fisica generale - alan giambattista, betty mccarty richardson copyright © 2008 – the mcgraw-hill...

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson

Copyright © 2008 – The McGraw-Hill Companies s.r.l.

Chapter 26: Relativity

•The Postulates of Special Relativity

•Simultaneity

•Time Dilation

•Length Contraction

•Relativistic Velocity Addition

•Relativistic Momentum

•Relativistic Rest Mass Energy, Kinetic Energy, and Total Energy

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§26.1 Postulates of Relativity

Postulate 1: The laws of physics are the same in all inertial reference frames (the principle of relativity).

An inertial reference frame is one in which no accelerations are observed in the absence of external forces. (Recall Newton’s first law).

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Scientists in the 19th century believed that light was a vibration in an invisible medium called the ether.

The Michelson-Morley experiment found that there was no observable change in the speed of light due to the Earth’s motion around the Sun. The conclusion was that the ether did not exist.

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Postulate 2: The speed of light in vacuum is the same in all inertial reference frames. The value is independent of the motion of the source of light or of the observer.

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What follows does not show that Newtonian physics is wrong; it is just not complete. Newtonian physics breaks down at speeds near c; this is where special relativity is needed. All of the familiar relationships from Newtonian physics are contained within special relativity in the limit of low speeds. This is known as the correspondence principle.

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§26.2 Simultaneity and Ideal Observers

The location of an event can be specified by four coordinates: the three spatial coordinates (x,y,z) and a time coordinate t. Taken together (x,y,z,t) are the four coordinates of space-time.

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Consider a high speed train with AC=BC & A’C’=B’C’. The marks ACB are on a stationary train platform and the marks A’C’B’ are on the moving train.

A BC

A’ B’C’v

At t=0: AA’, BB’, and CC’ are lined up. At this instant two lightning bolts strike at AA’ and BB’.

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An observer standing at point C (on the platform) will see both strikes simultaneously.

The observer at C’ (on the train) will see the strike at AA’ first followed by the strike at BB’.

The observers do not agree on what happened, but both are correct.

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This experiment shows that events that are simultaneous in one frame (the platform) are not simultaneous in another frame (the train).

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§26.3 Time Dilation

Consider a light clock. A light pulse reflects back and forth between two mirrors. One complete trip can be considered one “tick” on the clock.

The time interval for a round trip by the light pulse is

.2

0 c

L

v

dt

Mirror

L Light pulse

Mirror

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As long as an observer is stationary with respect to the clock, he will measure a time interval of t0 between clock ticks. Now put the light clock on a moving train. What does a stationary observer outside the train see?

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The motion of the train is to the right with speed v.

c

tvL

c

dt

22

21

2

Knowc

L

v

dt

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The light pulse appears to have the path shown below. How long does it take the light pulse to return to the bottom mirror?

L

vt

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Eliminating L gives .

1

02

0 t

cv

tt

Let,21

1 and

c

v(Lorentz Gamma factor)

The person outside the train will measure a longer time interval between ticks compared to the observer in the train.

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The relative motion between the observers (train and outside the train) changes the rate at which time passes. The effect is known as time dilation.

Moving clocks run slower.

The quantity t0 is known as the proper time. This is time interval between two events that occur in the same place.

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Example (text problem 26.6): An unstable particle called the pion has a mean lifetime of 25 ns in its own frame. A beam of pions travels through the laboratory at a speed of 0.60c. (a) What is the mean lifetime of the pion as measured in the laboratory?

Given: t0 = 25 ns and v = 0.60c (=0.60)

250.160.01

1

1

1

22

ns 31ns 25250.10 tt

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Example continued:

(b) How far does a pion travel (as measured by laboratory observers) during this time?

m 6.5ns 316.0 ctvd

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§26.4 Length Contraction

To measure the length of an object, its ends must be located simultaneously. The proper length is measured when an object is at rest relative to you.

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A person on a train platform will measure the length of the platform to be L0. A moving object will pass by the platform in a time t so L0 = vt (note: t is not the proper time).

A person riding on a train will measure the length of the platform to be L = vt0 (this person measures the proper time).

0

00

0

or L

L

t

t

tv

tv

L

L

where L0 is the proper length.

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The two observers measure different lengths for the platform.

Moving meter sticks are shorter.

This effect is known as length contraction and only applies to lengths parallel to the direction of motion.

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Example (text problem 26.15): A cosmic ray particle travels directly over a football field from one goal line to the other, at a speed of 0.50c. (a) If the length of the field between goal lines is 91.5 m, what is the length measured in the rest frame of the particle?

Given: L0 = 91.5 m and v = 0.50c (=0.50)

155.150.01

1

1

1

22

m. 79155.1

m 5.910 L

L

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Example continued:

(b) How long does it take the particle to go from one goal line to the other according to earth observers?

(c) How long does it take in the rest frame of the particle?

s 101.650.0

5.91 70 cv

Lt

s 103.550.0

m 79 70

cv

Lt

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Example (text problem 26.17): Two spaceships are moving directly toward each other with a relative velocity of 0.90c. If an astronaut measures the length of his own spaceship to be 30.0 m, how long is the spaceship as measured by an astronaut in the other ship?

Given: L0 = 30.0 m and v = 0.90c (=0.90)

294.290.01

1

1

1

22

m. 13294.2

30.0m0 L

L

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§26.5 Velocities in Different Reference Frames

What is the velocity of the probe as measured by Abe? According to Galilean relativity, it is vPA= vPB+vBA.

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Once the speeds get large enough, the relativistic velocity transformation formula must be used.

2BAPB

BAPBPA

1cvvvv

v

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Example (text problem 26.25): The rogue starship, Galaxa, is being chased by the battlecruiser, Millenia. The Millenia is catching up to the Galaxa at a rate of 0.55c when the captain of the Millenia decides it is time to fire a missile. First the captain shines a laser range finder to determine the distance to the Galaxa, and then he fires a missile that is moving at a speed of 0.45c with respect to Millenia. What speed does the Galaxa measure for (a) the laser beam and (b) the missile as they both approach the starship?

The laser beam will be measured to have speed c.

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Example continued:

The velocity of Millenia relative to Galaxa is vMG=0.55c. The velocity of the missile relative to Millenia is vmG = 0.45c. The velocity of the missile relative to Galaxa is

.802.055.045.0

1

55.045.0

1 22MGmM

MGmMmG c

ccccc

c

vvvv

v

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§26.6 Relativistic Momentum

vp m

Note: it is true that p = Ft, but Fma.

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Example: Show that the expression for relativistic momentum reduces to its classical form when v<<c.

mv

c

vmc

c

vmc

mvmvmv

mvmvp

3

22

2

2

1

2

1

2

11

1

1

When v<<c, <<1

To first order in v/c

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Example (text problem 26.36): A body has a mass of12.6 kg and a speed of 0.87c. (a) What is the magnitude of the momentum?

Given: m = 12.6 kg and v = 0.87c (=0.87)

028.287.01

1

1

1

22

m/s. kg 107.687.0kg 6.12028.2 9

c

mvp

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Example continued:

(b) If a constant force of 424.6 N acts in the direction opposite to the body’s motion, how long must the force act to bring the body to rest?

years 0.50s 1058.6

N 6.424

m/s kg 107.6

7

9

F

pt

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§26.7 Mass and Energy

The rest mass energy of a particle is its energy measured in its rest frame.

20 mcE

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1 eV (electron volt) is the change in energy that a charge e (the fundamental unit of charge) experiences when accelerated through a 1 volt potential difference.

eV 1Volt 1 eVqW

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A proton has m=1.6710-27 kg; its rest mass energy is

GeV 0.938

MeV 938

20

mcE

The mass of the proton can be written as 938 MeV/c2.

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A quantity is conserved if its value is unchanged in a given reference frame.

A quantity is invariant if its value is the same in all reference frames.

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§26.8 Relativistic Kinetic Energy

21 mcK Kinetic energy

2

22

0

1

mc

mcmc

KEE

Total energy

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Example:

Show that the expression for relativistic kinetic energy reduces to its classical form when v<<c.

.2

1

2

11

2

11

11

11KE

22222

2

2

2

mvmcmc

mcmc

When v<<c, <<1

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.2222 mcpcE

The relationship between relativistic momentum and total energy is

Other useful relationships are 0

22 2KEKpc

.E

c

c

pv

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Example (text problem 26.48): When an electron travels at 0.60c, what is its total energy in MeV?

Given: v = 0.60c (=0.60)

150.160.01

1

1

1

22

MeV. 65.0MeV/ 511.0250.1 22

2

cc

mcE

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Example (text problem 26.53): An electron has a total energy of 6.5 MeV. (a) What is its momentum (in MeV/c)?

.2222 mcpcE The energy-momentum relationship is

Solving for p:

cp

mcEpc

MeV/ 5.6

MeV 5.6MeV 511.0MeV 5.6 22

222

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Example continued:

(b) What is the value of for this electron?

7.12MeV/ 0.511

MeV 5.6222

ccmc

E

(c) What is the kinetic energy of this electron?

MeV 0.6MeV/ 511.07.11

122

2

cc

mcK

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Summary

•The Postulates of Relativity

•Simultaneity

•Time Dilation (moving clocks run slower)

•Length Contraction (moving meter sticks are shorter)

•Addition of Relativistic Velocities

•Relativistic Momentum

•Rest Mass Energy

•Relativistic Kinetic Energy

fisica generale - alan giambattista, betty mccarty richardson copyright © 2008 – the mcgraw-hill...

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