Gravitational Field Strength

1.1. To investigate how the gravitational field To investigate how the gravitational field strength varies as you move away from a strength varies as you move away from a planet and also inside the planetplanet and also inside the planet

2.2. To To beginbegin to explore the concept of to explore the concept of Gravitational potentialGravitational potential

HW : Read about Cavendish’s measurement of HW : Read about Cavendish’s measurement of GG::

http://en.wikipedia.org/wiki/Cavendish_experimenthttp://en.wikipedia.org/wiki/Cavendish_experiment

Book Reference : Pages 62-63Book Reference : Pages 62-63Book Reference : Pages 56-57Book Reference : Pages 56-57

Last lesson we used planetary data from our Last lesson we used planetary data from our solar system to confirm Newton’s law of solar system to confirm Newton’s law of gravitation. gravitation.

For a planet acting as a point mass M the For a planet acting as a point mass M the magnitude of the force of attraction on a mass m magnitude of the force of attraction on a mass m can be given by :can be given by :

F = GmMF = GmM

rr22

The magnitude of the gravitational field strength, The magnitude of the gravitational field strength, (Nkg(Nkg-1-1) is given by :) is given by :

g = F/m = GMg = F/m = GM

rr22

For a planet of mass M and Radius R, For a planet of mass M and Radius R, sketch a graph of the gravitational field sketch a graph of the gravitational field strength against distance from the strength against distance from the centre, R, 2R, 3R, 4R etccentre, R, 2R, 3R, 4R etc

But what happens if we go inside But what happens if we go inside the planet (between 0 and R)?the planet (between 0 and R)?

R 2R 3R 4R0

g

g/4

g/9

Distance from centre of planet with Radius RDistance from centre of planet with Radius R

Gravitational Field StrengthGravitational Field Strength

For a spherical planet with a density For a spherical planet with a density show that show that the gravitational field strength g inside the the gravitational field strength g inside the planet at a radius r is given by planet at a radius r is given by

g = 4g = 4GGr/3r/3

Where Where is the density of the planet is the density of the planet

Volume of a sphere = 4/3 Volume of a sphere = 4/3 r r33

Density = mass/volume (Density = mass/volume ( = m/v) = m/v)m = m = vvm = 4/3 m = 4/3 rr33

g = Gm/rg = Gm/r22

g = 4/3 Gg = 4/3 Grr

Note there is a linear relationship between g and Note there is a linear relationship between g and r. When r=0 g=0 r. When r=0 g=0

Tempting to think that within the Tempting to think that within the planet planet gg follows the curve and follows the curve and increases dramatically. However, increases dramatically. However, inside the planet at a radius inside the planet at a radius rr, only the , only the mass inside the smaller sphere with mass inside the smaller sphere with radius radius r r applies, the mass outside applies, the mass outside r r can can be ignoredbe ignored

At the centre the contributing mass is At the centre the contributing mass is 0 and so g is 00 and so g is 0

R 2R 3R 4R0

g

g/4

g/9

Distance from centre of planet with Radius RDistance from centre of planet with Radius R

Gravitational Field StrengthGravitational Field Strength

Imagine you’re onboard a rocket wishing to leave Imagine you’re onboard a rocket wishing to leave the surface of a planetthe surface of a planet

The gravitational field of the planet extends far The gravitational field of the planet extends far into space, (but weakens quickly with distance, into space, (but weakens quickly with distance, zero at zero at ))

The rocket must do work against gravityThe rocket must do work against gravity

If the engine and fuel does not provide enough If the engine and fuel does not provide enough energy to escape you will always fall backenergy to escape you will always fall back

As the rocket climbs the Gravitational Potential As the rocket climbs the Gravitational Potential Energy (GPE) increases and vice versaEnergy (GPE) increases and vice versa

If the rocket was equipped with a “GPE meter” it If the rocket was equipped with a “GPE meter” it could be set to read zero when at could be set to read zero when at

It would give a negative reading on the surface It would give a negative reading on the surface & increase towards zero as you move away& increase towards zero as you move away

For the rocket to escape it must increase it’s GPE For the rocket to escape it must increase it’s GPE to zero by doing workto zero by doing work

The Gravitational Potential at a point in a The Gravitational Potential at a point in a gravitational field is the GPE per unit mass and is gravitational field is the GPE per unit mass and is equal to the work done per unit mass in moving equal to the work done per unit mass in moving from infinity to that pointfrom infinity to that point

The Gravitational potential V at a point is the The Gravitational potential V at a point is the work done per unit mass to move a small object work done per unit mass to move a small object from infinity to that pointfrom infinity to that point

V = work done / massV = work done / mass

Units : JkgUnits : Jkg-1-1

An example : An example :

A rocket with a payload of 1000kg attempts to A rocket with a payload of 1000kg attempts to leave the surface of a planet with a gravitational leave the surface of a planet with a gravitational potential of -100MJkgpotential of -100MJkg-1-1. For the payload to . For the payload to escape completely the GPE must increase from escape completely the GPE must increase from 1000 x -100MJ to zero.1000 x -100MJ to zero.

If the payload is only given 40,000MJ of kinetic If the payload is only given 40,000MJ of kinetic energy by the fuel it will only be able to reach a energy by the fuel it will only be able to reach a position where the gravitational field potential position where the gravitational field potential is -60 MJkgis -60 MJkg-1-1

In general if a small mass m is moved from a In general if a small mass m is moved from a gravitational potential Vgravitational potential V11 to a gravitational to a gravitational potential Vpotential V22 then the change in GPE is given by then the change in GPE is given by

EEpp = m (V = m (V22 – V – V11))we can call (Vwe can call (V22 – V – V11) ) VV

EEpp = m = mVV

The work done to move the mass from VThe work done to move the mass from V11 to V to V22 is is equal to the change in GPEequal to the change in GPE

W = mW = mVV