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NCEA Level 3 Physics
Gravitational Fields
Intro- Force Fields
• The concept of a field is a valuable
‘thinking model’ to help scientists (like
yourselves) to explain the interaction
between objects which are not in
direct contact.
Types of fields
• Electrical- due to charged objects
• Magnetic- due to moving charges
• Gravitational- due to masses
Check out the demonstrations!!
Static moving coke can
Electromagnet
Slinky
Gravitational Fields
• Newton’s falling apple
• There is a gravitational attraction
between any two objects with massthis force is usually too weak to be
noticed unless one of the objects
considered has a massive amount of
mass (or just very large!).
Curved Space
Gravitational Force Field
Any object with
mass creates a
field around it-
The density of
the field lines
corresponds
with the field
strength
Force Field
(Gravitational Field)any object placed
within the field will
experience a force
Field lines- the
radial lines around
a sphere (planet)
show the direction
a small mass
would follow if in
the field
By definition
• Gravitational Field Strength, g, is the
force per unit mass on a small test
mass placed in a field
g=F/m
Why does it
-1
have to be
Nkg
N / kg
small?
Where F is the Force acting on a mass, m
in the field.
…eh because
• A small mass does not have a
significant force field of its own so it will
not affect the field which we are
concerned with.
…just so you know
• Field lines close to the Earth look like
this• This is an example of a uniform field
where the strength is the same
everywhere- we use this when
distances are considered to be
relatively small (couple of thousand
miles).
Black Hole Fun!
• Physics Ripped apart at the seems…
where all you’ve been studying falls to
pieces and no-one can save you!!
• The average sized black hole is 8 times
more massive than our own Sun all
concentrated into a space no larger
than your finger nail
Field strength on a point mass
m1
Consider Newton’s Laws on the point
mass (2 ways of expressing F)•F =
Gm1m2 (Law of gravitation)
r2
• F = m2 g (Newton’s 2nd Law- F = m a)
Equate
m2 g =
Gm1m2
r2
m2
Field strength on a point mass
g = Gm
r2
• m is the mass of the object causing the
gravitational field
• r is the distance from the centre of the
mass m to the point mass
• G is the universal constant of gravity
Brings us back… Gravity constant
g = Gm
r2
= F
m
g also follows an inverse square law
Draw a graph to show the relationship
between g and r
g
Inverse square Law
• Field strength g, weakens as distance r,
increases
12
22
32
Calculating the Mass of the Earth
• Value of g on the Earth’s surface
g = 9.81 Nkg-1
• Radius of the Earth
r = 6400km = 6.4x106m
Using the field strength equationg = Gm
Rearrange for mass=
r2
m =
g r2
G
9.81 x (6.4x106m)2 / 6.67x10-11
6.02 x 1024 kg
Did you Know… (probably not)
• The mass of the Earth increases every
year because of 3,000 tonnes of
meteorite debris that hits its surface
from space.
Get you thinking…
• Using the new formula and Newton’s
second law of motion, can you show
that the value for the gravitational
field strength is also the value for
acceleration due to free-fall?
Fields- syllabus reference
10 Gravitational force between point
masses (Newton’s law)
- Knowledge and use of F = Gm1m2/r2.
- G introduced as constant with units.
11 Gravitational field of a point mass
- Knowledge and use of g = Gm/r2.
- Calculation of the Earth’s mass.
Newton’s Law
• In 1689 an apple fell…
Well.. It wasn’t as
simple as that!
Newton’s Law
• If the force of gravity reaches to the
top of the highest tree, might it not
reach even further; in particular, might
it not reach all the way to the orbit of
the Moon?
Just like the acceleration of the apple…
• The acceleration due to gravity
changes the velocity of the Moon so
that it follows an orbit around the Earth
Sent into Orbit!
• At just the right speed and trajectory
the canon ball will fall towards the
Earth due to the gravitational force at
the same rate as the Earth’s surface
curves away from it.
• The canon ball has been sent into
orbit!
Newton’s Law of Gravitation
• Every object attracts every other
object along a line that joins their
centres. It is proportional to the
product of their masses and inversely
proportional to the square of their
distances apart
The equation
F
=
Gm1m2
r2
F is the gravitational force of attraction
m1m2 is the product of the two masses
r is the distance between the centre of the
masses
G is the gravitational constant with a value of
6.67 x 10-11
Wee Example
• The Moon’s orbit of the Earth is at a
radius 3.84x108m, the mass of the
moon is 7.35x1022kg and the Earth’s
mass is 6.00x1024kg
• Calculate the size of the force which
keeps the Moon in orbit of the Earth.
F =
Gm1m2
r2
= 1.99x1020 N
=
6.67x10-11 x 6.00x1024 x 7.35x1022
(3.84x108m)2
And what about this rocket…
• A rocket of mass 42 000kg is fired
towards the Moon. What is the net
gravitational force on the rocket when
it is 3.00x108m away from the centre of
the Earth?
• FE = 187N
• FM = 29.2N
• Net Force =187 – 29.2 = 158N
(Towards the Earth!)
How attractive are you?
• Use the equation to approximately
work out the force of attraction you
have with the person closest to you
• Why is the value so small?
An inverse square relationship
• As the distance
between two objects
increases the
attractive force
decreases (inverse) by
a directly square
proportion-
Comet Action
So…
• If the force between two objects is
200N when they are 5m apart, what is
the force between them when they
are• A) 10m apart? (2 times the distance)
• B) 20m apart? (4 times the distance)
• C) 30m apart? (6 times the distance)
200/4 = 50N
200/16= 12.5N
200/36= 5.56N
The unit of G (G unit)
• Re-arrange the N. Law of G. to make
G the subjectG = F r2 / m1m2
Unit of G is therefore = N m2 kg-2
Some extra notes to be aware of…
• Gravitational force is equal and
opposite for two masses (each ‘pull’
each other with the same force)
• The masses are considered to be point
masses with all their mass
concentrated at their centre
• Gravitational forces are very weak
unless we are looking at enormous
masses
For our previous rocket…
• At what distance from the Earth is their
no resultant forces acting on the
rocket?
• Force due to Earth = Force due to Moon
GmEmR
=
(3.84x108m- R)2
GmMmR
R2
3.84x108m
Earth
R
Moon
For our previous rocket…
GmEmR
=
(3.84x108m- R)2
mE x R2
GmMmR
R2
=
mM x (3.84x108m- R)2
Solve using quadratic equation…
3.84x108m
Earth
R
Moon
Or…
x
Earth
y
GmEmR
=
GmMmR
X2
Y2
X 2 mM
X2
Y2
Moon
=
Y 2 mE
=
mE
mM
Take the square root…
X is 9/10 of total distance
X
Y
=
9
Y is 1/10 of total distance
One TENTH of 3.84x108 is 0.384x108
Therefore, distance from the Earth where
the forces are balanced is 3.84x108 –
0.384x108
=
3.456x108m
Syllabus Reference
11 Gravitational field of a point mass
- Knowledge and use of g = Gm/r2.
- Calculation of the Earth’s mass.
Bit of Homewrok
• Try the past paper question…
Syllabus Reference
12
Planetary and satellite motion
• Quantitative treatment of circular orbits
only.
• A formal statement of Kepler’s laws is not
required, but candidates should be able to
show that the mathematical form of the
third law (t2 proportional to r3) is consistent
with Newton’s law of gravitation.
• Geostationary satellites.
The two equations for
F
=
Gm1m2
Newton’s Law for
Gravitation
r2
F = m2v2
r
Centripetal
force
The force required to keep a planet in circular orbit
2
Massive
object!
The force required to
keep a planet in
circular orbit
Acceleration
towards centre
2
• Sub into equation
g = Gm1 / r2
m2 g
g
=
2
v2
r
Geostationary Orbits
• Some satellites orbit exactly
over the Earth's equator and
make one orbit per day.
- Period of orbit T = 24hrs
• Since the Earth rotates once
on its axis per day, the
satellite seems to hover over
the same spot on Earth all the
time.
Satellites in Orbit
• The first satellite to be put
into Geostationary orbit was
way back in 1963
• Since then there are now
over 600 at a height of
35,000km above the Earth
• What would they be used
for?
• Any idea what the satellite
‘Hotbird 1’ is tracking?
Kepler’s Third Law… you gotta know this
• The ratio of the period of a mass’ orbit
squared (T2) to the mean radius of it’s
orbit cubed (R3) is the same constant
value k for all the planets which orbit the
Sun.
Period
Average
T /R
Planet
2
3
(s)
Dist. (m)
(s2/m3)
Earth
3.156 x 107 s
1.4957 x 1011
2.977 x 10-19
Mars
5.93 x 107 s
2.278 x 1011
2.975 x 10-19
What’s more
• Newton’s Law of gravitation is
consistent with Kepler’s 3rd Law
- Consider a planet with mass mp to
orbit in nearly circular motion about
the sun of mass mS
The Net Centripetal
Force F of the orbiting
planet
The Net Centripetal
Force F of the
gravitational field
p
Let’s do some substituting and rearranging…
=
v=2πr/T
Divide each side by r2
Multiply each side by T2
Divide each side by Gms
p
v2 = 4 π2 r2 / T2
Gms
4 π2 r2
=
r
T2
Gms
4 π2
=
r3
T2
T2 Gms
2
=
4
π
r3
T2
4 π2
=
3
r
Gms
This value will
be the same
for all
planets
What’s more
• Newton’s Law of gravitation is
consistent with Kepler’s 3rd Law
- Consider a planet with mass mp to
orbit in nearly circular motion about
the sun of mass mS
The Net Centripetal
Force F of the orbiting
planet
The Net Centripetal
Force F of the
gravitational field
p
Question 1 (Jan 2001)
• The planet Mars has radius 3.39x106m
and mass 6.50x1023kg. The length of a
day on Mars is 8.86x104s (24.6 days)
a) At what height above the surface of
Mars should a geostationary satellite
be placed?
b) Calculate the acceleration of free-fall
on the surface of Mars
Mars has two moons, Phobos (radius of
orbit 9.38x103km, 0.319 days) and
Deimos (radius of orbit 23.5x103km)
c) What is the period of Deimos’ orbit?
d) What are the three features of a
geostationary satellite?
Question 2 (Jan 2002)
a) State in words, Newton’s law of
Gravitation
b) The period T of a satellite around a
planet is related to its radius r by
T2
4 π2
=
Gms
r3
where M is the mass of the planet being
orbited. Show this is consistent with
Newton’s Law of Gravitation
Using the data radius of Earth 6.37x106m
and mass 5.98x1024kg, a satellite is in
geostationary orbit around the Earth.
c) What is meant by a geostationary
orbit?
d) Calculate the height of the satellite
above the Earth’s surface.
Circular Motion
• What is angular velocity (illustration?)
• What is linear velocity?
• Define frequency and period and
show how they are related to angular
velocity
• Centripetal acceleration and hence
force (with directions…)
• Again link in period, velocity and
angular velocity
Newton’s Law of gravitation
• State in words
• What does G stand for, what’s special
about it?
• Show in diagram form
• Value of g, equation for calculations
• Inverse square relationship
Geostationary Orbits
• Meaning of a Geostationary Orbit
• Newton’s second law for a circular
orbit in terms of angular velocity and
period T
• Equate Newton’s two equations
• Show how Kepler’s Law is consistent
with Newton’s Law