HCI 07 Gravitation Lecture Notes

Hwa Chong Institution (College) H2 Physics C1 2023 1 Chapter 7 Gravitational Field Playlist of Lecture Examples and Concept Videos can be found at https://youtube.com/playlist?list=PL_b5cjrUKDlbcIl2_xwdbabhxq0fs40Xw. Alternative resource (with videos) at xmphysics: https://xmphysics.com/xmGravitation/

Hwa Chong Institution (College) H2 Physics C1 2023 2 Chapter 7: Gravitational Field H2 Physics Syllabus 9749 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 fie ld as an example of field of force and define the gravitational field strength at a point as the gravi tational force exerted per unit mass placed at that point. (b) recognize the analogy between certain qualitative and quant itative aspects of gravitational and electric fields (will be done in the chapter on electric fields). (c) recall and use Newton’s law of gravitation in the form 2 21 r mGmF  . (d) derive from Newton’s law of gravitation and the definition of gravitational field strength, the equation 2r GMg  for the gravitational field strength of a point mass. (e) recall and apply the equation 2r GMg  for the gravitational field strength of a point mass to new situations or to solve related problems. (f) show an appreciation that near the surface of the Earth, gr avitational field strength is approximately constant and equal to the acceleration of free fall. (g) define the gravitational potential at a point as the work d one per unit mass in bringing a small test mass from infinity to the point. (h) solve problems by using the equation r GM for the gravitational potential in the field of a point mass. (i) analyse circular orbits in inverse square law fields by rel ating the gravitational force to the centripetal acceleration it causes. (j) show an understanding of geostationary orbits and their app lication.

Hwa Chong Institution (College) H2 Physics C1 2023 3 7.1 Newton’s Law of Universal Gravitation Newton’s Law of Universal Gravitation states that: Every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Consider two point masses of m 1 and m2 separated by a distance r. Each will exert a force F on the other, and its magnitude is given by 2 21 r mmGF  where F : gravitational force between the two point masses. m1, m2 : masses of the two point masses. r : distance between the two point masses. G : gravitational constant The two forces form an action-reaction pair and have the following characteristics,  are equal in magnitude,  are opposite in direction,  act on different bodies  are of the same type (gravitational force). G is called the gravitational constant (or constant of universal gravitation), which has been measure d experimentally to be: G = 6.67 x 10 -11 N m 2 kg-2. It is given in the list of constants provided on the second page of all tests and exams. Important points to note about Newton’s Law of Gravitation 1. Newton’s Law of Gravitation is a universal law. It applies everywhere in the universe. 2. Attractive nature of gravitational force: N o t e t h a t t h e m a s s e s i n t h i s c a s e a r e a l w a y s attracted to each other. 3. The gravitational force is a field force that always exists between two masses regardless of the medium that separates them and the two masses need not be in contact with other. In fact, it would still exist if there were no medium between them. m1 m2 F F r

Hwa Chong Institution (College) H2 Physics C1 2023 4 4 . Inverse-square law: Newton’s Law of Gravitation is an example of an inverse-square law, i.e., the force is inversely proportional to the square of the separation of the masses. F  2 1 r 5. Newton’s Law of Gravitation can be applied to spherical bod ies as though they were point masses with all their mass concentrated at their geometrical centres. Example 1: Finding Resultant Gravitational Force Three identical masses, each of mass m, are located on a table at the corners of an equilateral triangle of side d. Determine the resultant gravitational force on mass C due to masses A and B. Solution: A B C y x Concept: If several masses are present, each pair will experience a mutual gravitational attraction. The resultant gravitational force on a given mass is the vector sum of the separate attractive forces acting on it due to the masses interacting with it.

Hwa Chong Institution (College) H2 Physics C1 2023 5 7.2 Gravitational Field Strength The gravitational field is a region of space where a mass will experience a gravitational force. Consider a particle of mass m placed at a point in a gravitational field (due to another mass M). The particle would experience a gravitational force, F g, in the gravitational field, given by: Fg = mg Definition: Gravitational Field Strength, g The gravitational field strength g at a point is the gravitational force per unit mass acting on a small test mass placed at the point. g = gF m where g : gravitational field strength at that point Fg : gravitational force acting on mass m placed at that point in the gravitational field. Notes: 1. S.I. unit of g: N kg-1 or m s-2. 2. g is a vector quantity. Its direction is determined by the gravitational force acting on a mass m placed at the point in the gravitational field. 3. Mass m is a particle that is placed in the gravitational field. It is not the mass that is creating the gravitational field. Hence, mass m is often known as the test mass. The mass M that is creating the gravitational field is known as the source mass. The presence of the test mass is not necessary for the field to be present. 4. g is also commonly known as the acceleration due to gravity or gravitational acceleration. M m Fg

Hwa Chong Institution (College) H2 Physics C1 2023 6 7.2.1 Gravitational Field of a Point Mass If a point mass M is placed at some point in space, it will set up a gravitation al field around itself. Suppose we wish to determine the gravitational field strength g at a distance r away from this point mass M. We need to place a test mass m at the distance r away from this source mass M. The test mass would experience a gravitational force F as it is in the gravitational field created by M. By Newton’s Law of Gravitation,  The magnitude of the gravitational force acting on m due to the gravitational field set up by M, F = 2r MmG From the definition of gravitational field strength,  The magnitude of gravitational field strength due to M at a distance r away from M, g = Magnitude of gravitational force acting on test mass Mass of test mass m m = m F = 2 MmG r m   = 2r MG Gravitational Field Strength due to a Point Mass The magnitude of the gravitational field strength at a point due to a point mass M at a distance r away i s g i v e n b y : g = 2r MG Note: The equation is also valid for gravitational field strength outside a spherical body of uniform density or spherical shells of uniform density, where its whole mass is co ncentrated at its geometric centre. 1 r would then represent the distance between the test mass and the geometric centre of the source mass. 1 Read