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 square of the distance between them. F – F r M m Formula

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 3 | Page Note: • This law is applicable only between point masses. - Every spherical body with constant density can be considered as a point mass at the centre of the sphere. • The gravitational forces between two point masses are equal and opposite. As shown in Fig. 8.1, they - constitute an action and reaction pair of forces. - are attractive in nature, and - always act along the line joining the two point masses. • Newton’s law of gravitation is an example of an inverse square law - because the magnitude of the force varies inversely with the square of the separation of the two point masses. • Gravitational force is attractive in nature. - Some text/sources indicate this with a negative sign in the formula to highlight that the force is attractive. Example 1 A man of mass 85.0 kg is standing on the surface of the Earth. The Earth has a mass of 5.98 × 1024 kg and a radius of 6.37 × 106 m. Calculate the force that the Earth exerts on the man and the force that the man exerts on Earth. Solution

Page | 4 Example 2 On the surface of the Earth, the gravitational force acting on an object is 45 N. When the object is at a height h above the surface, the gravitational force acting on it is 5 N. Determine h in terms of R where R is the radius of the earth. Solution Example 3 The mass of the Earth is 5.98 × 1024 kg and that of the Moon is 7.35 × 1022 kg. A spacecraft travelling from the Earth to the Moon will reach a point X where it experiences no resultant gravitational force. The distance between the centre of the Earth and the centre of the Moon is 3.85 × 108 m. (a) Draw a free-body diagram of the spacecraft at X. (b) Calculate the distance from X to the centre of the Moon. (c) Describe the motion of the spacecraft before reaching X and after passing X if its engine is switched off. Solution (a) Moon X Earth 3.85 × 108 m Moon X Earth Fm FE Mm ME d 3.85 × 108 m

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 5 | Page 8.2 Gravitational Field Strength g  Gravitational Field  Gravitational Field Lines A region of gravitational field can be visualised as consisting of an array of imaginary field lines. The gravitational force on a mass placed at a point in a field, acts along the tangent to the field line at that point. • The direction of the field lines indicates the direction of the gravitational field. • The density of the field lines indicates its strength. - A region with a stronger gravitational field strength will have closer or denser field lines. • For a point mass or a uniform spherical mass, the field lines are directed towards its centre. • Fig. 8.2(a) shows the field lines around Earth. Zooming into a region near the surface of Earth, the field lines seem to be parallel to each other and evenly spaced as shown in Fig. 8.2(b). Gravitational Field A gravitational field is a region of space in which a mass placed in that region experiences a gravitational force.

Page | 6 • Hence, near the surface of the Earth, we can consider the gravitational field to be approximately uniform.  Gravitational Field Strength, g Fg m=  Gravitational Field Strength of a Point Mass Since the gravitational force between two point masses is given by 2 GMmF r= , 2 GMg r= Gravitational field strength is a vector quantity. Gravitational field strength is also known as gravitational acceleration or free fall acceleration. S.I. unit for gravitational field strength is N kg–1 or m s–2. Fig. 8.3 Gravitational Field Strength The gravitational field strength at a point in space is defined as the gravitational force experienced per unit mass at that point. Fig. 8.2(a) Field lines around the Earth Fig. 8.2(b) Field lines near the surface of the Earth r g point P M Formula Formula

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 7 | Page Note: Gravitational field strength points towards the mass which created it as shown in Fig. 8.3. Some books indicate this by including a negative sign, just like gravitational force. The resultant field strength at a point due to more than one mass can be found from the vector sum of the individual gravitational field strengths due to each mass at that point. Example 4 Determine the resultant gravitational field strength at points A and B due to the two masses, given that M2 > M1. Solution  Gravitational Field Strength of a Uniform Sphere For a uniform solid sphere of radius R, the variation of the gravitational field strength g with displacement r from the centre of