Chapter 08 Gravitational Fields

transcript

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

• 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.

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.”

� � ��

��

� �

• 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.

• 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?

• 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.

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.

• 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?

• 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?

• 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!

• 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.

� �

• Hence, the gravitational field

strength at a point, �

��;

• The unit of N kg-1

• 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

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.

• 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

• 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).

EXAMPLESEXAMPLESMay/Jun 2011, Paper 41, question 1.

EXAMPLESEXAMPLESMay/Jun 2011, Paper 41, question 1 (cont’d).

Oct/Nov 2010, Paper 43, question 1 (cont’d).

EXAMPLESEXAMPLES

Oct/Nov 2010, Paper 43, question 1 (cont’d).

EXAMPLESEXAMPLES

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

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.

E N E R G Y

• Recall also the equation for

gravitational potential energy

(GPE) .

• Noting that�

��, and setting

, we get GPE��

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.

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

• Definition: “The gravitational

potential at a point, , is

defined as the work done in

bringing an unit mass from

infinity to that point.”

P OT E N T I A L

2 � 3��

• 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

P OT E N T I 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.

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.

P OT E N T I A L

• 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).

HOMEWORKHOMEWORK1. Oct/Nov 2009, Paper 42, question 1.

Chapter 08 Gravitational Fields

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