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# 6.2 newtons law of gravitation

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### 6.2 newtons law of gravitation

1. 1. Topic 6: Circular motion & Gravitation 6.2 Newton’s law of gravitation
2. 2. Gravitational Force & Fields What do you know about gravity? What does a gravitational field look like? Can we defy gravity? What does gravity affect?
3. 3. Gravitational Force and Field Newton proposed that a force of attraction exists between any two masses. This force law applies to point masses not extended masses However the interaction between two spherical masses is the same as if the masses were concentrated at the centres of the spheres.
4. 4. Newton´s Law of Universal Gravitation Newton proposed that “every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses, and inversely proportional to the square of their distance apart”
5. 5. This can be written as F = G m1m2 r2 Where G is Newton´s constant of Universal Gravitation It has a value of 6.67 x 10-11 Nm2 kg-2
6. 6. Calculate the gravitational pull of the Earth on the moon Mm =0.01230*Me Me =5.976 x 1024 kg R = 384400 km
7. 7. Calculate the mass of the Earth How do you determine the mass of the Earth?
8. 8. Gravitational Field Strength A mass M creates a gravitational field in space around it. If a mass m is placed at some point in space around the mass M it will experience the existance of the field in the form of a gravitational force
9. 9. We define the gravitational field strength as the ratio of the force the mass m would experience to the mass, M That is the gravitational field strength at a point, is the force exerted per unit mass on a particle of small mass placed at that point
10. 10. The force experienced by a mass m placed a distance r from a mass M is F = G Mm r2 And so the gravitational field strength of the mass M is given by dividing both sides by m g = G M r2
11. 11. The units of gravitational field strength are N kg-1 The gravitational field strength is a vector quantity whose direction is given by the direction of the force a mass would experience if placed at the point of interest
12. 12. Field Strength at the Surface of a Planet If we replace the particle M with a sphere of mass M and radius R then relying on the fact that the sphere behaves as a point mass situated at its centre the field strength at the surface of the sphere will be given by g = G M R2
13. 13. If the sphere is the Earth then we have g = G Me Re 2 But the field strength is equal to the acceleration that is produced on the mass, hence we have that the acceleration of free fall at the surface of the Earth, g g = G Me Re 2
14. 14. Is g = 9.81 ? Use the data from before and verify whether the acceleration due to gravity is 9.81 ms-2
15. 15. Orbital speed The speed of an object can be found from V = 2 π r /T but can you show that GM = v2 r
16. 16. Orbital Motion Gravitation provides the centripetal force for circular orbital motion The behaviour of the solar system is summarised by Kepler´s laws Kepler´s law state  1. Each planet moves in an ellipse which has the sun at one focus  2. The line joining the sun to the moving planet sweeps out equal areas in equal times
17. 17. Deriving the Third Law Suppose a planet of mass m moves with speed v in a circle of radius r round the sun of mass M The gravitational attraction of the sun for the planet is = G Mm r2 From Newton’s Law of Universal Gravitation
18. 18. If this is the centripetal force keeping the planet in orbit then G Mm = mv2 (from centripetal equation) r2 r ∴GM = v2 r
19. 19. If T is the time for the planet to make one orbit v = 2π r v2 = 22 π2 r2 T T2 ∴ GM = 4π 2 r2 r T2 ∴ GM = 4π 2 r3 T2 ∴r3 = GM T2 4π 2 ∴r3 = a constant T2