RI Chap 8 Gravitational Fields Lecture Notes

8 GRAVITATIONAL FIELDS H2 Physics 9748 Content Page Introduction 2 8.1 Gravitational Force F 2 8.2 Gravitational Field Strength g 5 8.3 Gravitational Potential Energy U and Gravitational Potential φ 8 8.4 Graphs of φ − r and g − r between Two Masses 14 8.5 Escape Velocity 15 8.6 Rotation of Earth 16 8.7 Circular Orbits 20 Summary 28 Appendixes 29 Learning Outcomes Candidates should be able to: (a) recall and use Newton’s law of gravitation in the form 12 2 Gm mF r= . (b) derive, from Newton’s law of gravitation and the definition of gravitational field strength, the field strength due to a point mass = 2 MgG r . (c) recall and use = 2 MgG r for the gravitational field strength due to a point mass to solve problems. (d) 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. (e) define gravitational potential at a point as the work done per unit mass by an external force in bringing a small test mass from infinity to that point. (f) solve problems using the equation φ = − MG r for the gravitational potential in the field due to a point mass. (g) show an understanding that the gravitational potential energy of a system of two point masses is = −G MmUG r . (h) recall that gravitational field strength at a point is equal to the negative potential gradient at that point and use this to solve problems. (i) analyse problems related to escape velocity by considering energy stores and transfers. (j) analyse circular orbits in inverse square law fields by relating the gravitational force to the centripetal acceleration it causes. (k) show an understanding of satellites in geostationary orbit and their applications.

Page | 2 Introduction In 1687, Isaac Newton proposed in his Philosophiæ Naturalis Principia Mathematica that every mass attracts another mass with a force of gravity. According to him, two seemingly unrelated phenomena – the fall of an apple from an apple tree and the orbital motion of the planets around the Sun – are due to the same reason: gravitational attraction. He came up with the Newton’s Law of Gravitation. This law is universally valid and applies to any planet in the solar system and even between distant galaxies. How Newton discovered Newton’s Laws of Gravitation 8.1 Gravitational Force F  Newton’s Law of Gravitation The magnitude of the gravitational force F between two particles of masses M and m which are separated by a distance r is given by 2 GMmF r= where G is the gravitational constant and G = 6.67 × 10−11 N m2 kg−2. Gravitational force is a vector quantity. S.I. unit for gravitational force is the newton (N). Fig. 8.1 Newton’s Law of Gravitation It states that two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the