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Newton, Einstein, and Gravity

- Chapter 5

Guidepost

Astronomers are gravity experts. All of the

heavenly motions described in the preceding

chapters are dominated by gravitation. Isaac

Newton gets the credit for discovering gravity,

but even Newton couldnt explain what gravity

was. Einstein proposed that gravity is a

curvature of space, but that only pushes the

mystery further away. What is curvature? we

might ask. This chapter shows how scientists

build theories to explain and unify observations.

Theories can give us entirely new ways to

understand nature, but no theory is an end in

itself. Astronomers continue to study Einsteins

theory, and they wonder if there is an even

better way to understand the motions of the

heavens. The principles we discuss in this

chapter will be companions through the remaining

chapters. Gravity is universal.

Outline

I. Galileo and Newton A. Galileo and Motion B.

Newton and the Laws of Motion C. Mutual

Gravitation II. Orbital Motion A. Orbits B.

Orbital Velocity C. Calculating Escape

Velocity D. Kepler's Laws Re-examined E.

Newton's Version of Kepler's Third Law F.

Astronomy After Newton III. Einstein and

Relativity A. Special Relativity B. The General

Theory of Relativity C. Confirmation of the

Curvature of Space-Time

A New Era of Science

Mathematics as a tool for understanding physics

Isaac Newton (1643 - 1727)

- Building on the results of Galileo and Kepler

- Adding physics interpretations to the

mathematical descriptions of astronomy by

Copernicus, Galileo and Kepler

Major achievements

- Invented Calculus as a necessary tool to solve

mathematical problems related to motion

- Discovered the three laws of motion

- Discovered the universal law of mutual gravitation

Velocity and Acceleration

Acceleration (a) is the change of a bodys

velocity (v) with time (t)

a

a Dv/Dt

Velocity and acceleration are directed quantities

(vectors)!

v

Different cases of acceleration

- Acceleration in the conventional sense (i.e.

increasing speed)

- Deceleration (i.e. decreasing speed)

- Change of the direction of motion (e.g., in

circular motion)

Acceleration of Gravity

Acceleration of gravity is independent of the

mass (weight) of the falling object!

Iron ball

Wood ball

Newtons Laws of Motion (1)

- A body continues at rest or in uniform motion in

a straight line unless acted upon by some net

force.

An astronaut floating in space will continue to

float forever in a straight line unless some

external force is accelerating him/her.

Newtons Laws of Motion (2)

- The acceleration a of a body is inversely

proportional to its mass m, directly proportional

to the net force F, and in the same direction as

the net force.

a F/m ? F m a

Newtons Laws of Motion (3)

- To every action, there is an equal and opposite

reaction.

M 70 kg

V ?

The same force that is accelerating the boy

forward, is accelerating the skateboard backward.

m 1 kg

v 7 m/s

The Universal Law of Gravity

- Any two bodies are attracting each other through

gravitation, with a force proportional to the

product of their masses and inversely

proportional to the square of their distance

Mm

F - G

r2

(G is the Universal constant of gravity.)

Understanding Orbital Motion

The universal law of gravity allows us to

understand orbital motion of planets and moons

Example

- Earth and moon attract each other through

gravitation.

Dv

- Since Earth is much more massive than the moon,

the moons effect on Earth is small.

v

v

- Earths gravitational force constantly

accelerates the moon towards Earth.

Moon

F

- This acceleration is constantly changing the

moons direction of motion, holding it on its

almost circular orbit.

Earth

Center of Mass

(SLIDESHOW MODE ONLY)

Orbital Motion (2)

In order to stay on a closed orbit, an object has

to be within a certain range of velocities

Too slow gt Object falls back down to Earth

Too fast gt Object escapes Earths gravity

Orbital Motion (3)

Geosynchronous Orbits

Newtons Cannon

(SLIDESHOW MODE ONLY)

Geosynchronous Orbit

(SLIDESHOW MODE ONLY)

Keplers Third Law Explained by Newton

Balancing the force (called centripetal force)

necessary to keep an object in circular motion

with the gravitational force ? expression

equivalent to Keplers third law,

Py2 aAU3

Einstein and Relativity

Einstein (1879 1955) noticed that Newtons laws

of motion are only correct in the limit of low

velocities, much less than the speed of light.

? Theory of Special Relativity

Also, revised understanding of gravity

? Theory of General Relativity

Two Postulates Leading to Special Relativity (1)

- Observers can never detect their uniform motion,

except relative to other objects.

This is equivalent to

The laws of physics are the same for all

observers, no matter what their motion, as long

as they are not accelerated.

Two Postulates Leading to Special Relativity (2)

- The velocity of light, c, is constant and will be

the same for all observers, independent of their

motion relative to the light source.

Basics of Special Relativity

The two postulates of special relativity have

some amazing consequences. Consider thought

experiment

Motion of stationary observer

Assume a light source moving with velocity v

relative to a stationary observer

v

v

v

c Dt

c Dt

Light source

c Dt

v Dt

Seen by an observer moving along with the light

source

Seen by the stationary observer

Basics of Special Relativity (2)

Now, recall that the velocity of light, c, is the

same for all observers.

? The times Dt and Dt must be different!

Then, the Pythagorean Theorem gives

(cDt)2 (cDt)2 (vDt)2

or Dt (Dt)/g where g 1/(1 v/c2)1/2 is

the Lorentz factor.

c Dt

c Dt

v Dt

This effect is called time dilation.

Other Effects of Special Relativity

- Length contraction Length scales on a rapidly

moving object appear shortened.

- Relativistic aberration Distortion of angles

- The energy of a body at rest is not 0. Instead,

we find - E0 m c2

General Relativity

A new description of gravity

Postulate Equivalence Principle Observers can

not distinguish locally between inertial forces

due to acceleration and uniform gravitational

forces due to the presence of massive bodies.

Another Thought Experiment

Imagine a light source on board a rapidly

accelerated space ship

Time

Time

a

Light source

a

a

a

g

As seen by a stationary observer

As seen by an observer on board the space ship

Thought Experiment (2)

For the accelerated observer, the light ray

appears to bend downward!

Now, we cant distinguish between this inertial

effect and the effect of gravitational forces

Thus, a gravitational force equivalent to the

inertial force must also be able to bend light!

Thought Experiment (Conclusion)

This bending of light by the gravitation of

massive bodies has indeed been observed

During total solar eclipses The positions of

stars apparently close to the sun are shifted

away from the position of the sun.

? New description of gravity as curvature of

space-time!

Another manifestation of bending of light

Gravitational lenses

A massive galaxy cluster is bending and focusing

the light from a background object.

Other Effects of General Relativity

- Perihelion advance (in particular, of Mercury)

- Gravitational red shift Light from sources near

massive bodies seems shifted towards longer

wavelengths (red).

New Terms

natural motion violent motion acceleration of

gravity momentum mass acceleration velocity invers

e square law field circular velocity geosynchronou

s satellite center of mass closed orbit escape

velocity open orbit angular momentum energy joule

(J)

special relativity general theory of relativity

Discussion Questions

1. How did Galileo idealize his inclines to

conclude that an object in motion stays in motion

until it is acted on by some force? 2. Give an

example from everyday life to illustrate each of

Newtons laws.

Quiz Questions

1. According to Aristotle, where is the proper

place of the classical elements earth and water

that is, what location do they seek? a. The

center of Earth. b. The center of the

Universe. c. The Heavens. d. Both a and b

above. e. Both b and c above.

Quiz Questions

2. According to the principles of Aristotle, what

part of the motion of an arrow that is fired

vertically upward is natural motion and what part

is violent motion? a. Both the upward and

downward parts are natural motion. b. Both the

upward and downward parts are violent motion. c.

The upward part is natural motion and the

downward part is violent motion. d. The upward

part is violent motion and the downward part is

natural motion. e. Neither the upward nor the

downward parts are natural or violent motion.

Quiz Questions

3. If we drop a feather and a hammer at the same

moment and from the same height, on Earth we see

the hammer strike the ground first, whereas on

the Moon both strike the ground at the same time.

Why? a. The surface gravity of Earth is

stronger than the gravity of the Moon. b. In

strong gravity fields heavier objects fall

faster. c. The is no air resistance effect on the

Moon. d. Both a and b above. e. All of the above.

Quiz Questions

4. Which statement below best describes the

difference between your mass and your weight? a.

Your mass is constant and your weight varies

throughout your entire life. b. Your mass is a

measure of the amount of matter that you contain

and your weight is a measure of the amount of

gravitational pull that you experience. c. Your

mass is a measure of your inertia, whereas your

weight is a measure of the amount of material you

contain. d. The only difference is the unit used

to measure these two physical quantities. Mass

is measured in kilograms and weight is measured

in pounds. e. There is no difference between your

mass and your weight.

Quiz Questions

5. Which of the following is true for an object

in uniform circular motion? a. The velocity of

the object is constant. b. The acceleration of

the object is zero. c. The acceleration of the

object is toward the center of motion. d. The

angular momentum of the object is zero. e. The

speed of the object is changing.

Quiz Questions

6. If a 1-kilogram rock and a 6-kilogram rock are

dropped from the same height above the Moon's

surface at the same time, they both strike the

Moon's surface at the same time. The

gravitational force with which the Moon pulls on

the 6-kg rock is 6 times greater than on the 1-kg

rock. Why then do the two rocks strike the

Moon's surface at the same time? a. The

acceleration of each rock is inversely

proportional to its mass. b. The Moon's surface

gravity is one-sixth the surface gravity at

Earth's surface. c. The 1-kg rock is attracted

less by the nearby Earth. d. Both a and b

above. e. All of the above.

Quiz Questions

7. Why did Newton conclude that some force had to

pull the Moon toward Earth? a. The Moon's

orbital motion is a curved fall around Earth. b.

The Moon has an acceleration toward Earth. c. The

force and acceleration in Newton's second law

must have the same direction. d. Both b and c

above. e. All of the above.

Quiz Questions

8. What did Newton determine is necessary for the

force exerted by the Sun on the planets to yield

elliptical orbits? a. The force must be

attractive. b. The force must be repulsive. c.

The force must vary inversely with distance. d.

The force must vary inversely with distance

squared. e. Both a and d above.

Quiz Questions

9. Which of Kepler's laws of planetary motion is

a consequence of the conservation of angular

momentum? a. The planets orbit the Sun in

elliptical paths with the Sun at one focus. b. A

planet-Sun line sweeps out equal areas in equal

intervals of time. c. The orbital period of a

planet squared is proportional to its semimajor

axis cubed. d. Both b and c above. e. All of the

above.

Quiz Questions

10. How did Galileo slow down time in his falling

body experiments? a. He performed the

experiments near the speed of light. b. He

measured the time objects took to fall through

water. c. He used a stopwatch. d. He rolled

objects down inclines at low angles. e. He began

each fall with an upward toss.

Quiz Questions

11. Which of Newton's laws was first worked out

by Galileo? a. The law of inertia. b. The net

force on an object is equal to the product of its

mass and its acceleration. c. The law of action

and reaction. d. The law of universal mutual

gravitation. e. Both c and d above.

Quiz Questions

12. According to Newton's laws, how does the

amount of gravitational force on Earth by the Sun

compare to the amount of gravitational force on

the Sun by Earth? a. The amount of force on

Earth by the Sun is greater by the ratio of the

Sun's mass to Earth's mass. b. The amount of

force on the Sun by Earth is negligible. c. The

amount of force on the Sun by Earth is the same

as the amount of force on Earth by the Sun. d.

The amount of force on the Sun by Earth is

greater by the ratio of the Sun's mass to Earth's

mass. e. It is impossible to compare these two

vastly different amounts of force.

Quiz Questions

13. Suppose that Planet Q exists such that it is

identical to planet Earth yet orbits the Sun at a

distance of 5 AU. How does the amount of

gravitational force on Planet Q by the Sun

compare to the amount of gravitational force on

Earth by the Sun? a. The amount of the two

forces is the same. b. The amount of force on

Planet Q is one-fifth the force on Earth. c. The

amount of force on Planet Q is 5 times the force

on Earth. d. The amount of force on Planet Q is

one twenty-fifth the force on Earth. e. The

amount of force on Planet Q is 25 times the force

on Earth.

Quiz Questions

14. Newton's form of Kepler's law can be written

as (Msun Mplanet) Py2 aAU3, where the masses

of the Sun and planet are in units of solar

masses, the period is in units of years, and the

semimajor axis in astronomical units. Why is

Kepler's form of his third law nearly identical

to Newton's form? a. Both forms are very similar

in that they have periods and semimajor axes in

units of years and astronomical units

respectively. b. The mass of the Sun plus the

mass of a planet is nearly one. c. The mass of

each planet is very large. d. Both b and c

above. e. All of the above.

Quiz Questions

15. How does the orbital speed of an asteroid in

a circular solar orbit with a radius of 4.0 AU

compare to a circular solar orbit with a radius

of 1.0 AU? a. The two orbital speeds are the

same. b. The circular orbital speed at 4.0 AU is

four times that at 1.0 AU. c. The circular

orbital speed at 4.0 AU is twice that at 1.0

AU. d. The circular orbital speed at 4.0 AU is

one-half that at 1.0 AU. e. The circular orbital

speed at 4.0 AU is one-fourth that at 1.0 AU.

Quiz Questions

16. In the 1960s television program "Space 1999"

an accident on the Moon causes the Moon to be

accelerated such that it escapes Earth and

travels into interstellar space. If you assume

that the Moon's orbit was nearly circular prior

to the accident, by what minimum factor is the

Moon's orbital speed increased? a. The Moon's

speed must be increased by a factor of 4 to

escape Earth. b. The Moon's speed must be

increased by a factor of pi to escape Earth. c.

The Moon's speed must be increased by a factor of

2 to escape Earth. d. The Moon's speed must be

increased by a factor of 1.4 to escape Earth. e.

It cannot be determined from the given

information.

Quiz Questions

17. Just after a alien spaceship travels past

Earth at one-half the speed of light, a person on

Earth sends a beam of light past the ship in the

same direction that the ship is traveling. How

fast does an alien on the ship measure the light

beam to be traveling as it zips past the

spaceship? a. At the speed of light, or 300,000

km/s. b. At one-half the speed of light, or

150,000 km/s. c. At one and one-half the speed of

light, or 450,000 km/s. d. At twice the speed of

light, or 600,000 km/s. e. The measured speed

depends on the method of measurement.

Quiz Questions

18. Who first proposed that gravity is the

bending of space-time due to the presence of

matter? a. Tycho Brahe (1546 - 1601) b. Johannes

Kepler (1571 - 1630) c. Galileo Galilei (1564 -

1642) d. Isaac Newton (1642 - 1727) e. Albert

Einstein (1879 - 1955)

Quiz Questions

19. What major orbital problem of the late 1800s

is solved by general relativity? a. The reason

for the elliptical shape of planetary orbits. b.

The relationship between circular and escape

velocity. c. The periods of parabolic and

hyperbolic orbits. d. The excess precession of

Mercury's perihelion. e. The three-body problem.

Quiz Questions

20. What is significant about the May 29, 1919

solar eclipse? a. It was an annular eclipse

visible from South America, the South Atlantic,

and central Africa in 1919. b. The bending of

light by gravity was observed, thus verifying

general relativity. c. The Moon was at New phase

and at one node of its orbit during this

eclipse. d. It marked the end of the first

complete Saros cycle of the 20th century. e. It

was not predicted.

Answers

1. d 2. d 3. c 4. b 5. c 6. a 7. e 8. e 9. b 10. d

11. a 12. c 13. d 14. b 15. d 16. d 17. a 18. e 19

. d 20. b