Gravitational Field JPJC Notes

1 JURONG PIONEER JUNIOR COLLEGE 9749 H2 PHYSICS Gravitational Field Learning Outcomes Candidates should be able to: Gravitational field (a) show an understanding of the concept of a gravitational field as an example of field of force and define the gravitational field strength at a point as the gravitational force exerted per unit mass placed at that point. (b) recognise the analogy between cer tain qualitative and quantitative aspects of gravitational and electric fields (*To be taught in Electric Fields) Gravitational force between point masses (c) recall and use Newton's law of gravitation in the form F = 12 2 Gm m r Gravitational field of a point mass (d) derive, from Newton's law of gravitation and the definition of gravitational field strength, the equation g = 2 GM r for the gravitational field strength of a point mass (e) recall and apply t he equation g = 2r GM for the gravitational field strength of a point mass to new situations or to solve related problems Gravitational field near to the surface of the Earth (f) 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 Gravitational potential (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  = r GM− for the gravitational potential in the field of a point mass Circular orbits (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.

2 Introduction ▪ Isaac Newton (1642 –1726), published Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy") in 1687. In it, he formulated the Newton’s laws of motion (Dynamics), and the Newton’s law of gravitation (Gravitational Field). ▪ Newton’s studies in gravitation started when he observed an apple falling to the ground and he began to ask himself questions. Why does a falling apple always descend perpendicularly to the ground? Did the force that pulled an apple to the Earth also extend further and pull bodies like the Moon towards the Earth? Would this force affect the orbit of the moon? We will answer these questions and more in this chapter. 1 GRAVITATIONAL FORCE (c) Candidates should be able to recall and use Newton’s law of gravitation in the form 12 2 Gm mF r= . . 1.1 Newton’s law of gravitation Newton’s law of gravitation states that the force of attraction between two point masses is directly proportion al to the product of the masses and inversely proportional to the square of the distance between them. Fig. 1 Gravitational force between two bodies ▪ Mathematically, it is written as 2 21 r mmGFg = where where G is the gravitational constant with a value of 11 2 26.67 10 N m kg−− , and are the masses of the two bodies respectively, r is the distance between their centres of masses, and Fg is the magnitude of the gravitational force acting between them. ▪ This is known as an inverse square law, as the magnitude of the force varies with the inverse square of the separation of the particles. ▪ The gravitational forces of attraction between two bodies form an action -reaction pair, consistent with Newton’s third law. ▪ For spherical masses, the distance r is measured between their respective centres of mass (assume mass concentrated at centre). 1m 2m r

3 ▪ In vector form, the equation is written as 2 21 r mmGF   −= to indicate the attractive nature of the force. The negative sign is normally ignored when only magnitude is required. Example 1 A man of mass 85.0 kg is standing on the surface of the Earth. Given that the mass of the Earth is 5.98 × 1024 kg with a radius of 6.37 × 106 m, calculate the gravitational force the Earth exerts on the man. Hence, state the gravitational force the man exerts on the Earth. Solution: 12 2 11 24 62 (6.67 )(5.98 )(85.0) (6.37 10 ) 10 10 835.539... N 840 N g Gm mF r = =   =  The gravitational force that the Earth exerts on the man is 840 N towards Earth. The gravitational force that the man exerts on the Earth is 840 N towards the man. Example 2 Two stationary objects of masses 35.0×10 kg and 3101.0 kg are placed at a dist ance of 10 m apart. A third object of mass 1.0 kg, placed between the two masses, experiences no resultant force. (a) On Fig. 1.1, draw the forces acting on the 1.0 kg object. (b) Calculate the distance x between the 1.0 kg and the 1000 kg objects. Solution: (a) Fig. 1.1 10 m kg x kg .0 kg

4 (b) Since the 1.0 kg object experiences zero resultant force, ( ) 2 32 2 31 10 x MMG x MMG = − ( ) 22 1000 10 5000 xx = − ( ) 5000 1000 10 2 2 = − x x Taking square root on both sides of the equation, ( ) 5 1 10 =− x x xx 510 =− 1.3x m Therefore, the distance x between the 1.0 kg and 1000 kg objects is 3.1 m. 2 GRAVITATIONAL FIELD (a) Candidates should be able to show an understanding of the concept of a gravitational field as an example of field of force and define gravitational field strength as force per unit mass.. . 2.1 Concept of gravitational field ▪ A field is a model used by physicists to help them understand how objects not in direct contact with each other can affect each other’s behaviour. Common types of field are gravitational, electric and magnetic fields. ▪ A gravitational field is a region of space in which a ma ss experiences a force due to the presence of another mass. ▪ The gravitational field of a mass can be represented by field lines. The field is directed towards the centre of the mass and gets stronger as one gets closer to the surface. Fig. 2 Gravitational field lines of a mass

5 (d) Candidates should be able to 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. . 2.2 Gravitational field strength Gravitational field strength at a point is defined as the gravitational force per unit mass acting at that point. Fig. 3 Gravitational field strength at point X due to an object of mass M is 2r GMg X = ▪ It is a vector having both direction and magnitude, therefore addition of gravitational field strengths must be done by vector addition. The direction is the same as the gravitational force. Units for g is N kg−1. ▪ Note that the gravitational field at point X in Fig. 3 is independent of mass m. The value of g only depends on the distance r from the object causing the gravitational field. Example 3 Using Newton’s law of gravitation and the definition of gravitational field strength, derive the equation 2r GMg = . Solution: Consider a mass M and another mass m placed at point X, which is at a distance r away from mass M. By Newton’s law of gravitation, the force gF experienced by mass m is 2r GMm . M point X r 2r GM m F g g == M m r point X

6 From the definition of gravitational field strength at point X, = gFmg m experienced by massat point X mass 2 MmG rg= m 2 GMg= r Hence, the gravitational field strength at point X due to mass M is