Far-Sighted Correction

Section 26.1

no ssf

111

Near-Sighted Correction

isf

111

zero

Compound Microscope

Section 26.2

i Ntotal obj eyepiece

obj eyepiece

s sm m m , ƒ ƒ

Refracting Telescope

Section 26.3

Tm

obj

eyepiece

mƒ

ƒ

Shutter Speed and ƒ-NumberThere is a trade-off

between shutter speed and ƒ-number If you reduce shutter speed,

you need to compensate by increasing the ƒ-number

Same Exposure Value (Camera settings) can have different f-number and time

Halving f-number reduces EV by sqrt(2)

Section 26.4

time

fEV n

2

Relativity

Chapter 27

Historical Development1600s

Newton discovered his laws of mechanicsApplied to a wide variety of problems over the next two

decadesWorked well

Late 1800sMaxwell’s equations explained the physics of

electromagnetism and lightEarly 1900s

RelativityQuantum Mechanics

Types of RelativitySpecial relativity

Concerned with objects and observers moving at a constant velocity

Topic of this chapterGeneral relativity

Applies to situations when the object or the observer is accelerated

Has implications for understanding gravitation

Relativity

The term relativity arises when a situation is described from two different points of view

When the railroad car moves with a constant velocity, Ted and Alice see different motions of the ball

Section 27.1

Reference FramesA reference frame can be thought of as a set of

coordinate axesInertial reference frames move with a constant

velocityThe principle of Galilean relativity is the idea that

the laws of motion should be the same in all inertial framesFor example, adding or subtracting a constant velocity

does not change the acceleration of an object and if Newton’s Second Law is obeyed in one inertial frame, it is obeyed in all inertial frames

Section 27.1

Interpretation by Ted and AliceTed observes the ball’s motion purely along the vertical

directionAlice sees the ball undergo projectile motion with a

nonzero displacement in both the x- and y-directionsTed would think the ball’s horizontal velocity is zero, but

Alice would disagreeBoth agree that the ball’s acceleration is downward with a

magnitude gBoth agree the ball’s horizontal acceleration is zeroBoth agree that the only force acting on the ball is the

force of gravity and that Newton’s second law is obeyed

Section 27.1

Galilean Relativity and LightAccording to Maxwell’s equations, the speed of light

is a constantHe also showed the speed of light is independent of

the motion of both the source and the observerAssume Ted is moving with a constant velocity v

relative to Alice when he turns on a flashlight Newton’s mechanics predict that speed of the light

wave relative to Alice should be c + vAccording to Maxwell’s theory, Ted and Alice should

both observe the light wave to move with speed c

Galilean Relativity and Light, cont.Galilean Relativity and

electromagnetism predict different results for observers in different inertial frames

Experiments showed that Maxwell’s theory was correct

The speed of light in a vacuum is always cGalilean relativity for how

the speed of light depends on the motion of the source is wrong

Section 27.1

Special RelativityEinstein developed a theory to analyze the Ted and Alice

situation called special relativityHis work was not motivated by any particular experimentHe suspected the speed of light is the same in all reference

frames Maxwell was correct

He then worked out what that implies for all the other laws of physics

The basics of the theory were stated in two postulates about the laws of physicsFor fast-moving objects Newton’s theory breaks down and

Einstein’s theory gives a correct description of motion in this regime

Section 27.2

Postulates of Special RelativityAll the laws of physics are the same in all inertial

reference frames The speed of light in a vacuum is a constant,

independent of the motion of the light source and all observers

Section 27.2

Postulates – Details First postulate is traced to the ideas of Galileo and

Newton on relativityThe postulate goes further than Galileo because it

applies to all physical laws Not just mechanics

The second postulate is motivated by Maxwell’s theory of lightThis is not consistent with Newton’s mechanics

The postulates will lead to a new theory of mechanics that corrects and extends Newton’s Laws

Section 27.2

More About LightOur everyday experience with conventional waves

cannot be applied to lightLight does not depend on having a conventional

material medium in which to travelA light wave essentially carries its medium with it as

it propagatesIn the electric and magnetic fields

The lack of a conventional medium was surprising and hard to reconcile with conventional intuition

Section 27.2

Inertial Reference FramesInertial reference frames play an important role in

special relativityA definition of what it means to be inertial is neededThe modern definition of an inertial reference is one

in which Newton’s First Law holdsYou can test for an inertial frame by observing the

motion of a particle is zero If the particle moves with a constant velocity, the reference

frame is inertialNewton’s other laws should also apply in all inertial

frames

Section 27.2

Earth as a Reference FrameSince the Earth spins about its axis as it orbits the

Sun, all points on the Earth’s surface have a nonzero acceleration

Technically, a person standing on the surface of the Earth is not in an inertial reference frame

However, the Earth’s acceleration is small enough that it can generally be ignored

In most situations we can consider the Earth to be an inertial reference frame

Section 27.2

Light ClockThe two postulates lead to

a surprising result concerning the nature of time

A light clock keeps time by using a pulse of light that travels back and forth between two mirrors

The time for the clock to “tick” once is the time needed for one round trip: 2ℓ / c

Section 27.3

Moving Light Clock

The clock moves with a constant velocity v relative to the ground

From Ted’s reference frame, the light pulse travels up and down between the two mirrors

Section 27.3

Moving Light Clock, cont.The time for the clock to make one tick as measured

by Ted is

Alice sees the light pulse travel a longer distanceThe speed of light is the same for Alice as for TedBecause of the longer distance, according to Alice

the light will take longer to travel between the mirrors

ot c2

Section 27.3

Moving Clock, Alice’s TimeFor Alice, the time for one tick of the clock is

The time for Ted is different from the time for AliceThe operation of the clock depends on the motion of

the observer

ottv

c2

21

Section 27.3

Moving Clocks Run SlowAlice’s measures a longer time than TedPostulate 1 states that all the laws of physics must

be the same in all inertial reference framesTherefore the result must hold for any clock

Special relativity predicts that moving clocks run slow

This effect is called time dilationFor typical terrestrial speeds, the difference between Δt and Δto is negligible

Section 27.3

Time DilationWhen the speed is

small compared to c, the factor is very close to 1

Approximations given in Insight 27.1 may be used in many terrestrial cases

v c2 21

Section 27.3

Speeds Greater the cIf the value of the speed is greater than the speed of

light, Δt / Δto will be imaginary

Speeds greater than the speed of light have never been observed in nature

Experiments have shown that the time dilation predicted by special relativity is correct

The result applies to all clocks, even biological ones

Section 27.3

Proper TimeThe time interval Δto is measured by the observer at

rest relative to the clockThis quantity is called the proper timeThe time interval measured by a moving observer is

always longer than the proper timeThe proper time is always the shortest possible time

that can be measured for a process, by any observer

Section 27.3

Twin Paradox

An astronaut, Ted, visits a nearby star, Sirius, and returns to EarthSirius is 8.6 light-years from EarthTed is traveling at 0.90 c

Alice, Ted’s twin, stays on Earth and monitors Ted’s trip

Section 27.3

Twin Paradox, TimesAlice measures the trip as taking 19 years

Ted’s body measures the proper time of 8.3 years

Alice concludes that Ted will be younger than she isTed calculates the Earth (and Alice) move away from him

at 0.90 cTed concludes Alice will age 8.3 years while he ages 19

yearsTed concludes that Alice will be younger than he is

lyt years

c17.2

190.90

ot t v c years c c years22 2 21 19 1 0.90 / 8.3

Section 27.3

Twin Paradox, ResolutionTime dilation appears to lead to contradictory resultsAlice’s analysis is correct

She remains in an inertial frame and so can apply the results of special relativity

Ted is incorrectHe accelerates when he turns around at SiriusSpecial relativity cannot be applied during this time

spent in an accelerating frame

Section 27.3

Time Dilation and GPSEach GPS satellite

contains a very accurate clock

The satellite clocks are moving in orbit, so they experience time dilationThey run slow by about

7µs per dayTo accurately determine a

position, the effect of time dilation must be accounted for

Section 27.3

Simultaneity

Two events are simultaneous if they occur at the same timeTed is standing the middle of his railroad carHe moves at a speed v relative to AliceTwo lightning bolts strike the ends of the car and leave burn

marks on the ground which indicate the location of the two ends of the car where the bolts strike

Section 27.4

Simultaneity, cont.Did the two lightning bolts strike simultaneously?According to Alice

She is midway between the burn marksThe light pulses reach her at the same timeShe sees the bolts as simultaneous

According to TedThe light pulse from at B struck before the bolt at A

Since he is moving toward B

The two bolts are not simultaneous in Ted’s view

Section 27.4

Simultaneity, finalSimultaneity is relative and can be different in

different reference framesThis is different from Newton’s theory, in which time

is an absolute, objective quantityIt is the same for all observers

All observers agree on the order of the events

Section 27.4

Length Contraction

Alice marks two points on the ground and measures length Lo between them

Ted travels in the x-direction at constant velocity v and reads his clock as he passes point A and point BThis is the proper time interval of the motion

Section 27.5

Length Contraction, cont.Distance measured by Alice = Lo = v Δt

Distance measured by Ted = L = v Δto

Since Δt ≠ Δto, L ≠ Lo

The difference is due to time dilation and

The length measured by Ted is smaller than Alice’s length

oL L v c2 21 /

Proper LengthTed is at restAlice moves on the

meterstick with speed v relative to Ted

Ted measures a length shorter than Alice

Moving metersticks are shortened

The proper length, Lo, is the length measured by the observer at rest relative to the meterstick

Length Contraction EquationLength contraction is

described by

When the speed is very small, the contraction factor is very close to 1This is the case for

typical terrestrial speeds

o

L vL c

2

21

Section 27.5

Proper Length and Time, ReviewProper time is measured by an observer who is at

rest relative to the clock used for the measurementProper length is measured by an observer who is at

rest relative to the object whose length is being measured

Section 27.5

Experimental SupportA large number of experiments have shown that

time dilation and length contraction actually do occurAt ordinary terrestrial speeds the effects are

negligibly smallFor objects moving at speeds approaching the

speed of light, the effects become significant

Section 27.5