Jw Fundamentals of Physics 1 GRAVITY. jw Fundamentals of Physics 2 Chapter 13: Newton, Einstein, and Gravity Isaac Newton 1642 - 1727Albert Einstein 1872

jw Fundamentals of Physics 1

GRAVITY


Chapter 13: Newton, Einstein, and Gravity

Isaac Newton 1642 - 1727Albert Einstein 1872 - 1955


Fundamentals of Physics

Chapter 13 Gravitation

1. Our Galaxy & the Gravitational Force

2. Newton’s Law of Gravitation

3. Gravitation & the Principle of Superposition

4. Gravitation Near Earth’s Surface

5. Gravitation Inside Earth

6. Gravitational Potential Energy

Path Independence

Potential Energy & Force

Escape Speed

7. Planets &Satellites: Kepler’s Laws

8. Satellites: Orbits & Energy

9. Einstein & Gravitation

Principle of Equivalence

Curvature of Space Review & SummaryExercises & Problems


The World & the Gravitational Force

– Allowed matter to spread out after the Big Bang.

– Eventually pulled large amounts of mass together:

– Clouds of gas

– Brown dwarfs

– Stars

– Galaxies (The Milky Way, Andromeda)

– Clusters of Galaxies (“The Local Group”)

– Super Clusters

• “billions & billions” of galaxies and stars

– Big Bang - hydrogen, helium, lithium, & very little else

– Big Stars - burned out & exploded creating heavier elements - carbon, oxygen, iron, gold, . . .

– Gravity then pulled the sun & earth together

Gravity is a very weak force, but essential for us to exist!


Physics ~1680

By now the world knew:

• Bodies of different weights fall at the same speed

• Bodies in motion did not necessarily come to rest

• Moons could orbit different planets

• Planets moved around the Sun in ellipses with the Sun at one focus

• The orbital speeds of the planets obeyed “Kepler’s Laws”

But why??? Isaac Newton put it all together.


Newton’s Law of Gravitation

Every massive body in the universe attracts every other massive body through the gravitational force!

Newton (1665):

Gravity Force is weak!

G = 6.67 x 10-11 N m2 / kg2

Principle of superposition: net effect is the sum of the individual forces.(added vectorially).

221~

r

mmF

221

r

mmGF




Measurement of the Gravitational Force

If two masses are brought very close together in the laboratory, the gravitational attraction between them can be detected.

G = 6.673 x 10-11 N m2 / kg2

(relatively poorly known: 1 part in 10,000)

Cavendish Experiment(~1760):

221

r

mmGF


Gravitation Near Earth’s Surface

• The Earth is not uniform.

• The Earth is not a sphere.– An ellipsoid ~0.3%

• Earth is rotating.– Centripetal acceleration

FN-mag = m(-2R)

mg = mag-m(2R)

(measured wt.) = (mag. of gravitational force) – (mass x centripetal acceleration)

g = ag –2R

on equator difference ~ 0.034 m/s2


Gravitation Near Earth’s Surface

r is the distance between the centers of the two bodies

Newton: The force exerted by any spherically symmetric object on a point

mass is the same as if all the mass were concentrated at its center.

221

r

mmGF

Newton had to invent calculus to prove it!

r



Newton: The gravitational force provides the centripetal acceleration to hold the earth in its orbit around the sun:

2

2

r

mMG

r

vmF ES


Newton: The gravitational attraction between the Earth and the Moon causes the Moon to orbit around the Earth rather than moving in a straight line.

Orbital Motion


Gravitation Inside A Shell

Newton: gravitational force inside a uniform spherical shell of matter:

1 0 2 00 2 2

1 2

21 1

22 2

0

But:

0

m m m mF G G

r r

m r

m r

F

The forces on m0 due to m1 and m2 vary as ~1/r2, but the masses of m1 and m2 grow as r2, hence the two forces cancel out!

A uniform spherical shell of matter exerts no net gravitational force on a particle located anywhere inside it.


Gravitation Outside A Shell

Newton invents calculus!!!

g dg

dgG dM

s

dM MdAA

MR R d

R

s r R r R

sds r R d

R s r sr

r r

r

2

2

2 2 2

2 2 2

2

4

2

2 2

2

cos

sin

cos

sin

cos

Gravitational acceleration at a distance r from a point particle!

Integrating from s = r - R to s = r + R:

g r G M

r 2

Newton: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s matter were concentrated at its center.


Gravitation Inside & Outside a Uniform Sphere

Newton: Consider the sphere as a set of concentric shells:

The gravitational force inside a uniform spherical shell of matter is zero.

Only the matter inside radius r attracts an object at that radius.


Gravitation Inside Earth

Newton: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s matter were concentrated at its center.

Hooke’s Law

F r~

m oscillates back & forth!

Newton: The gravitational force inside a uniform spherical shell of matter is zero.

rR

MGgr 3

rR

mMGFr 3


Gravitational Potential Energy of a 2-Particle System

Gravity is a conservative force.

The work done by the gravitational force on a particle moving from an initial point A to a final point G is independent of the path taken between the points.

Gravity is a conservative force.

Potential Energy:

(attractive force) 2

M mF r G

r

r rU U W F r dr

r

mMGrU

Only U is important; the location of U = 0 is arbitrary.

Choose U = 0 to be the point at which the masses are far apart.

0U

M

m

r

G M mU U

r


Escape Velocity

Consider a projectile of mass m, leaving the surface of a planet. The loses kinetic energy and gains gravitational potential energy.

Does not depend on the mass of the projectile nor on its initial direction (i.e. escape speed not escape velocity).

12

2 0

2

mv GM m

R

vG M

R

U GM m

R 0

E K U constant

K U 0

K mv 12

2 0

Escape Speed: When the projectile reaches infinity, it has no potential energy (U = 0) and no kinetic energy (stopped: v = 0): the total mechanical energy is zero.


Escape Velocity

~ 25,000 mi/hr

E K U constant

rU

E

Escapes

Bound Orbit

No hydrogen & helium in the earth’s atmosphere: escvv

skmmsmgrve /2.11)1038.6)(/81.9(22 62

U rG M m

rE Choosing U 0

Escape Speed


http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html

http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html


Tycho Brahe (1546-1601)

Johannes Kepler (1571-1630)

Planets & Satellites: Kepler’s Laws

Sun

Path of Mars across the sky.


Planets & Satellites: Kepler’s Laws

– Equal Areas in Equal Times by sun-planet line.

Sun

Kepler’s Laws:

– Elliptical orbits with the Sun at one focus.

– T2 ~ r3 (Period & Mean Distance from sun)


Kepler’s 1st Law

The Law of Orbits: All planets move in elliptical orbits, with the Sun at one focus.

ellipse - sum of distances from 2 foci is constant (2a).

eccentricity of earth = 0.0167 e = 0 circle (1 focus)

M m

Sun is very near one focus.

perihelion - closest point to Sum

aphelion - farthest point


Kepler’s 2nd Law

The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant.

slower faster

http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/Kepler/Kepler.html

http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/Kepler/Kepler.html


Kepler’s 2nd Law

The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant.

Conservation of Angular Momentum:

dA r v dt

dAm

r mv dt

dAm

L dt

12

12

12

dAdt m

L

L

12

constant

constant

Any Central Force


Kepler’s 3rd Law

The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Consider the special case of a circular orbit:

See Table 13-3.

r

MGv

r

mMG

r

vmF

2

2

2

v

rT

2 :Period 3

22 4

rMG

T

32 ~ rT

Determine the mass of a planet by measuring period and mean orbital distance of a moon orbiting it.


What did Newton know?

Kepler’s 3rd Law:32 ~ rT

acceleration of the moon: r

va

2

velocity of the moon:T

rv

2

His own law! amF

A little algebra by Newton:2

1~

rF

The force holding the moon in orbit depends on the square of its distance from earth.


Einstein & Gravitation

“Science is a perfectionist” - Narlikar

– Newton’s Law of Gravity works great!

– But, how & why does it work?

– A small problem with the planet Mercury.



A “Small” problem with the planet Mercury:

• Mercury takes 88 Earth days to orbit the sun.

• But, measurements show the perihelion “advancing”:

– 0.159o per 100 years! (Newton’s Law says zero!)

• Attractions by the other planets almost explain it.

– 0.147o per 100 years!

• 43 arc-seconds per 100 years unexplained.

Einstein explains it!



• Special Theory of Relativity

– Space & Time are intimately connected.

c = constant everywhere

– Mass & Energy are equivalent.

E = m c2

• General Theory of Relativity

– A theory of gravitation

space–time geometry mass (material bodies)



• Galileo / Newton Universe

– No interdependence of time, space & mass.

• Time flows uniformly.

• Space is immutable.

– Euclidean geometry - “straight lines”

• Mass of a body is constant.

(shape & dimension of rigid bodies is also constant)

• Einstein Universe

• Gravitational forces reach out to infinity.

– All bodies are moving in the gravitational field of other bodies.

Not moving in a straight line.

• Space is curved!



The General Theory of Relativity:

The Equivalence Principle:

Is it gravity or rockets that causes a?

No experiment can tell the difference!

Einstein’s Strong Equivalence Principle:


If light appears to follow a curved path in the elevator, gravity must also cause it to curve.

Gravity & Light:

Einstein’s Strong Equivalence Principle:

Observe light in an accelerating elevator:

Einstein: Light does not travel in a straight line!


In his General Theory of Relativity, Einstein explained the force of attraction between massive objects in this way:

“Mass tells space-time how to curve, and the curvature of space-time tells masses how to accelerate.”

Einstein’s View of Gravitation

“space-time” refers to 4-dimensional space: x, y, z, ct

Einstein: Space-Time is curved by the presence of mass!


Bending of Space Time

• Newton said the forces due to the mass of the Earth and the mass of the Moon kept the Moon in orbit.

• Einstein said the Moon was trapped in the funnel of the Earths gravity well.

– A gravity well is formed when a mass bends the fabric of space-time.

• General Theory of Relativity

– Orbit is not caused by forces but by the curvature of space-time (funnel curve)



Einstein: Space-Time is curved by the presence of mass!


Einstein proposed a radical experiment to test his theory:

1919: Einstein’s prediction verified by Eddington during a solar eclipse by the moon.

1915

Einstein becomes the most famous scientist of the 20th century!


Gravitational Lensing

“An Einstein Ring”


Gravitational Lensing

These usually involve light paths from quasars & galaxies being bent by intervening galaxies & clusters.

a galaxy behind a galaxy

“Einstein Ring” “Einstein Cross”

multiple images


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Jw Fundamentals of Physics 1 GRAVITY. jw Fundamentals of Physics 2 Chapter 13: Newton, Einstein, and Gravity Isaac Newton 1642 - 1727Albert Einstein 1872