EJC Physics H207 Gravitational Field 2023 1. Notes (FULL)

9749(2023) H2 Physics H207 Gravitational Field – Notes Page 1 of 26 H2 Topic 07 – Gravitational Field Despite the apparent “weightlessness” of astronauts at the International Space Station (ISS), they are still subject to a gravitational acceleration of 8.7 m s-1, or about 89% of “1 g”. If the Earth’s gravity were to suddenly stop acting on ISS, the ISS will move away from Earth tangentially at a constant velocity instead. Find out more here in this topic. 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 Objectives: 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 place at that point. (b) recognise the analogy between certain qualitative and quantitative aspect of gravitational and electric fields. (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 on the surface of the 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 small test mass from infinity to the point. (h) solve problems using the equation GM r φ = − for the 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. (bold) items are needed for 8867 H1 Physics

9749(2023) H2 Physics H207 Gravitational Field – Notes Page 2 of 26 7.1 Gravitational Field A “field” is a region of space where a force is experienced by an “entity” without contact. A mass experiences a gravitational force when placed in a gravitational field. Forces are vector quantities so the direction has to be well-defined. Gravitational force is a purely attractive force. An isolated mass generates its gravitational field in the region surrounding the mass and it permeates all of space (“to infinity”). When another mass is inside this region, that mass interacts with the existing gravitational field and both masses experience gravitational force. 7.1.1 Field Lines Field lines represent a field visually. Arrows show the direction of force acting on a test mass. A test particle is an idealized object which does not alter the behaviour of the rest of the system. The value of a test particle is shown when ther e is more than one body of significant mass . The diagram below shows the resultant field lines of the gravitational field jointly generated by both the Earth and the moon: The moon is not a test particle. It itself generates a gravitational field of its own and alters the behaviour of Earth’s gravitational field. If we replace the moon with a test particle, the resulting field will be purely radial towards Earth’s centre. Both the moon and Earth will attract any test masses in this left-hand- side region. So the field lines here generally point to the right. At this point (location), a test mass is equally attracted to both the moon and the Earth: In this right -hand-side region, the smaller mass of the moon results in negligible changes to the resultant field pattern; as if only Earth’s gravitational field is present. At this point (location), a test mass will be attracted to both the moon (weaker) and the Earth (mor e strongly). The resultant direction is due to the vector sum of forces: The tangent at a point on a gravitational field line is the direction of gravitational force acting on a test mass placed at that point. Field lines will never cross each other. A gravitational field is a region of space where a mass experiences a gravitational force. Gravity is an attractive force. Unfortunately, the anti -gravity hover-board from the “Back to the Future” franchise is one piece of technology that will stay as science- fiction. A line of force in a gravitational field is the direction of the gravitational force acting on a small test mass.

9749(2023) H2 Physics H207 Gravitational Field – Notes Page 3 of 26 Example 1 Notes: (i) we represent uniform fields using equally-spaced parallel field lines pointing in the same direction. (ii) A uniform field is one where the field strength has the same magnitude and same direction everywhere within the field. (iii) Near Earth’s surface, by Newton’s 2nd Law, net mmF ag= = so the gravitational field strength g is equal to the acceleration of free fall. There is a formal definition for field strength which we will introduce later. Visually, from the above example we can already see that gravitational field strength is • stronger where gravitational field lines are closer together • weaker where gravitational field lines are further apart from each other Now that we are familiar with 1. the direction of gravitational force (mutual attraction between masses) 2. the general variation (the further away, the weaker the force of attraction) We now look at how to quantify the gravitational force. By reference to lines of gravitational force near the Earth’s surface, explain why the acceleration of free fall near the Earth’s surface is approximately constant. Solution • The lines of force are radial and appear to converge at the centre of Earth. • Earth is a large radius, relatively the heights near and above the surface is small so the lines are approximately parallel. • The parallel field lines indicate uniform field streng th hence constant acceleration of free fall. comparatively weaker field comparatively stronger field

9749(2023) H2 Physics H207 Gravitational Field – Notes Page 4 of 26 7.2 Newton’s Law of Gravitation The gravitational constant G = 6.67 × 10-11 N m2 kg-2. m1 and m2 are the masses of the two point masses and r is the distance (separation) between the two point masses (i.e. the distance between the centres of the mases). The direction of the gravitational force acts along the line joining the two point masses. By Newton’s 3 rd Law, the gravitational attraction that m1 exerts on m 2 is equal in magnitude and opposite in direction as the gravitational force that m2 exerts on m1. Newton’s Law of Gravitation works between point masses. Even though planets and stars are massive, Newton’s Law of Gravitation still applies as the distances between planetary bodies are significantly much larger than their diameters so the bodies can still be regarded as point masses. Note: In some textbooks, Newton’s Law of Gravitation is given as 2F GMm r=− ; the negative sign denotes that the direction of the force is opposite to the direction of r, the displacement (or position) vector. Newton’s law of gravitation states that the [type of force] gravitational force of attraction between two point masses [magnitude] is directly proportional to the product of the masses and inversely proportional to the square of separation between the masses 12 2 mFG m r= r m1 m2 F F force on m by M M R r m + ve direction for r - ve direction for F

9749(2023) H2 Physics H207 Gravitational Field – Notes Page 5 of 26 Example 2 The Sun (mass of 3010 g1 99 k. × ) is 111 0 m1 0 5