Newton, Einstein, and Gravity

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Newton, Einstein, and Gravity. Chapter 5. Guidepost.

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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 couldn’t 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 Einstein’s 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

Galileo and Inertia

Forefather of modern science: conducts experiments using scientific method.

Used inclined planes and tried to eliminate friction to study motion.

Determined that objects (without friction) naturally maintain motion-they have inertia

Galileo and Gravity

Acceleration of gravity is independent of the mass (weight) of the falling object.

Friction interferes with falling bodies so they fall differently.

Iron ball

Wood ball

Without friction, all bodies fall at same rate near Earth’s surface.

Isaac Newton (1643 - 1727)
• Builds on the results of Galileo and Kepler
• Adds physics to the mathematical descriptions of astronomy by Copernicus, Galileo and Kepler

• “If I have seen farther than others, it has been by standing on the shoulders of giants.”

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
Newton’s Laws of Inertia

1st Law: A body continues at rest or in uniform motion in a straight line unless acted upon by some net force.

Example: A spacecraft moving in space will continue to travel forever in a straight line unless some external force acts on it.

Newton’s Laws of Acceleration

2nd Law: The accelerationa of a body is inversely proportional to its mass m, directly proportional to the net forceF, and in the same direction as the net force.

a = F/m F = m a

Acceleration is the rate at which velocity changes: a race car goes from 0 to 200 mph in a few seconds!

Aristotle’s “natural” and “violent” motion are rejected. Newton says that constant speed in a straight line is natural, and any accelerated motion is forced.

Newton’s Laws of Action/Reaction

3rd Law: To every action, there is an equal and opposite reaction.

A rifle pushes on a bullet and the bullet pushes on the rifle

The same force that is accelerating the boy forward, is accelerating the skateboard backward.

A rocket is pushed up by forcing exhaust out of engine.

The Universal Law of Gravity
• Newton assumed the laws of the universe apply to terrestrial (Earth) objects and celestial (above Earth) objects alike. He compared a falling apple (downward acceleration) with the orbiting moon (circular acceleration).
• Newton found that any two bodies attract each other through gravitation, with a force equal to the product of their masses divided by the square of their distance. There’s a constant too.

For astronomy, a body of mass m orbits another body of mass M.

Gravity & Inverse Square Law

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.
• Since Earth is much more massive than the moon, the moon’s effect on Earth is small (tides!).

v

v

• Earth’s gravitational force constantly accelerates the moon towards Earth (not straight).

moon

F

• This acceleration is constantly changing the moon’s direction of motion, keeping it on an 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  object falls back down to Earth

Too fast  object escapes Earth’s gravity

Satellite projection animation

Newton’s Cannon

(SLIDESHOW MODE ONLY)

Orbital Motion (3)

Geosynchronous Orbits

Geosynchronous Orbit

(SLIDESHOW MODE ONLY)

Kepler’s Laws Explained by Newton

1st Law: The orbits of the planets are ellipses with the sun at one focus.

2nd Law: A line from a planet to the sun sweeps over equal areas in equal intervals of time.

3rd Law: A planet’s orbital period (P) squared is proportional to its average distance from the sun (a) cubed.

Py2 = aAU3

All laws of planetary motion are proved using law of gravitation!

Planet Platonic Solid

Mercury Inside the Octahedron

Venus Octahedron

Earth Icosahedron

Mars Dodecahedron

Jupiter Tetrahedron

Saturn Cube

Kepler

Newton

Einstein and Relativity

Albert Einstein (1879 – 1955) showed that Newton’s laws of motion are approximate. For low velocities they work well, but not for high velocities (near the speed of light.)

• Einstein developed two theories:
• Theory of Special Relativity
• for constant velocities
• revised ideas of space and time
• Theory of General Relativity
• for acceleration
• revises concept of gravitation
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
• Time Dilation
• Time is not absolute! It is “relative”. Depends on motion of an observer.

Light clock animation

• Examples
• Atomic clocks keep precise time. When a clock is flown on an airplane, it slows down compared with another atomic clock that remained at rest.
• Global Positioning Satellites (GPS)
• require relativity for exact results.
Basics of Special Relativity
• Length Contraction
• Length is not absolute! It’s “relative” - depends on motion of on observer.

Length contraction animation

• Energy/Mass Equivalence
• Mass is not absolute – it’s relative too!
• Objects that move have kinetic energy.
• But so do objects at rest - they have “rest energy”
• Nuclear energy utilizes the conversion of mass to energy with radioactive elements.
General Relativity

A new description of gravity

Equivalence Principle:

“Observers can not tell the difference between acceleration and gravitational forces.”

Which also means:

“Mass tells space-time how to curve, and the curvature of space-time (gravity) tells mass how to accelerate.”

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 can’t 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!

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

inverse square law

field

circular velocity

geosynchronous 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 Newton’s laws.