Energi Gravitasi

March 10, 2009 Physics 114A - Lecture 30 1/30

Physics 114A - MechanicsLecture 30 (Walker: Ch. 12.4-

5)Gravitational Energy

March 10, 2009

Physics 114A - MechanicsLecture 30 (Walker: Ch. 12.4-

5)Gravitational Energy

March 10, 2009

John G. CramerProfessor of Physics

B451 [email protected]


AnnouncementsAnnouncements Homework Assignment 8 is due at 11:59 PM on

Thursday, March 12. Homework Assignment 9 (the last one!) is due at 11:59 PM on Monday, March 16 (after the Final). Homework up to 24 hours late will receive 70% credit. Also, complete the Course Survey on Tycho by March 16.

There are now 193/197 clicker registrations, so 4 people still need to register their clickers.

Exam 3 will graded and the grades posted on Wednesday and returned on Thursday. The grades for Part 1 (multiple choice) are posted on Tycho. Check Tycho for your grades as they appear. If any grades are missing (or have an underline in place of a number), see Susan Hong (room C136 PAB) immediately to identify your paper and get credit for your work.


The Physics 114A Final Exam

The Physics 114A Final Exam

On Monday, March 16 at 2:30 PM we will have the Final Exam, covering Chapters 1 through 13 of Walker. It will be a 100 point exam similar in format to the previous ones, with 60 pts of multiple choice questions, a 20 pt free-response question based on Tycho homework, and 20 pts of free-response qualitative questions. There will be some emphasis on Chapters 12 and 13 and on material covered in Chapters 1-11 but not tested on previous exams.

If you have successfully taken Exams 1-3 and are satisfied with your grades, the Final is optional. I plan to post “Estimated Course Grades” tomorrow.

There will be assigned seating for the Final, so send me E-mail if you have not already done so and need or want a left-handed, aisle, table, or front row seat. I will post seat assignments on Tycho tomorrow.


Physics 114A - Introduction to Mechanics

Lecture: Professor John G. Cramer

Textbook: Physics, Vol. 1 (UW Edition), James S. Walker

Week Date L# Lecture Topic Pages Slides Reading HW Due Lab

9

2-Mar-09 27 Angular Momentum 10 25 11-5 to 11-7

Rotational Motion, Mom.of Inertia

3-Mar-09 28 Rotational Dynamics 3 26 11-8 to 11-9

5-Mar-09 R3 Review & Extension - 49 -

6-Mar-09 E3 EXAM 3 - Chapters 9-11

10

9-Mar-09 29 Gravity Revisited 16 29 12-1 to 12-3

Make-up labs10-Mar-09 30 Gravitational Energy 10 30 12-4 to 12-5

12-Mar-09 31 Oscillations I 14 30 13-1 to 13-4 HW8

13-Mar-09 32 Oscillations II 12 32 13-5 to 13-8 HW9

16-Mar-09 FE Final Examination, 2:30 - 4:20 PM, Monday, March 16 (Comprehensive)

Lecture Schedule (Part 4)

Lecture Schedule (Part 4)

We are here.


Gravitational Potential EnergyGravitational Potential Energy Gravitational potential energy of an object of mass m a distance r from the Earth’s center:


Gravitational Potential EnergyGravitational Potential Energy

Very close to the Earth’s surface, the gravitational potential increases linearly with altitude:

Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.

Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.


Gravitational Potential Energy

Gravitational Potential Energy

1 2g

GmmU

r

A plot of the gravitational potential energy Ug looks like this:


Energy ConservationEnergy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth is:

This confirms what we already know – as an object approaches the Earth, it moves faster and faster.


Energy ConservationEnergy Conservation

An object falling into the Earth’s gravity well, (e.g., a comet), initially with zero mechanical energy.

0E U K


Energy ConservationEnergy Conservation Another way of visualizing the gravitational potential well:


Escape SpeedEscape Speed

Escape speed: the initial upward speed a projectile must have in order to escape from the Earth’s gravity. How much speed is required to climb out of the well?


Example: Escape SpeedExample: Escape Speed

A 1000 kg rocket is fired straight away from the surface of the Earth. What speed does it need to “escape” from the gravitational pull of the Earth and never return? (Assume a non-rotating Earth.)

2 2 1 1K U K U 1 212

0 0 E

E

GM mmv

R

escape

211,200 m/s 25,000 mphE

E

GMv

R

This is also the speed at which (in the absence of atmosphere) a meteor, falling from very far away, would strike the surface of the Earth. It is called “escape velocity.”


Energy ConservationEnergy ConservationSpeed of a projectile as it leaves

the Earth, for various launch speeds


Clicker Question 1Clicker Question 1 An astronaut is transported by a series of rockets from the surface of the Earth to the surface of the Moon and then returned to the surface of the Earth. How does the energy UEM required to transport him from Earth to Moon compare with UME required to transport him from Moon to Earth?

(a) UEM>UME (b) UEM=UME (c) UEM<UME


Example: Crashing into the Sun

Example: Crashing into the Sun

Suppose the Earth were suddenly to halt its motion in orbiting the Sun. The gravitational force would pull it directly into the Sun. What would be its speed as it crashed?

2 2 1 1K U K U

1 222

1

0( )

E s E sE

E S

GM M GM MM v

R R r

52

1

1 12 6.13 10 m/ss

E S

v GMR R r


Example:The Height of a Projectile

Example:The Height of a Projectile

A projectile is fired straight up from the Earth’s South Pole with an initial speed vi = 8.0 km/s.

Find the maximum height it reaches, neglecting air drag.

f f i iK U K U 1 22

0 E Ei

f E

M m M mG mv G

r R

2 2

6 11 2 2 24

8 -1

1 1 1 (8000 m/s)

2 (6.37 10 m) 2(6.67 10 N m /kg )(5.98 10 kg)

7.68 10 m

i

f E E

v

r R GM

71.30 10 mfr 7 6 6(1.30 10 m) (6.37 10 m) 6.7 10 mf Eh r R


Satellite Orbits and Energies

Satellite Orbits and Energies

2

M on m 2 r

GMm mvF ma

r r

= gUGMv

r m

The tangential velocity v needed for acircular orbit depends on the gravitationalpotential energy Ug of the satellite at theradius of the orbit. The needed tangentialvelocity v is independent of the mass m ofthe satellite (provided m<<M). Notice that to make v larger, you need to go deeper into the gravity well, i.e., to a lower orbit where –Ug is larger and r is smaller.


Orbital EnergeticsOrbital Energetics2

M on m

2

2

g

mv KF

r rUGMm

r r

1

2 gK U

The equation K = ½Ug is called “The Virial Theorem”. In effect, it says that for a planet in orbit around the Sun, if you turned its velocity by 90o, so that it pointed straight out of the Solar System, you would have only half the kinetic energy needed to escape the Sun’s gravity well.


Example:The Total Energy of a

Satellite

Example:The Total Energy of a

Satellite Show that the total energy of a satellite in a circular orbit around the Earth is half of its gravitational potential energy.

21

2EGM m

E K U mvr

22

2E EGM m GMmv

vr r r

1

2 2E E EGM GM m GM m

E mr r r

EGM mU

r

1

2E U

Although derived for this particular case, this is a general result, and is called the Virial Theorem. The factor of ½ is a consequence of the inverse square law.


Black HolesBlack Holes

Black holes:

If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface.

This is a black hole.


Gravitational LensingGravitational Lensing Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster.

This is called gravitational lensing, and many examples have been found.


Gravitational LensingGravitational Lensing


Example:A Gravity Map of the Earth

Example:A Gravity Map of the Earth

Twin satellites launched March 2002 are making detailed measurements of the Earth’s gravitational field. They are in identical orbits, with one satellite in front of the other by 220 km. The distance between the satellites is continuously monitored with micrometer accuracy using onboard microwave telemetry equipment. How does the distance between the satellites change as the satellites approach a region of increased mass? As the satellites approach a region of increased mass, they are pulled forward by it, but the leading satellite experiences a greater force that the trailing satellite. Therefore, the distance between them increases, providing an indication of a mass concentration.


g for a Solid Sphereg for a Solid Sphere

2

3

ˆoutside

inside

GMg r r R

rGM

g r r RR

Gravitational field of a solid uniform sphere

4 333

4 3 33

'r r

M M MR R

3

2 2 3 3 3

'ˆ ˆ ˆinside

GM G r GMr GMg r M r r r

r r R R R


g for a Hollow Sphereg for a Hollow Sphere

2ˆ

0

outside

inside

GMg r r R

rg r R

21 1 1 1 2

2 2 22 2 2 1 2

so and the forces cancel.m A r m m

m A r r r

Gravitational field of a uniform spherical shell


TidesTides Usually we can treat planets, moons, and stars as though they were point objects, but in fact they are not.

When two large objects exert gravitational forces on each other, the force on the near side is larger than the force on the far side, because the near side is closer to the other object.

This difference in gravitational force across an object due to its size is called a tidal force.


TidesTides

Tidal forces can result in orbital locking, where the moon always has the same face towards the planet – as does Earth’s Moon.

If a moon gets too close to a large planet, the tidal forces can be strong enough to tear the moon apart. This occurs inside the Roche limit; closer to the planet we have rings, not moons.


TidesTides This figure illustrates a general tidal force on the left, and the result of lunar tidal forces on the Earth on the right.


TidesTides Which has a larger effect on the Earth’s ocean tides, the Sun or the Moon?

It turns out that there are two factors that control the tidal effect of a given celestial object:(1) The size of the object in the sky (solid angle), and

(2) The mass density of the object.

By a celestial accident, the Sun and Moon have almost exactly the same size in the sky (solar eclipses), but the Moon has 3 times the mass density of the Sun. Therefore, the Moon’s tidal effect on Earth’s oceans is 3 times that of the Sun.


End of Lecture 30End of Lecture 30 For Thursday, read Walker, Chapter 13.1-4.

Homework Assignment 8 is due at 11:59 PM on Thursday, March 12. Homework Assignment 9 is due at 11:59 PM on Monday, March 16 (after the Final). You must also complete the Course Surve7 on Tycho by March 16. Homework up to 24 hours late will receive 70% credit.

There are now 193/197 clicker registrations, so 4 people still need to register their clickers. Register yours before the end of the quarter, if you have not already done so.

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