NEA, Haze updates. Retrieved from [http://www.haze.gov.sg/haze -updates]. Name: ________________________________ ( ) Class: 25 / ___

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-2 Additional Notes Before 1686, much data had been collected on the motions of the Moon and the planets, but scientists had yet to understand the forces which governed those motions. Isaac Newton knew that, from his laws of motion, a net force had to be acting on the Moon. Otherwise, the Moon would move in a straight -line path rather than in its almost circular orbit. Along with other reasons, this led Newton to postulate t he Universal Law of Gravitation. • Mathematically, the magnitude of the gravitational force F between two masses M and m, separated by a distance r, is given as where G is the universal gravitational constant. (G = 6.67 x 10-11 N m2 kg-2) • Note that: o When stating the law, you must specify that the masses are point masses. In practice, point masses are not possible, but the law applies to masses whose separation is much greater than their dimensions (separation  radius). The law also applies to spheres of uniform densities, given their separation is much greater than their dimensions . In which case, the mass of the sphere is taken to be concentrated at its centre. M m r 2 MmF r 2 MmFG r= Introduction A.1 Newton’s Law of Gravitation Newton’s Law of Gravitation: The mutual force of attraction between any two point masses is proportional to the product of the masses and inversely proportional to the square of their separation. For spherical mass of uniform densities, r will be centre to centre distance.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-3 Additional Notes o Gravitational forces are always attractive. Sometimes a negative sign is used to indicate the attractive nature of the force. The magnitude of the gravitational force between two masses is actually given by: 2 MmFG r= o The force acts along the line joining the centres of the bodies. o The mutual attractive forces between the two bodies form an action-reaction pair. M is attracted towards m with a force F towards the right. At the same time, m is attracted by M with an equal force F but in the opposite direction. (Newton’s 3rd Law) o Since the value of G is very small, gravitational forces are significant only when the bodies have significant masses. Worked example 1 Two identical lead spheres of mass 3.00 kg each are placed such that the separation is 10.0 m from the centre of one sphere to another. They attract each other with a gravitational force F. Calculate the force F. (6.00 x 10-12 N) Example 1 Find the force between (a) the Earth and a 50 kg girl (b) a 60 kg boy and a 50 kg girl, one metre apart. [mass of Earth = 6.0  1024 kg, radius of Earth = 6.4  106 m] Solution 2 11 2 -12 3.00 3.006.67 10 (10.0) 6.00 x 10 N MmFG r − = =   = F F r M m

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-4 Additional Notes Example 2 The average distance separating the centre of Earth and the centre of Moon is 3.84 x 108 m. A spaceship of mass 3.00 x 10 4 kg travels along the line joining the centre of mass of the Earth and the Moon. (a) Determine the net gravitational force on the spaceship due to the Earth and the Moon when it is located halfway between them. (322 N towards the Earth) (b) Determine the distance between the spaceship and the Earth where no net gravitational force acts on the spaceship. (3.46 x 108 m) [mass of Earth = 6.00 x 1024 kg, mass of Moon = 7.36 x 10 22 kg] Thinking Process Key question on objective: Is force a vector or scalar? So does the direction matter? • Vector. Direction matters. Key questions to solve the question: (a) What is the question asking for? • Net gravitational force on spaceship What is the approach to take? • Find gravitational force between spaceship and earth; between spaceship and moon • Then take vector sum of the two forces to find the net force (b) What is the net force at the required point? Where is the point? • Need the point where =ES MSFF Express in terms of r, where distance from centre of earth to the spaceship is r. • I will need a point, where distance from centre of earth to the spaceship is r, and the distance from the centre of the moon to the spaceship will be (3.48 x 10 8 – r) m

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-5 Additional Notes Learning Point: F12 = 𝐺𝑀1𝑀2 𝑟2 is the gravitational force that M1 exerts on M2. If there is a third mass M3, then M3 also exerts a gravitational force on M2. To find the net gravitational force acting on M2, the resultant of the 2 forces must be found. M1 M3 M2 F12 F32

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 7-6 Additional Notes • A field is a model used by physicists to help them understand how objects not in direct contact with each other (ca