REVIEW NOTES IN PHYSICS 2Prepared by: Engr. Luzviminda A. Lescano

HEAT

Thermal energy – energy resulting from heat flow; also known as internal energy Internal energy – total potential and kinetic energy of the particles of a substance. Heat – thermal energy that is transferred from a hot body to a cold body; thermal energy in

motion. Q = m c ( t2 - t1 )Q = + when heat is absorbedQ = - when heat is liberated

Example 1:How much heat does 25 g of aluminum give off as it cools from 100 oC to 20 oC? For aluminum c = 0.21 cal/g Co Ans. – 420 cal

Temperature – measures the average kinetic energy of the particles in a body. Specific heat of a substance ( c )– amount of heat needed to raise the temperature of a unit mass

of a substance by one degree; substances with high specific heat heas up more slowly or cool down at a slower rate; with low specific heat heat up quickly and cool off quickly

Calorimetry – involves the measurement of the heat transferred between substances; calorimetry problems involve the sharing of heat energy among initially hot objects and cold objects. Since energy must be conserved:

Heat lost by hotter substances = heat gained by the cooler substances Example 2: A thermos bottle contains 150 g of water at 4 oC. Into this is placed 90 g of metal at 100 oC. After equilibrium is established, the temperature of the water and metal is 21 oC. What is the specific heat of the metal? Assume no heat loss to the thermos bottle . Specific heat of water = 1 cal/g Co Ans. 0.36 cal/g Co

Example 3: A 255-g block of gold at 85 oC is immersed in 155 g of water at 25 oC. Find the equilibrium temperature, assuming the system is isolated and the heat capacity of the cup can be neglected. Sp. ht. of Au = 0.0308 cal/g Co Ans. 27.9 oCExample 4: A 20-kg gold bar at 35oC is placed in a large, insulated 0.80-kg glass container at 15 oC and 2.0 kg of water at 25 oC. Calculate the final temperature of mixture. Sp. ht. of glass= 0.2 cal/g Co Ans. 26.6 oC

Abnormal expansion of water:Cooling water from 4 C to 0 C – it expandsAt 0 C – it freezes to ice, volume increases, density decreasesBelow 0 C – it contracts

Vaporization – process of changing from liquid to gaseous state at the boiling Melting of fusion – process of changing a solid to liquid. Freezing – changing liquid to solid Condensation – vapor to liquid Sublimation – solid to vapor Latent Heat of fusion – heat required to melt 1 kg of a solid in to a liquid phase at the solid melting

point; also equal to the heat given off when a substance changes from liquid to solid. For water: 80 cal/g

Latent Heat of vaporization – energy required to change 1 kg of a substance from the liquid to the vapor state at the liquid’s boiling point; also the energy given off when the substance changes from vapor to liquid. For water: 540 cal/g

Heat involved in a change of phase:For heat absorption processes: Q =+ mL ( m=mass , L=latent heat )For heat liberation process: Q = -mL

Example 5: How much heat is required to change 40 g of ice cube from ice at -10oC to steam at 110oC? Specific heats: ice = 0.5 cal/g Co ; steam = 0.48 cal/g Co Ans. 29 192 cal Example 6: How much heat is given up when 20 g of steam at 100 oC is condensed to 20 oC?

Ans. -12 400 cal

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Example 7: A 20-g piece of aluminum at 90 oC is dropped into a cavity in a large block of ice at 0 oC? How much ice does the aluminum melt? Sp. ht. of Al= 0.215 cal/g Co Ans. 4.7 g

Example 8: Determine the resulting temperature, when 150 g of ice at 0 oC is mixed with 300 g of water at 50 oC. Ans. 6.7 oC

Evaporation – process in which a liquid changes to the gas phase at room temperature. Conduction – transfer of heat due to atomic ( or molecular ) collisions within a substance

(solid, liquid and gas) or from one object to another when they are in contact. The rate of heat flow within a substance is

k A TH =

dWhere: H = rate of heat flow ; k is the thermal conductivity of the material, A is the cross-

sectional area, d is the thickness (or length) of the material , and T is the temperature difference between one side and the other.

Convection – transfer of heat in fluids by means of fluid currents within the heated fluids that carry heat from one place to another; the material itself moves from one place to another.

Radiation – transfer of heat by means of electromagnetic waves ; James Prescot Joule- showed the quantitative relationship between heat and work If an object has an initial length Lo at some temperature to and undergoes a change in

temperature t , its linear dimension changes by the amount L, which is proportional to the object’s initial length and the temperature change. L = Lo t = Lo ( t – to ) where = coefficient of linear expansion of the material V = Vo t = Vo ( t – to ) = coefficient of volume expansion of the material

= 3Example 9: A steel railroad track has a length of 30.000 m when the temperature is 0 oC. What is its length on a hot day when the temperature is 40 oC? of steel = 11 x 10-6 ( Co)-1

Ans. 30.013 mExample 10. A 1.00-liter aluminum cylinder at 5.00 oC is filled to the brim with gasoline at the same temperature. If the aluminum and gasoline are warmed to 65 oC, how much gasoline spills out? of Al = 24 x 10-6 ( Co)-1 ; of gasoline = 9.6 x 10-4 ( Co)-1 Ans. 53.3 cm3

NATURE AND TYPES OF WAVES

Waves are created when objects vibrate or oscillate; disturbance that travels through a medium or through empty space.

Mechanical waves – need a material medium through which they can travel; water waves, sound waves; rope waves

Medium does not travel with the wave but it is the energy which is being transferred by the traveling disturbance.

Transverse waves– waves which can move in such a way that the motion of the medium is perpendicular to the motion of the waves, that is, the motion of the particles of the medium move up and down while the wave move forward. Examples :rope waves, water waves, electromagnetic waves

Longitudinal waves– ( or compressional waves ) – the medium vibrates in a direction parallel to the direction in which the waves travel. Examples: sound waves, waves in stretched spring

In a liquid, the motion of the particles may be neither purely transverse nor purely longitudinal but a combination of the two.

Periodic waves – a series of waves created by a continuous disturbance.

Characteristics of Waves

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Crests – highest points of a transverse wave Troughs – lowest points of a transverse wave Wavelength ( ) – distance from crest to crest; or from trough to trough. Amplitude ( A ) – distance of the crest or trough from the midpoint of the wave. Frequency ( f ) – number of crests or waves that passes a fixed point per second; the unit of

frequency is hertz (Hz) ; higher frequencies are measured in kHz ( AM radio waves ), MHz ( FM radio waves and GHz ( radar or microwave ovens ). AM – amplitude modulation ; FM – frequency modulation

Period ( T ) – time it takes a wave to travel a distance equal to a wavelength and is the reciprocal of frequency. T = 1 / f

crest

● ● ●

amplitude

amplitude trough

● ●

Wave velocity: v = f Example 11: An FM station broadcast at a frequency of 99.9 MHz; with a radio wave having a wavelength of 3.0 m . Find the speed of the radio wave. Ans. 3 x 108 m/s

Wave speed i a medium: constant in a given medium ; depends on the properties of that medium; on strings speed of waves depends on the tension ( T ) in the string and the linear density µ

T T v = ---- = ------- µ m/L

In a given medium if the frequency of the wave is changed, then the wavelength of the wave change ; the amplitude of a wave does not affect its wave ; Long wavelength – low frequency ; short wavelength – high frequency Example 12: A uniform string has a mass of 0.30 kg and a length of 6 m. Tension is maintained in the string by suspending a 2.0-kg block from one end. Find the speed of a pulse in this string. Ans.19.8 m/s

Characteristics of wave motion Rectilinear propagation – In a uniform medium, waves travel in straight lines perpendicular to

the advancing wave fronts ; Wave front – surface passing through the point of a wave that have the same phase

and amplitude; parallel lines representing the wave crest ;perpendicular to the direction of a wave

Ray – straight line drawn in the direction of the wave motion perpendicular to the wave fronts; for spherical wave front, like water waves produced by dropping a

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pebble itno a pool of water, the ripples of concentric circles are the wave front and the rays are radially outward from the point source.

Reflection – bouncing back of waves when it strikes a surface they hit; wavelength and frequency of the waves are not affected by the reflection when the reflecting surface is stationary; Examples: echoes of sound waves, reflection of light from a mirror or any smooth surface

Refraction – bending of waves as they pass obliquely from one medium into another of different speed of propagation

Diffraction – spreading of waves around a barrier in its path; ( Huygen’s Principle- when a wave meets a very small obstacle like a small slit, the wave front in the small slit acts as a new source of light; It is diffraction of sound waves which enables you to hear your teacher even if she still outside the room near the door

Interference – refers to the effect of two or more waves moving simultaneously through a medium. Interference of incoming radio waves with those reflected from clouds or airplanes can be sometimes observed in your television or radio set ; the rainbow colors observed in soap bubbles and in oil floating in the street on rainy days are produced by interference of light waves.

Constructive interference – occurs when two waves with the same frequency and amplitude traveling in the same direction are in phase ( particles which are in the same relative positions and move in the same direction are said to be in the same phase ) ; individual effects add together to form a wave having the same frequency as the individual waves but twice their amplitude; there is reinforcement. When two crests, or two troughs meet, there is constructive interference. Example: Constructive interference of light waves, the resulting effect is the production of brighter light

Destructive interference – occurs when two waves with the same frequency and amplitude are 180o out of phase; the result when they combine is complete cancellation ;The crest of one wave coincide with the troughs of the other ; the individual amplitudes subtract.

Standing waves – stationary wave pattern formed in a medium when two waves having the same frequency, amplitude and wavelength travel in opposite directions through a medium. Standing waves can be set up in a stretched string by connecting one end of the string to a stationary clamp and connecting the other end to a vibrating object such as the end of a tuning fork or by shaking the hand holding the string up and down at a steady rate; set up when you pluck a guitar string.

A standing wave on a string causes the string to sweep out a pattern which shows the displacement nodes and the displacement antinodes.

Displacement node – point on a standing wave where the amplitude of vibration is zero.; points of destructive interference.

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Displacement antinodes – point on a standing wave where the amplitude of vibration is greatest; points of constructive interference.

The distance between two adjacent nodes ( or antinodes) is half wavelength. The region between two nodes is called a segment. The wavelength of the standing wave bears a simple relationship to the length L of the string.

n n where n = number of harmonicL =

2

Superposition principle – states that when two waves meet while traveling through a medium at the same time, the result is a wave whose displacement is equal to the vector sum of the displacement of the two waves. Example: Sound wave from all the musical instrument of an orchestra move simultaneously through the air to our ears, yet we can still listen to the sound of a particular instrument.

Natural frequency – frequency at which an elastic object naturally tends to vibrate if it is disturbed and the disturbing force is removed.

Fundamental frequency – lowest natural frequency of the vibrating string or the first harmonic; corresponds to one antinode ( or one loop or segment )

V where V = speed of the wavef1 =

2 L The natural frequencies of vibration of a stretched string of length L, fixed at both ends

n Tfn = n f1 = ---- ----- where n= 1,2,3,……

2 L µ

When n = 1 (one segment) first harmonic = 2Ln = 2 (two segments) second harmonic ( also called the first overtone ) = Ln = 3 (three segments) third harmonic ( second overtone ) = 2/3 L

Example 13: A piano string is 1.10 m long and a mass of 9 g. (a) How much tension must the string be under if it is to vibrate at a fundamental frequency of 131 Hz? (b) what are the frequencies of the first four harmonics? Ans. 747.5 N , 131 Hz, 262 Hz, 393 Hz, 524 Hz.Example 14: A string vibrates in 5 segments to a frequency of 460 Hz. What frequency will cause it to vibrate in 2 segments? Ans. 184 Hz

In a stationary wave there are points called nodes at which the amplitude is always zero; at other point, called antinodes, the amplitude is a maximum and equal to the sum of the amplitudes of the individual waves ; there is no flow of energy through the medium in a stationary wave

ELECTROMAGNETIC WAVES

Electromagnetic waves – or electromagnetic radiation are oscillations in the electric field of an accelerated charged particle which could travel across space.;need no medium to travel since these waves can travel through a vacuum; the medium through these waves are the electric and magnetic field;

Predicted by James Clerk Maxwell and experimentally confirmed by Heinrich Hertz Are created by accelerating electric charges Are transverse waves because the electric and magnetic fields are perpendicular to the

direction of the waves Travel at the speed of light The electromagnetic spectrum includes waves covering a broad range of frequencies and

wavelengths

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Radio waves – result of charges accelerating through conducting wires; used in radio and TV communication systems; have the longest wavelengths but lowest frequency

Microwaves – (short-wavelength radio waves ) – have wavelengths generated by electronic devices; their short wavelengths make them well suited for radar systems used in aircraft navigation and for the study of atomic and molecular properties of matter

Infrared waves – ( sometimes incorrectly called “heat wave”, produced by hot objects and molecules

Visible light- part of the spectrum that is detected by the human eye; wavelengths of visible light are classified as colors

Colors of lightViolet – shortest wavelength, highest frequencyIndigoBlueGreenYellowOrangeRed – longest wavelength, lowest frequency

Ultraviolet ( UV) light – the Sun is an important source; ozone O3 is produced from reactions of oxygen with ultraviolet radiation

X-rays – common source of x-rays is the acceleration of high-energy electrons bombarding a metal target

Gamma rays – emitted by radioactive nuclei; have the shortest wavelengths but highest frequency

SOUND

Sound waves are longitudinal waves traveling through a medium, such as air. Sound is a disturbance of the type capable of being detected by the ear. Audible waves – lie within the range of sensitivity of the human ear, approximately 20 to 20

000 Hz Infrasonic waves – with frequencies below the audible range ; ex. Earthquake Ultrasonic waves – with frequencies above the audible range for humans; can be very

penetrating due to their short wavelength Supersonic – refers to an object traveling faster than sound Compressions – regions where the air molecules are close to each other Rarefactions – where the air molecules are far apart ; Speed of sound: We see lightning flash and then hear a thunder a few seconds later

because sound travels more slowly than light; generally greater in liquids than in gases and still greater in solids( solids have greater elasticity than liquids or gases) ; it is four times faster in water than in air, and about 15 times faster in steel than in air; that’s why the sound of a distant train can be heard more clearly by putting your ear against the rail ; sound of motor boats can be heard more clearly under water than above water. The speed of sound in air at 0 C is about 331.5 m/s; in water at 25 oC is about 1500 m/s ; in steel rod at 25 oC is 5200 m/s; increases by about 0.6 m/s for each oC

The speed of sound in liquids of bulk modulus B and density is

Bv = ------

The speed of sound in solids of Young modulus Y and density is

Yv = ------

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The speed of sound also depends on the temperature of the medium

T T is the absolute (Kelvin) tempv = (331 m/s) --------

273 K

Or: v = 331 m/s + 0.6 ( t )

The speed of sound in gases

Where: R = 8.314 J/mol K

v = k R T M = Molecular weight in kg/molM T = absolute temperature in K

k = cp / cv = 1.4 for air , oxygen, nitrogen

Example 15: What is the speed of sound waves in water? The bulk modulus for water is 2.2 x 109 N/m2. Ans. 1480 m/s

Example 16: The speed of sound in a metal rod is 6000 m/s. What is the Young’s Modulus for the material of the rod if the density of the material is 8.2 g/cc?Ans. 3 x 1011 N/m2

Example 17: Compute the speed of sound in air at 20 oC. Mean molar mass of air is 28.8 x 10-3 kg/mol Ans. 344 m/s

Example 18: Compute the speed of sound in He (at. wt. =4 ) at 800oC. For He: k=1.66Ans. 1924 m/s

Reflection of Sound Waves

Echo – reflection of sound waves ; occurs when reflected sound waves return to the observer 0.1 s or more after the original wave reaches him

Reverberation – the persistence of sound in an enclosed space, due to repeated reflection of sound.; the successive echoes that can be heard ; continuous reflection of sound

Reverberation time – total time during which a sound remains audible due to repeated reflection.

Smooth surfaces are good reflectors of sound; in such case the reverberation time is long; In lecture rooms and conference halls, reverberation is undesirable; to minimize walls are made rough or draperies are placed (good sound absorbing materials)

Physical Properties of Sound

Pitch – degree of highness or lowness of the sound as perceived by the listeners; pitch of sound can be expressed in terms of its frequency; high pitch-high frequency

Doppler effect – change in pitch produced by the relative motion of the source and observer

fs ( v + vo )

fo = (v – vs )

where: v = speed of soundvo = speed of observer; (+) when observer moves toward the source,(-) when observer moves away

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vs = speed of source (+) when source moves toward the observer,(-) when moves awayfo = observed frequencyfs = frequency of source of sound

Example 19: An automobile moving at 30 m/s is approaching a factory whistle that has a frequency of 500 Hz. (a) If the speed of sound in air is 340 m/s, what is the frequency of the whistle as heard by the driver. (b) Repeat for the case of the car leaving the factory at the same speed. Ans. 544 Hz , 456 HzExample 20: A car moving at 20 m/s with its horn blowing ( f=1200 Hz) is chasing another car going at 15 m/s. What is the frequency of the horn as heard by the driver being chased? Take the speed of sound to be 340 m/s. Ans. 1219 Hz

Intensity ( I ) – known as the “loudness” of sound ; rate at which energy flow through a unit area expressed in decibel ; amplitude dependent ; bigger amplitude – loud sound

Loudness – measure of the amount of auditory sensation sound waves produced; sensory effect of intensity;

Threshold of hearing – faintest sound the human ear can detect at a frequency of 1 000 Hz have an intensity of about 10 -12 W/m2 ( 0 db )

Threshold of pain – loudest sound the ear can tolerate have an intensity of about 1 W/m2 ( 120 dB)

Spherical Waves : Intensity of sound at a distance r from a source is

P PI = = where: P = power in watts

A 4 r2 A = surface area in cm2 I = intensity in decibel (dB)

W1 dB = 1

cm2

Ratio of the intensities of two spherical surfaces

I1 r22

=I2 r1

2

Example 21: A small source emits sound waves with a power output of 80 W. Find the intensity 3.0 m from the source. Ans. 0.707 W/m2

The intensity level of a sound wave is I where: Io = reference intensity

= 10 log------ = 1 x 10-12 W/m2 Io = measured in decibel (dB)

I = any intensity

Source sound Intensity Level, dB Intensity W/m2

whisper 20 10-10

busy street traffic 70 10-5

military jet 30 m away 140 102

Example 22: A sound has an intensity of 3 x 10-8 W/m2 . What is the sound level in dB?Ans. 44.8 dB

Example 23: A noise-level meter reads the sound level in a room to be 85 dB. What is the sound intensity in the room? Ans. 3.16 x 10-4 W/m2

Quality of sound – also known as timbre ; depends upon the complexity of the wave; upon the number and relative prominence of the overtones; two sounds coming from different sources

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may have the same frequency and the same loudness but because of quality, one source can be distinguished from the other; the quality of sound depends on vibrations of wave forms.

Beats – fluctuations in amplitude produced by two sound waves of slightly different frequency; Beat frequency - the number of beats per second is equal to the difference in frequencies of

the two sound waves that recombined. Shock wave – the cone-shaped wave made by an object moving at supersonic speed through

a fluid. Sonic boom – the loud sound resulting from the incidence of a shock wave. Sonar – procedure for underwater detection and navigation based on the emission from and

return to a tracking ship of a pulse signal. Resonance – the reinforcing of sound; occurs when the frequency of forced vibration on an

object matches the object’s natural frequency. Acoustics – science that ties together the production and transmission of sound to our sense

of hearing Musical tone – produced by a regular succession of compression and the following

rarefactions Music – produced by regular vibration pleasing to the ear. Unpitched sound – produced by an irregular succession of compression and rarefactions or

by a disturbance of such short duration that the ear is unable to distinguish a regular succession.

Noise – produced by irregular vibrations that corresponds to an irregular vibrations of the eardrums ; undesired sound

Sensory effects(may vary among individual and

Subjective

Physical Property(measurable and objective)

loudness intensitypitch frequency

quality waveform

Carrier Wave – a wave usually of radio frequency, whose characteristics are modified in the process of modulation.

Modulation – the process of impressing one wave system upon another of higher frequency.

OPTICS

Geometric optics – branch of optics for which the ray description is adequate. Physical optics – branch dealing with wave behavior Corpuscular theory – light consists of tiny particles called corpuscles that travel in a straight

line through space ; proposed by Isaac Newton Wave theory – proposed by Christian Huygens ; suggested that light consists of a series of

waves. Electromagnetic theory – light is an electromagnetic wave which can travel in space without

any medium, proposed by James Clark Maxwell Heinrich Rudolf Hertz – verified experimentally the existence of electromagnetic wave as

described by Maxwell. Quantum theory –light consists of packets of energy called photons or quanta; explained the

photoelectric effect Modern theory – light exhibits a dual characteristics that of wave and of particle. Speed of light : Light travels in empty space (vacuum) at the speed of 3 x 108 m/s Dispersion – spreading of light into colors. Dispersion in water droplets is responsible for

rainbows. Transparent materials – materials which transmit light because objects can be seen through

them Translucent materials – transmit little light through them

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Opaque – materials which absorb or reflect light Polarization – process of affecting radiation especially light so that the vibrations of the waves

assume a (s) is(are) permitted to pass through a polarizer; process of filtering light such that only one, two or definite direction The reflected light is polarized when the reflected and transmitted rays are perpendicular to each other.

Color – property of light that reaches the eyes; an object appears a particular color because it reflects light of that color; black is the absence of reflected light; White objects reflect all the wavelengths while black objects absorb all the wavelengths.

Luminous intensity ( I ) – amount of light that a source give out measured in candela (cd).; brightness of light source

Luminous flux ( F ) – rate at which light energy is emitted from a source expressed in lumen (lm)

F = 4 I Illumination ( E ) – rate at which light energy falls on a unit area some distance from a light

source; brightness on surface; luminous flux per unit areaF

E =A

Illumination from an isotropic light source with a given intensityI

E = when point source is directly above the surfacer2 or the flux is perpendicular to the surface

I cos E = when point source is not directly above the surface

r2 or the flux makes an angle with the normal

where : r = distance of point source from surface

Units of E: lm/m2 = lux = m-candlelm/cm2 = cm-candlelm/ft2 = foot-candle

Luminous efficiency – total luminous flux (F ) radiated by the source divided by the power ( P )of the source expressed in lm/W.

FLuminous efficiency =

PExample 24: A 60 watt incandescent lamp has a luminous intensity of 66.5 cd. Determine the total luminous flux radiated by the lamp and the luminous efficiency of the lamp.Ans. 836 lm ; 13.9 lm/W

Example 25: Compute the illumination of a small surface at a distance of 120 cm from an isotropic point source of luminous intensity 72 cd (a) if the surface is normal to the luminous flux and (b) if the normal to the surface makes an angle of 30o with the light rays.Ans. 50 lux ; 43 lux

Photometry - deals with the measurement of the intensity of a source of light. Photometer – instrument for comparing the luminous intensities of light sources.

I1 I2 Where: I1 and I2 are the luminous intensities of the two light sources = r1 and r2 are their respective distances from the screen of the photometerr1

2 r22

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Example 26. In an experiment with a bar photometer , it was found that two lamps produced the same illumination when the screen was 40 cm from a standard lamp and 160 cm from a second lamp. The standard lamp was rated at 8 cd. Determine the luminous intensity of the second lamp. Ans. 128 cd

Reflection of Light Kinds of reflection of light:

Diffuse reflection – reflection on a rough surface; light is reflected in irregular directions

Specular or regular reflection – reflection on smooth-polished surface

Law of reflection: In specular ( or mirror) reflection: (1) the incident ray, reflected ray and normal to the surface lie in the same plane. (2) the angle of incidence equals the angle of reflection.

normal

Incident ray reflected ray1 2

Reflecting surface 1 = 2

Mirror – any highly polished surface that forms images by regular reflection of light. Plane Mirror – form images that are erect, of the same size as the object, and as far behind

the reflecting surface as the object is in front of it. ; the images are virtual. Real image – image formed by converging rays of light actually passing through the image

point; can be projected on a screen; appear in front of the mirror Virtual image – image formed by rays of light that appear to have diverged from the image

point but do not actually pass through that point; cannot be projected on the screen Spherical mirror – or curve mirror – is a small portion of the surface of a sphere, one side of

which is polished with a reflective material ; when viewed from the inside, the mirror is called concave mirror (converging mirror), and viewed from the outside, it is called convex mirror ( or diverging mirror)

Ray Diagram Rays parallel to the principal axis of a spherical mirror pass the focus after reflection Rays that pass the focus are reflected parallel to the principal axis Rays that proceeds along a radius of the mirror is reflected back along its original

path Parts of a spherical mirror

Center of curvature ( C ) – center of sphere Radius of curvature ( R ) – Vertex ( V ) – center of mirror Principal axis – line that passes through the vertex and the center Secondary axis – line passing through the center of curvature normal to the mirror. Focus ( F ) – point where rays parallel to the principal axis meet or seem to meet

after reflection. Focal length ( f ) – distance from the focus to the mirror: f = ½

For concave mirror or converging mirror: rays parallel to the principal axis actually converge at the focus after reflection

mirror

Incident ray

object

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● ● ●V principal avisC F V

image

Reflected rays

f

R =radius of curvature

Image formed by a concave mirrorCase 1. Object at an infinite distance

Image is a point at the focal point FCase 2. object at a finite distance beyond the center of curvature C

Real, inverted, smaller than object, located between the focalpf point and the center of curvature

Case 3 object at the center of curvature CImage is at C, inverted, real and same size as object

Case 4 Object between C and FImage is beyond C, inverted, real and bigger

pf Case 5 object is between F and mirrorImage is virtual, bigger , erect

P=f Case 6 object at FNo image

For convex mirror or diverging mirror: rays parallel to the principal axis diverge after reflection Reflected ray

Incident ray

Object image

principal axis ● ● F C

Mirror

Image formed by convex mirrorVirtual, erect, behind the mirror, smaller

Mirror equation for both concave and convex spherical mirrors:1 1 2 1 + = =p q R f

where: p = object distance from the mirrorq = image distance from the mirrorR = radius of curvature of the mirrorf = focal length of the mirror = R/2

p = + when object is in front of mirrorq = + when the image is real ( in front of mirror)q = - when the image is virtual ( behind the mirror)R and f are (+) for a concave mirror and (-) for a convex mirror

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Size of Image formed by a spherical mirror:height of image image distance q

Linear magnification = = = height of object object distance p

qIn general: M = -

PWhen M = ( - ) the image is inverted M = ( + ) the image is upright or erect M 1 the image is smaller than object M 1 the image is bigger than object M = 1 the image is of the same size as object

Example 27: Describe the image of an object positioned 20 cm from a concave spherical mirror of radius 60 cm. Ans. virtual, erect, 60 cm behind the mirror , magnified 3 times. Example 28; An object 7 cm high is placed 15 cm from a convex mirror of radius 45 cm. Describe its image. Ans. virtual, erect, 9 cm behind mirror, 4.2 cm highExample 29; How far should an object be from a concave mirror of radius 36 cm to form a real image 1/9 its size? Ans. 180 cmExample 30: What is the focal length of a convex mirror which produces an image 1/6 the size of an object located 12 cm from the mirror? Ans. -2.4 cm

Refraction of Light Bending of light rays as they pass obliquely from one medium to another of different optical

density Bending of light caused by a change in the velocity of light due to a change in the optical

density of the medium It is because of refraction that we are able to see transparent objects. This phenomenon is a direct result of the fact that light travels slower in a denser medium and

that it travels faster when it passes into a medium that is less dense. Optical density – property of a transparent substance which measure the speed of light

through the medium. The greater the optical density of a medium, the slower the speed of light through the medium.

When a ray of light passes obliquely from a less dense medium where its speed is greater to a denser medium where its speed is less, the ray of light is bent towards the normal as it enters the denser medium.

normal

incident ray less dense ( air ) 1

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2 denser (glass )

Refracted rayIf v1 v2 then 1 2

When a ray of light passes from a denser medium where its speed is less to a less dense medium where its speed is greater, the ray will bent away from the normal

normal

incident ray denser (glass ) 1

2 refracted ray

less dense (air )

If v1 v2 then 1 2

Absolute Index of Refraction ( n ) of a medium – ratio of speed of light in vacuum or air to the speed of light in medium

Speed of light in vacuum cn = =

Speed of light in medium v

Relative index of refraction ( nr ) – ratio of speed of light in the first medium of the speed in the second medium

v1

nr = v2

Snell’s Law of Refraction : For any two given transparent mediums ,the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant

sin 1 v1 n2

= = sin 2 v2 n1

For air: n = 1.0

Critical angle for Total Internal Reflection (c) : When light passes from an optically denser medium out into air , as the angle of incidence is increased the angle of refraction increases and approaches the limiting value of 90o, beyond which there could be no light refracted into the air, or there is no refracted wave but the wave is totally reflected at the boundary between the media. The limiting angle of incidence in the denser medium, which makes the angle of refraction 90o, is called the critical angle of incidence c.

Total internal reflection – principle behind fiber optics Optical fiber – far superior to copper wire in its capacity to carry information because of the

higher frequency of the infrared light used to carry the information .

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Denser mediumc

90o

less dense medium

n2 n1 = index of refraction of the denser mediumSin c = n2 = index of refraction of the less dense medium

n1

Index of refraction and wavelength o where: = wavelength of light in a material = o = wavelength of light in vacuum = 633 nm N n = index of refraction of material

Example 31 The angle of incidence of a ray of light in air is 45 o and the angle of refraction in a medium is 30o. What is the index of refraction of the second medium? Ans. 1.414Example 32: What is the speed of light in a substance having an index of refraction of 1.5?Ans. 2 x 108 m/sExample 33: A ray of light in water (n=4/3) is incident upon a plate of crown glass (n=1.517) at an angle of 45o. What is the angle of refraction for the ray in the glass? Ans. 38.42o

Example 34: What is the critical angle for light passing from glass (n=1.54) to water (n=1.33) Ans. 59.73o Example 35: The wavelength of red light from a helium-neon laser is 633 nm in air but 474 nm in the aqueous humor inside your eyeball. Calculate the index of refraction of the aqueous humor and the speed and frequency of the light in this substance. Ans. 1.34 ; 2.25 x 108 m/s ; 4.74 x 1014 Hz

Shallowing Effect of Refraction of lightThe ratio of the actual depth to the apparent depth is 4/3 , which is the index of refraction of water.

Actual depthn =

Apparent depthExample 36: A pool of water is 60 cm deep. Find its apparent depth when viewed vertically through air. Ans. 45 cm

Lens – transparent medium bounded by spherical surfaces; transparent material which refracts light rays in such as way as to form an image.

Convex or converging lenses or positive lenses – is one where the center portion is thicker than the edge; lens that brings parallel light into a single real focal point; with two foci one on each side of the lens.

Double convex (biconvex)

Incident ray

● ● principal axis

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F F

Real focusVirtual focus (principal focus)(secondary focus)

Case 1: Object from infinity: Image is a point at the focus F

Images formed by a converging lens

    Characteristics of the Image

Case 2 Distant object

RealInvertedSmaller than objectAt F

Case 3 Object at 2F

RealInvertedSame sizeAt 2F

Case 4 Object between 2F and F

RealInvertedLarger than objectBeyond 2F

Case 5 Object at F

No imageRefracted raysare parallel

Case 6 Object between F and lens

VirtualErectLarger than objectBehind the object on the same side of the lens

Concave or diverging lenses or negative lens – one in which the center portion is thinner than the edge ; deviates parallel light outwards as though it originated at a single virtual focal point.

Double concave (biconcave)

Incident ray

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● ●F F

Virtual focus(principal focus)

Image formed by a diverging lens

e) Object at F

Characteristics of the image regardless of object positionVirtualErectSmaller than objectBetween object and lens

Thin Lens – one whose thickness is small compared with the radii of curvature R1 and R2.

Lens Equation – equation that relates the positions of the image and the object of a thin lens to the focal length.

1 1 1 + = p q f

where: p = object distance from the lensq = image distance from the lensf = focal length of the lens

p = + for real object and negative for virtual objectq = + for real image and negative for virtual imagef is (+) for a converging lens and (-) for diverging lens

Size of Image formed by a spherical mirror:height of image image distance q

Linear magnification (M ) = = = Height of object object distance p

qIn general: M = -

PWhen M = ( - ) the image is inverted M = ( + ) the image is upright or erect M 1 the image is smaller than object M 1 the image is bigger than object M = 1 the image is of the same size as object

Example 36: Locate and describe the image formed by a converging lens of focal length 20 cm if the object is placed 30 cm from the lens.

Ans. real image , 60 cm away from the lens, inverted , bigger

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Example 37: An object 9 cm high is 27 cm in front of a concave lens of focal length 18 cm. Determine the position and height of its image. Ans. 10.8 cm in front of the lens ; 3.6 cm

The Lensmaker’s Equation – equation used to obtain the focal length of thins lens

1 1 1 = ( n – 1 ) +f R1 R2

where: n = index of refraction of the lens materialR1 and R2 = radii of curvature of the two lens syrfaces

R1 = ( + ) for convex surfacesR2 = ( - ) for concave surfacesf = (+ ) for converging lensf = ( - ) for diverging lens

Lenses in Contact: When two thin lens having focal lengths f1and f2 are in contact, the focal length of the combination is

1 1 1 = +f f1 f2

Example 38: A lens has a convex surface of radius 20 cm and a concave surface of radius 40 cm , and is made of glass of refractive index 1.54. Compute the focal length of the lens and state whether it is a converging or a diverging lens. Ans. f = +74.1 cm , convergingExample 39: A double convex lens has faces of radii 18 and 20 cm. When an object is 24 cm from the lens, a real image is formed 32 cm from the lens. Determine the focal length of the lens and the refractive index of the lens material. Ans. +13.7 cm , n=1.69

Lens Power – amount by which it can change the curvature of a wave ; reciprocal of the frequency; expressed in diopers ( m-1 ).

Aberrations – responsible for the formation of imperfect images by lenses and mirrors. Spherical aberration – failure of parallel rays to meet at a single point on a spherical surface

after reflection or refraction Chromatic aberration – arises from the fact that light rays of different wavelengths focus at

different points when refracted by a lens Mirage – images seen in the desert, or on a hot road in summer caused by refraction of light

in the air Rainbows – formed by the dispersion of light in water; refraction in the atmosphere; produced

by bending of light rays in the atmosphere when there are large differences in temperature between the ground and the air.

Dispersion – the dependence of the index of refraction on wavelength; separation of white light into its component colors

Spectrum – band of continuous spread of colors from the longest to the shortest wavelength. White light – mixture of the waves of different wavelengths in the visible region of the

spectrum. The brilliance of diamond is due to its large refractive index and its large dispersion Prism – refracts a light ray and deviates the light through an angle called angle of deviation;

when a beam of white light ( a combination of all visible wavelengths is incident on a prism, because of dispersion, the different colors refract through different angle of deviation ( red, orange, yellow, green, blue and violet) ; violet deviates the most, red light the least.

red

White light

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violet

MODERN PHYSICS

I RELATIVITY Two basic postulates of the special theory of relativity ( theory predicts the results of experiments involving speed near the speed of light); formulated by Albert Einstein

Postulate 1 ( Principle of Relativity ) - The laws of physics are the same in all inertial frames of reference, thus all motion is relative ; velocities of object can only be given relative to some other objects and its impossible to determine the absolute velocity of the object.

Postulate 2 ( Principle of Constancy of the Speed of Light - The speed of light is the same for all inertial observers , independent of their motion or of the motion of the source of light.

Relativistic Mass: The moving body increases its mass ; where any change in mass is equivalent to a change in kinetic energy

mo where: m = relativistic massm = mo = rest mass

v 2 v = velocity of object1 - c = speed of light = 3 x 108 m/s c

Examples 40: Compute the mass of an electron traveling at half the speed of light. The rest mass of an electron is 9.1 x 10-31 kg.Ans. 1.05 x 10-30 kg

Some of the consequences of the special theory of relativity are as follows Clocks in motion relative to an observer slow down. This is known as time dilation.

Moving clocks are observed to run more slowly than clocks at rest in the observer’s own frame of reference.

v 2 where: tm = time of moving clock tm = ts 1 - ts = time of stationary clock

c tm ts

Example 41: (Twin Paradox- space traveler ages more slowly than his twin who remains on earth ). Two twins are 25 years old when one of them sets out on a journey through space at nearly constant speed. The twin in the spaceship measures time with an accurate watch. When he returns to the earth, he claims to be 31 years old, while the twin left on earth is then 43 years old. What was the speed of the spaceship? Ans. 2.83 x 108 m/s

The length of an object in motion is contracted in the direction of motion. This effect is called length contraction ; the length of an object is longest for the observer at rest with respect to the object and shorter for observer in motion with respect to it ( in the direction of the relative motion of two observers )

v 2 where: Lo = length at rest L = Lo 1 - L = length in motion

C L Lo

Example 42: An observer on earth sees a spaceship at an altitude of 4350 km moving downward toward earth with a speed of 0.97 c. What is the distance from the spaceship to earth as measured by the spaceship’s captain? Ans. 1.06 x 103 km

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Events that are simultaneous for one observer are not simultaneous for another observer in motion relative to the first.

Relativistic Velocity of Two BodiesWhen a spacecraft (A) is moving in the x-direction with a velocity VAE relative to the earth and shoots out a rocket (B) in the x-direction at velocity VBA relative to the spacecraft , then the velocity of the rocket as measured by an observer on the earth is:

VAE + VBA

VBE = VAE VBA

1 + c2

Example 43: Suppose that Bob’s spacecraft is traveling at 0.60 c in the positive x-direction, as measured by a nearby observer , while Mike is traveling in his own vehicle directly toward Bob in the negative x-direction at -0.80 c relative the nearby observer. What is the velocity of Bob relative to Mike? Ans. 0.946 c

Relativistic Energy – sum of the kinetic energy and the rest energyE = K + moc2 or

mo c2

E = v 2 1 - c

Mass-Energy ConversionE = moc2 where : E = rest energy of the object

Example 44: How much rest energy is contained in 0.500 mm3 of water? Ans.4.5 x 1010 J

II QUANTUM PHYSICS ( modern version of mechanics or wave mechanics ; explainsthe behavior of atoms, molecules and nuclei)

Planck hypothesis – blackbody radiation was produced by submicroscopic charged oscillators called resonators

Blackbody – ideal system that absorbs all radiation incident on it.

The energy E of an oscillating charge with a natural frequency is not continuous but quantized. The quantum of electromagnetic radiation is called the photon. The energy of a photon with frequency f is

hc E = hf = where h=Planck’s constant= 6.626 x 10-34 J s

Pair production- process in which energy of a photon is converted into mass; the photon

disappears as an electron-positron pair Pair annihilation – process in which an electron-positron pair produces two photons; inverse

of pair production Positron – particle similar to the electron but opposite charge; antiparticle of the electron; first

antiparticle discovered Photoelectric effect – emission of electrons from a metal surface due to an incident radiation

of sufficient wavelength ; led to the establishment of dualistic theory of light Photoelectrons – emitted electrons from the radiated metal surface Einstein Photoelectric equation – energy of the ejected electron

Photon energy = KEmax + Wmin

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Or ½ mvmax2 = h f - Wmin

Example 45: Compute the energy of a photon of blue light of wavelength 450 nm.Ans. 4.42 x10-19 J = 2.76 eV ( 1 eV=1.6 x 10-19 J ) eV=electron volt

Compton scattering – x-rays of a single wavelength impinged on matter are scattered in various directions having longer wavelength than the incident radiation

Laser – device which utilizes the effect for electromagnetic radiation; an acronym for light amplification by stimulated emission of radiation

III NUCLEAR PHYSICS Nucleons – particles in the nucleus Atomic number – number of protons in a nucleus Mass number – protons plus neutrons Isotopes – nuclei with the same atomic number but different neutron number Mass defect : when protons and neutrons join to form helium nucleus, the mass is decreased

in the process. This difference is called mass defect. Binding energy – energy required to break a nucleus apart into its constituent particles;

energy required to remove all the electrons. Ionization energy – energy required to remove one electrons Disintegration energy – amount of rest energy that is converted to other forms of energy Radioactivity – nuclear process characterized by the spontaneous emission of certain

radiation and particles from the nucleus of an atom Radioactive decay – process in which radiation and small particles are emitted from a

nucleus Decay rate – number of decays per second expressed in Becquerel ( Bq) Half-life – time it takes for half of a given number of radioactive nuclei to decay Alpha decay – emissions of energetic positively charged particles called alpha particles

( nuclei of helium atoms ) the original (parent) nuclide is converted to a “daughter” by the emission of an alpha particle; if a nucleus emits alpha particles, it loses 2 protons and 2 neutrons, therefore the neutron of a single nucleus decreases by 2

Beta decay – emissions of negatively charged particles called beta particles ( electrons) ; the daughter nucleus has the same number of nucleons as the parent nucleus, but the atomic number is changed by 1

Beta particles are electrons or positrons (sometimes called beta-minus and beta-plus particles.

Beta-minus decay – an electron is emitted and a neutron in the nucleus is converted into a proton. Thus, the mass number does not change, but the charge of the nucleus increases by one

Beta-plus decay – a positron is emitted and a proton in the nucleus is converted into a neutron

Gamma decay – results from the transition of the nucleus from a higher energy state to a lower energy state.

Gamma rays – high-energy photons Radiocarbon dating or carbon dating – technique based on the radioactive decay of a rare

isotope of carbon ( carbon-14) Nuclear reactions – occur when a bombarding particle strikes another nucleus Nuclear fission – splitting of a heavy nucleus to form daughter nuclei of about equal mass Nuclear fusion – combination o flight nuclei to form a heavy nucleus; process going on in the

sun and other stars. Nuclear reactors : system designed to maintain self-sustained chain reaction; fuel is uranium

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