Gravitational Field Strength

1. Learning Objectives Book Reference : Pages 62-63 Book Reference : Pages 56-57 Gravitational Field Strength To investigate how the gravitational field strength varies as you move away from a planet and also inside the planet To begin to explore the concept of Gravitational potential HW : Read about Cavendish’s measurement of G: http://en.wikipedia.org/wiki/Cavendish_experiment

2. Last lesson we used planetary data from our solar system to confirm Newton’s law of gravitation. • For a planet acting as a point mass M the magnitude of the force of attraction on a mass m can be given by : • F = GmM • r2 Planetary Gravitational Field Strength

3. The magnitude of the gravitational field strength, (Nkg-1) is given by : • g = F/m = GM • r2 • For a planet of mass M and Radius R, sketch a graph of the gravitational field strength against distance from the centre, R, 2R, 3R, 4R etc Planetary Gravitational Field Strength

4. Gravitational Field Strength Planetary Gravitational Field Strength g But what happens if we go inside the planet (between 0 and R)? g/4 g/9 0 R 2R 3R 4R Distance from centre of planet with Radius R

5. For a spherical planet with a density  show that the gravitational field strength g inside the planet at a radius r is given by • g = 4Gr/3 • Where  is the density of the planet Planetary Gravitational Field Strength

6. Volume of a sphere = 4/3  r3 • Density = mass/volume ( = m/v) • m = v • m = 4/3 r3 • g = Gm/r2 • g = 4/3 Gr • Note there is a linear relationship between g and r. When r=0 g=0 Planetary Gravitational Field Strength

7. Gravitational Field Strength Tempting to think that within the planet g follows the curve and increases dramatically. However, inside the planet at a radius r, only the mass inside the smaller sphere with radius r applies, the mass outside r can be ignored At the centre the contributing mass is 0 and so g is 0 Planetary Gravitational Field Strength g g/4 g/9 0 R 2R 3R 4R Distance from centre of planet with Radius R

8. Imagine you’re onboard a rocket wishing to leave the surface of a planet The gravitational field of the planet extends far into space, (but weakens quickly with distance, zero at ) The rocket must do work against gravity If the engine and fuel does not provide enough energy to escape you will always fall back As the rocket climbs the Gravitational Potential Energy (GPE) increases and vice versa Gravitational Potential 1

9. If the rocket was equipped with a “GPE meter” it could be set to read zero when at  It would give a negative reading on the surface & increase towards zero as you move away For the rocket to escape it must increase it’s GPE to zero by doing work Gravitational Potential 2

10. The Gravitational Potential at a point in a gravitational field is the GPE per unit mass and is equal to the work done per unit mass in moving from infinity to that point • The Gravitational potential V at a point is the work done per unit mass to move a small object from infinity to that point • V = work done / mass • Units : Jkg-1 Gravitational Potential 3

11. An example : A rocket with a payload of 1000kg attempts to leave the surface of a planet with a gravitational potential of -100MJkg-1. For the payload to escape completely the GPE must increase from 1000 x -100MJ to zero. If the payload is only given 40,000MJ of kinetic energy by the fuel it will only be able to reach a position where the gravitational field potential is -60 MJkg-1 Gravitational Potential 4

12. In general if a small mass m is moved from a gravitational potential V1 to a gravitational potential V2 then the change in GPE is given by • Ep = m (V2 – V1) we can call (V2 – V1) V • Ep = mV • The work done to move the mass from V1 to V2 is equal to the change in GPE • W = mV Gravitational Potential 5