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Newton, Einstein, and Gravity

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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.

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

Mathematics as a tool for understanding physics

- 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

Acceleration (a) is the change of a body’s 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 is independent of the mass (weight) of the falling object!

Iron ball

Wood ball

- 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.

- 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

- 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

- 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.)

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 moon’s effect on Earth is small.

v

v’

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

Moon

F

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

Earth

(SLIDESHOW MODE ONLY)

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

(SLIDESHOW MODE ONLY)

(SLIDESHOW MODE ONLY)

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 = aAU3

Einstein (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

- 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.

- The velocity of light, c, is constant and will be the same for all observers, independent of their motion relative to the light source.

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

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/c]2)1/2

is the Lorentz factor.

c Dt

c Dt’

v Dt

This effect is called time dilation.

- 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

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.”

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

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!

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.

- Perihelion advance (in particular, of Mercury)

- Gravitational red shift: Light from sources near massive bodies seems shifted towards longer wavelengths (red).

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)

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.

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.

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