Newton, Einstein, and Gravity

Chapter 5

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.

Guidepost

I. Galileo and NewtonA. Galileo and MotionB. Newton and the Laws of MotionC. Mutual Gravitation

II. Orbital Motion and TidesA. OrbitsB. Orbital VelocityC. Calculating Escape VelocityD. Kepler's Laws Re-examinedE. Newton's Version of Kepler's Third LawF. Tides and Tidal ForcesG. Astronomy After Newton

III. Einstein and RelativityA. Special RelativityB. The General Theory of RelativityC. Confirmation of the Curvature of Space-Time

Outline

Essential Questions1. What happens when an object

falls?

2. How did Newton discover gravity?

3. How does gravity explain orbital motion?

4. How does gravity explain the tides?

5. How did Einstein refine the understanding of motion and gravity?

A New Era of Science

Sir Isaac Newton used mathematics as a tool for understanding physics.

Isaac Newton (1642 - 1727)• Sir Isaac built on the results of Galileo and

Kepler

Major achievements:

1. Invented Calculus as a necessary tool to solve mathematical problems related to motion

• He added physics interpretations (reasons) to the mathematical descriptions of astronomy by Copernicus, Galileo and Kepler

2. Discovered the three laws of motion

3. Discovered the universal law of mutual gravitation

Galileo Galilei (1594 – 1642)

• Aristotle taught that falling bodies fell with a constant speed, but Galileo showed that they did not!

• Let’s watch a short video on the work of Galileo and then summarize what Galileo did…

Velocity and Acceleration

Acceleration (a) is the change of a body’s velocity (v) with time (t):

1. Acceleration in the conventional sense (i.e. increasing speed)

a = Dv/Dt

Different cases of acceleration:

Velocity and acceleration are directed quantities (vectors)!

3. Change of the direction of motion (e.g., in circular motion)

2. Deceleration (i.e. decreasing speed)

a

v

• 65 miles/hour is a speed• 65 miles/hour South is a velocity• Velocity has a direction associated

with it, speed does not• Acceleration is the rate of change in

velocity (speed and/or direction)• What three controls on a car will

cause acceleration?• Gas & brake pedals, steering wheel

Examples of Velocity and Acceleration

Acceleration of GravityAcceleration of

gravity is independent of the mass (weight) of the falling object!Iron ball

Wood ball

Difficult to verify on Earth because of air resistance; but astronauts could verify it easily on the moon.

Newton’s Laws of Motion (1)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.

Mass & Weight• Mass and weight have different meanings in

science:• Mass is a measure of how much matter an object

contains, or a measure of it’s resistance to change in motion.

• Weight is the force of gravity acting on that mass.• Think of an example where an object’s weight

changes, but its mass is the remains the same…• We measure force/weight in units of Newtons.• A Newton is about the weight of an apple!• Here’s how: • Weight = mass x acceleration of gravity• W = m x g … let’s practice …

Newton’s Laws of Motion (2)

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

• In other words…• If you double the net force on an

object, you’ll double the acceleration.• If you double the mass of an object,

the acceleration will be half of the original.

• Let’s demonstrate with the equipment in the lab…

Newton’s Laws of Motion (2)

Newton’s Laws of Motion (3)

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

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

• It might be better to write Newton’s 3rd Law of Motion:

• “For every force there is an equal and oppositely directed force.”

• Be careful though, the forces are be equal but the effects of those forces are often not!

• List some examples in your notes:

Newton’s Laws of Motion (3)

Newton and Gravity

• Did Isaac Newton discover gravity?

• Certainly not, everyone knew gravity existed!

• Then what did good ‘ole Isaac do?

• Well, that brings us to the story of the apple and the Moon…

The apple and the Moon• One day, Isaac saw an apple fall from

a tree and had a flash of insight…• The Moon orbited the Earth, but

obeyed the three laws of motion…• It ‘wanted’ to move in a straight line,

something pulled it so it moved in a circle (well ellipse actually)…

• Newton said it’s the same force that pulls the apple to the ground!

• Newton discovered gravity is universal, it affects even the heavenly objects!

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:

F = - G Mm

r2

G is the Universal constant of gravity.

G = 6.67 x 10-11 N m2/kg2 (it’s small!)

We should practice a bit.

Understanding Orbital MotionThe universal law of gravity allows us to understand orbital motion of planets and moons:

• Earth and moon attract each other through gravitation.

Example:

Earth

Moon

v v’

Dv

F

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

• Earth’s gravitational force constantly accelerates the moon towards Earth.

• This acceleration is constantly changing the moon’s direction of motion, holding it on its almost circular orbit.

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

Geosynchronous Orbits

Kepler’s 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 Kepler’s third law,

Py2 = aAU

3

The TidesCaused by the difference of the Moon’s gravitational attraction on the water on Earth

2 tidal maxima

Excess gravity pulls water towards the

moon on the near side

12-hour cycle

The unequal gravitational pulls on the Earth, stretch the Earth and oceans in the direction of the Moon

Spring and Neap TidesThe Sun is also producing tidal effects, about half as strong as the Moon.

• Near Full and New Moon, those two effects add up to cause spring tides.

• Near first and third quarter, the two effects work at a right angle, causing neap tides.

Spring tides

Neap tides

Acceleration of the Moon’s Orbital Motion

Earth’s tidal bulges are slightly tilted in the direction of Earth’s rotation.

Gravitational force pulls the moon slightly forward along its orbit.

Einstein and RelativityEinstein (1879 – 1955) noticed

that Newton’s 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)

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)

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 RelativityThe two postulates of special relativity

have some amazing consequences.Consider this thought experiment:

v

c Dt’Light source

Seen by an observer moving along with the light source

v’

• The two postulates of special relativity have some amazing consequences.

• Consider these thought experiments:• Imagine two identical ‘light clocks’, one at

rest and the other moving at a very large speed.

• You can draw what Mr. Bryant now draws on the board…

• Now imagine twins Speedy and Sluggo, one traveling in a very fast spaceship towards Alpha Centauri.

Basics of Special Relativity

Speedy and Sluggo• Sluggo, on Earth, sees Speedy (travelling

at 99% c) aging much less than himself, this is called Time Dilation. Speedy sees himself aging, talking, and walking normally.

• Speedy does see something odd though, the distance to Alpha Centauri is now much less! This is called Length Contraction.

• Consequently Speedy will arrive at Alpha Centauri in much less time (from his perspective) than originally thought!

Effects of Special Relativity

• So, clocks run slower when traveling very fast and lengths get smaller…

• Also, Einstein found an odd effect on energy as well…

• Even at rest, objects still have some ‘rest energy,’ given by E = mc2

• This simple equation allows us the use of nuclear energy and explains how the Sun and stars shine!

General Relativity

Einstein’s 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 ExperimentImagine a light source on board a rapidly accelerated space ship:

As seen by a “stationary” observer

As seen by an observer on board the space ship

Light source

Time Time

a aa

a

g

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)

New description of gravity as curvature of space-time!

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.

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).