23
Department of Physics and Applied Physics 95.141, Spring 2014, Lecture 10 Lecture 10 Chapter 6 Gravitation and Newton’s Synthesis 02.24.2014 Physics I Course website: http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI Lecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html

Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Lecture 10

Chapter 6

Gravitation and Newton’s Synthesis

02.24.2014Physics I

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html

Page 2: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Chapter 6

Newton’s Law of Universal Gravitation Kepler’s Laws

Outline

Page 3: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

universal gravitational constant

M

m

Fr

Gravitational force is central, attractive, proportional to masses, and inversely proportional to the square of the distance

rr

GmMF ˆ2

Newton’s Law of Universal Gravitation

G 6.671011 N m2 kg2

A gravitational force of attraction exists between all objects that have mass

Page 4: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Cavendish experiment.

The magnitude of the gravitational constant G can be measured in the laboratory.

G 6.671011 N m2 kg2

100 years later in 1798, Cavendish measured gravitational force between two known masses using a “torsion

balance” and calculated G. G is the same for all materials,

universal

Page 5: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Gravitational attraction to another personEstimate the attractive Force of Gravity between two average persons 1 m apart.

F Gm1m2

R2

6.671011 Nm2

kg2 70kg 70kg

1m1m 3.3107 N

Compare this to the weight (700 N) of each person.So, Gravitational force between two “small objects” can often be neglected

1 m

F

Gravitational forces between two objects are only significant when at least one of the objects is very massive.

Page 6: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Superposition of gravitational forcesIf there are many particles, the total force is the

vector sum of the individual forces:

1

2 3

m1

m2 m3

F12

F13

F21

F23 F32

F31

Example:Three objects, of equal mass, areat three corners of a square.Draw the gravitational forcevectors acting on each block.

Page 7: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Gravity near the Earth’s SurfaceWhat should we get for the acceleration due to gravity at the

Earth’s surface?

MEARTH 5.981024 kg REARTH 6.38106 m G 6.671011 Nm2

kg2

mg

g GME

RE2 9.80 m/s2

2rmGMF E

2E

E

RmGMF

ME

mLet’s apply N.2nd law for gravitational force

maF

2rmGMma E

Denote the acceleration due to gravity as 2)(

rGMrga E

If r=RE (on the surface)

Page 8: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Variations in “g” on the surface

The acceleration due to gravityvaries over the Earth’s surface dueto altitude, local geology, and theshape of the Earth, which is notquite spherical.

Page 9: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

The motion of Satellites

Page 10: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Newton’s Cannon on a Mountain

http://waowen.screaming.net/revision/force&motion/ncananim.htm

Page 11: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Geosynchronous Orbit

A satellite in a geosynchronous orbit stays at the same position with respect to Earth as it orbits.

Such satellites are used for TV and radio transmission, for weather forecasting, and as communication relays.

http://sphere.ssec.wisc.edu/images/geo_orbit.animated.gif

Page 12: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Geosynchronous OrbitTo calculate the height of a satellite of mass min geosynchronous orbit:

T 1 day 24 hrs 86, 400s

v 2RT R

mvF2

Final caution: Height is often given by ERRh

v

FGR

RE

2RmGMF EGravitational force provides

centripetal acceleration

Satellite’s velocity

22

24R

mGMT

RmF E

32

2

4TGMR E

h

Distance from the Earth’s center

22

TR

Rm

Page 13: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Satellite in an orbit just above ground level

If a cannonball is launched horizontally just above ground level, what speed would make it travel in a circular orbit at the surface of the Earth?(Radius of earth: 6400 km, acceleration due to gravity at earth’s surface: 10 m/s2)

g v2

RE 8000 m/s v gRE v 106400103

v

mgFG

Gravitational force provides centripetal acceleration

ERvm

2

FG

Page 14: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Weight is defined as the magnitude of the force of gravity on an object. At the surface of Earth this is mg.But we measure weight, by the force a mass exerts on, say, a spring scale.

“Weightlessness”

The “weightlessness” in a freely falling elevator is because everything is accelerating equally independent of mass. So, there is no normal force.

She does have a gravitational force acting on her, though!

Page 15: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

The “weightlessness” in orbit is exactly the same, everything is accelerating equally independent of mass.

The satellite and all its contents are in free fall, except with a huge tangential velocity. So, there is no normal force. This is what leads to the experience of weightlessness.

“Weightlessness”

They do have a gravitational force acting on them, though!

Page 16: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

“Weightlessness”

More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:

Page 17: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Kepler’s Laws

A classic example of scientific advancement

Observations pattern model

1546-1601

TychoBrahe

1571-1630

Johannes Kepler

1642-1727

Isaac Newton

Page 18: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Kepler’s Laws: 1

Planets move in planar elliptical paths with the Sun at one focus of the ellipse.

s semi-major axisb semi-minor axis

http://phys23p.sl.psu.edu/phys_anim/astro/embederQ4.40100.html

Page 19: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

During equal time intervals the radius vector from the Sun to a planet sweeps out equal areas.

Kepler’s Laws: 2

t

t

t

t

http://phys23p.sl.psu.edu/phys_anim/astro/kepler_2.avi

Page 20: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

where the constant C is the same for all planets.

If a planet makes one revolution around the Sun in time T, and if R is the semi-major axis of the ellipse (R is radius if orbit circular):

T 2 R3

T 2 R3 C

or

Kepler’s Laws: 3

Page 21: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Look at circular orbit for simplicity

v2r GMGravitational force provides centripetal acceleration

vrT 2

rmv

rGMm 2

2

T 2 4 2r2

v2

Constant for all planets

T 2

r3 4 2

v2rT 2

r3 4 2

GM

Kepler’s Third LawFG

aR

v

Period of rotation

m

M r

Page 22: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 10

Thank youSee you on Wednesday

Page 23: Gravitation and Newton’s Synthesisfaculty.uml.edu/Andriy_Danylov/Teaching/documents/pre...Newton’s Law of Universal Gravitation G 6.67 10 11 N m2 kg2 rˆ A gravitational force

The Moon does not crash into Earth because of its high

speed. If it stopped moving, it would, of course, fall

directly into Earth. With its high speed, the Moon would

fly off into space if it weren’t for gravity providing the

centripetal force.

ConcepTest 3 Averting Disaster

The Moon does not

crash into Earth

because:

A) it’s in Earth’s gravitational fieldB) the net force on it is zeroC) it is beyond the main pull of Earth’s

gravityD) it’s being pulled by the Sun as well as by

EarthE) none of the above