chapter 08 gravitational fields
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CIE A2 Gravitational FieldsTRANSCRIPT
CAMBRIDGE A – LEVEL
PHYSICS
GRAVITATIONAL GRAVITATIONAL
FIELDS
LEARNING OUTCOMES
No. LEARNING OUTCOME
i Understand the relationship between gravitational fields and
gravitational forces.
ii Look at the gravitational forces between two point masses and
extend it to larger uniform spheres.
iii Define gravitational field strength and derive an equation for the
gravitational field strength at a point.
iv Understand the role of gravitational forces for motion in circular
orbits. What are geostationary orbits?
v What is gravitational potential and what is its relationship with
gravitational potential energy?
G R AV I TAT I O N A L F I E L D S
A N D F O R C E S
G R AV I TAT I O N A L F I E L D S
A N D F O R C E S
• An object with mass is capable of
exerting an attractive force known
as a gravitational force on another
object that has mass.
• This is because any object that has a
mass has a gravitational field
around it.
• The gravitational field is an example
of a field of a force.
• A field of a force is a region in space
in which the force can act.
G R AV I TAT I O N A L F I E L D S
A N D F O R C E S
G R AV I TAT I O N A L F I E L D S
A N D F O R C E S
G R AV I TAT I O N A L FO R C ES
• Definition: “Newton’s Law of
Gravitation states that two point
masses attract each other with a
force that is proportional to the
product of their masses and
inversely proportional to the square
of their separation.”
G R AV I TAT I O N A L FO R C ES
� � �����
��
� �
• In the form of an equation, � � �����
��
where:
• � � magnitude of the gravitational force between
the two point masses, in N
• � � Universal Gravitational Constant, 6.67 �
10����������
• ��,�� � the masses of the two point masses
respectively, in kg,
• � � distance between the point masses, in m.
G R AV I TAT I O N A L FO R C ES
• The gravitational force is the weakest force
known, but is the most important force with
regards to planetary motion.
• The definition of gravitational forces is applied
to point masses. However, planets and their
satellites are not point masses.
• How then can we use the equations for
problems in which planets are involved?
G R AV I TAT I O N A L FO R C ES
• What we do is that we consider the masses of
these larger objects to be uniform and hence
for all points outside the large sphere, the
mass of the large sphere is considered to be
concentrated at the centre of its mass.
• Therefore, these spheres are also considered
to be point masses.
G R AV I TAT I O N A L FO R C ES
Source: http://images.slideplayer.com/1/273184/slides/slide_2.jpg
EXAMPLESEXAMPLESQuestions 1 and 2, page 274,
Chapter 18: GRAVITATIONAL
FIELDS; Cambridge International
AS and A Level Physics
Coursebook, Sang, Jones,
Chadha and Woodside, 2nd
edition, Cambridge University
Press, Cambridge, UK,2014.
G R AV I TAT I O N A L F I E L D S
• The gravitational field around an object
is the field of the gravitational force.
• In other words, any object that has a
mass has a gravitational field around it.
• How do we measure how strong a
gravitational field is? How does a
gravitational field look like?
G R AV I TAT I O N A L F I E L D S
• The gravitational field around an object
is the field of the gravitational force.
• In other words, any object that has a
mass has a gravitational field around it.
• How do we measure how strong a
gravitational field is? How does a
gravitational field look like?
G R AV I TAT I O N A L F I E L D S
• Definition: “The gravitational field
strength at a point is the
gravitational force acting on per unit
mass of a small mass placed there.”
• Let’s derive an equation for the
gravitational field strength!
G R AV I TAT I O N A L F I E L D S
• The gravitational force between
�
• The gravitational force between the object that produces the gravitational field and a small
mass placed there, ��
��;
–where �=mass of the object that produces the gravitational field,
–�=mass of the small mass.
G R AV I TAT I O N A L F I E L D S
� �
• Hence, the gravitational field
strength at a point, �
�
�
��;
• The unit of N kg-1
G R AV I TAT I O N A L F I E L D S
• The diagram on the left shows
how the gravitational field
around a point mass looks like.
• Consider the direction of the
field lines and look at how they
are spaced nearer the point,
and further from the point.
• The circle is an equipotential
surface. All points on the
surface will have the same
gravitational field strength.
Source: http://www.splung.com/kinematics/images/gravitation/field3.gif
G R AV I TAT I O N A L F I E L D S
We can calculate ����� � 9.81�����.This• We can calculate ����� � 9.81�����.Thismeans that the Earth will exert agravitational force of 9.81 N on every 1 kg ofmass on its surface.
• Since r does not change by much as we go upin altitude, we can safely take #$%�&' �(. )�*+#�� close to Earth’s surface.
• As a matter of fact, this value is constant forabout up to 1 km above the Earth’s surface.
G R AV I TAT I O N A L F I E L D S
• In a situation of free fall without air
resistance, the only force acting on an
object is the gravitational force.
• Therefore, the object’s free fall
acceleration has the same value as the
gravitational field strength, but with
different units.
W E I G H T A N D
G R AV I TAT I O N A L F O R C E
W E I G H T A N D
G R AV I TAT I O N A L F O R C E• The weight we experience on Earth’s
surface is due to the effect of Earth’s
gravitational field exerting a
gravitational force on us.
• Our weight is therefore the
gravitational force that the Earth exerts
on us.
EXAMPLESEXAMPLESTable 18.1 and question, page
275, Chapter 18:
GRAVITATIONAL FIELDS;
Cambridge International AS
and A Level Physics
Coursebook, Sang, Jones,
Chadha and Woodside, 2nd
edition, Cambridge University
Press, Cambridge, UK,2014.
EXAMPLESEXAMPLESQuestions 5 and 6, page 275,
Chapter 18: GRAVITATIONAL
FIELDS; Cambridge
International AS and A Level
Physics Coursebook, Sang,
Jones, Chadha and Woodside,
2nd edition, Cambridge
University Press, Cambridge,
UK,2014.
CIRCULAR ORBITS
, � �- ��.�
/
• Satellites are objects that orbit a larger mass.
• Satellites have an elliptical orbit, but to
simplify discussion, we assume circular
orbits.
• Satellites have uniform orbital periods, i.e.
/= constant.
• Recall from the previous chapter that
, � �- ��.�
/
Source:
http://img.brothersoft.com/screenshots/softimage/s/satellite_orbit_problems-
68298-1.jpeg
Centre of larger mass M
CIRCULAR ORBITS
�0
�
• The centripetal force, �0of the satellite is
provided by the gravitational force, � that
the larger mass exerts on the satellite.
• Mathematically, ���
��� �
,�
�� ��-� �
�1.��
/�
• We can use the equations above to obtain
quantities like T and r.
CIRCULAR ORBITS
G EO S TAT I O N A RY O R B I T S
• A geostationary orbit is a circular orbit 42,
300 kilometres from the Earth’s centre and
located at a point exactly above the Earth’s
equator.
• An object in such an orbit has an orbital
period equal to the Earth's rotational
period, and thus appears motionless, at a
fixed position in the sky, to ground observers.
G EO S TAT I O N A RY O R B I T S
• Satellites that have geostationary orbits have
an orbital period of revolution equal to the
period of rotation of the larger mass.
• For example, an artificial satellite, in
geostationary orbit, orbiting the Earth will
have an orbital period of 24.0 hours (equal to
the orbital period of rotation of Earth).
EXAMPLESEXAMPLESOct/Nov 2008, Paper 4, question 1.
EXAMPLESEXAMPLESOct/Nov 2008, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2008, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2008, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2008, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESMay/Jun 2011, Paper 41, question 1.
EXAMPLESEXAMPLESMay/Jun 2011, Paper 41, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2010, Paper 43, question 1.
EXAMPLESEXAMPLESOct/Nov 2010, Paper 43, question 1 (cont’d).
Oct/Nov 2010, Paper 43, question 1 (cont’d).
EXAMPLESEXAMPLES
Oct/Nov 2010, Paper 43, question 1 (cont’d).
EXAMPLESEXAMPLES
EXAMPLESEXAMPLESOct/Nov 2010, Paper 43, question 1 (cont’d).
HOMEWORKHOMEWORK1. Oct/Nov 2009, Paper 41, question 1.
2. Oct/Nov 2011, Paper 41, question 1.
3. May/June 2012, Paper 42, question 1.
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
When we place an object that mass in• When we place an object that mass ina gravitational field, that object willstore a amount gravitational potentialenergy.
• Recall that gravitational potentialenergy is the energy stored by anobject due to its position in agravitational field.
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
�
• Recall also the equation for
gravitational potential energy
(GPE) .
• Noting that�
��, and setting
, we get GPE��
�
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
• What happens to the value of• What happens to the value ofGPE when ?
• At points infinitely far awayfrom the centre of Earth,
, hence the gravitationalpotential energy (GPE) = 0 atthese points.
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
G R AV I TAT I O N A L P O T E N T I A L
E N E R G Y
• We now have a new reference level
to set GPE = 0.
• As we get nearer to the centre of
mass of , GPE decreases, or GPE
becomes more negative.
G R A V I T A T I O N A L P O T E N T I A L E N E R G Y
A N D G R A V I T A T I O N A L P O T E N T I A L
�
• Since it is easier to work with per unit of
mass for the smaller mass, �, we now
arrive at a new quantity.
• This quantity is called gravitational
potential and it deals with changes in
energy per unit mass of object.
G R AV I TAT I O N A L
P OT E N T I A L
G R AV I TAT I O N A L
P OT E N T I A L
• Definition: “The gravitational
potential at a point, , is
defined as the work done in
bringing an unit mass from
infinity to that point.”
G R AV I TAT I O N A L
P OT E N T I A L
G R AV I TAT I O N A L
P OT E N T I A L
2 � 3��
4 �
• Mathematically, the gravitational
potential at a point, 2 � 3��
�, where:
• 5 �the gravitational field strength at that
point, J kg-1
• � � the mass of the object that produces the
gravitational field, kg, and
• 4 � the distance between the centre of mass
to the point, m
G R AV I TAT I O N A L
P OT E N T I A L
G R AV I TAT I O N A L
P OT E N T I A Lϕ � 0 4 � ∞
�5Jpotential, ∆2 �2L 32M.
• ϕ � 0 at 4 � ∞ and decreases
(becomes more negative) as the value of
4 decreases (move closer to centre).
• If an object is moved from a point, A
with gravitational potential �5N to a
point B with gravitational potential
�5J , the change in gravitational
potential, ∆2 �2L 32M.
G R AV I TAT I O N A L
P OT E N T I A L
G R AV I TAT I O N A L
P OT E N T I A L
• An object that undergoes a change
in gravitational potential will when
moving from one point to another
will also have its gravitational
potential energy and kinetic energy
changed.
G R AV I TAT I O N A L
P OT E N T I A L
G R AV I TAT I O N A L
P OT E N T I A L
O
• If , then the object’s
GPE will have decreased, while
its O will have increased.
• If , then the object’s
GPE will have increased, while
its O will have decreased.
EXAMPLESEXAMPLESMay/Jun 2009, Paper 4, question 1.
EXAMPLESEXAMPLESMay/Jun 2009, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESMay/Jun 2009, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESMay/Jun 2009, Paper 4, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2011, Paper 43, question 1.
EXAMPLESEXAMPLESOct/Nov 2011, Paper 43, question 1 (cont’d).
EXAMPLESEXAMPLESOct/Nov 2011, Paper 43, question 1 (cont’d).
HOMEWORKHOMEWORK1. Oct/Nov 2009, Paper 42, question 1.
2. May/June 2010, Paper 42, question 1.
3. Oct/Nov 2010, Paper 41, question 1.
4. May/June 2012, Paper 41, question 1.
5. Oct/Nov 2012, Paper 41, question 1.