Think About This Why would the image you see of yourself in the lake be upright, while the image of the mountain is upside-down? physicspp.

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1 What Yu ll Learn Yu will learn hw light reflects ff different surfaces. Yu will learn abut the different types f mirrrs and their uses. Yu will use ray tracing and mathematical mdels t describe images frmed by mirrrs. Why It s Imprtant Hw light reflects ff a surface int yur eyes determines the reflectin that yu see. When yu lk dwn at the surface f a lake, yu see an upright reflectin f yurself. Muntain Scene When yu lk acrss a lake, yu might see a scene like the ne in this pht. The image f the trees and muntains in the lake appears t yu t be upside-dwn. Think Abut This Why wuld the image yu see f yurself in the lake be upright, while the image f the muntain is upside-dwn? physicspp.cm 456 Gerge Matchneer

2 Hw is an image shwn n a screen? Questin What types f mirrrs are able t reflect an image nt a screen? Prcedure 1. Obtain an index card, a plane mirrr, a cncave mirrr, a cnvex mirrr, and a flashlight frm yur teacher. 2. Turn ff the rm lights and stand near the windw. 3. Hld the index carn ne hand. Hld the flat, plane mirrr in the ther hand. 4. Reflect the light cming thrugh the windw nt the index card. CAUTION: D nt lk directly at the Sun r at the reflectin f the Sun in a mirrr. Slwly mve the index card clser t and then farther away frm the mirrr and try t make a clear image f bjects utside the windw. 5. If yu can prject a clear image, this is called a real image. If yu nly see a fuzzy light n the index card then n real image is frmed. Recrd yur bservatins. 6. Repeat steps 3 5 with the cncave and cnvex mirrr. 7. Perfrm step 4 fr each mirrr with a flashlight and bserve the reflectin n the index card. Analysis Which mirrr(s) prduced a real image? What are sme things yu ntice abut the image(s) yu see? Critical Thinking Based upn yur bservatin f the flashlight images, prpse an explanatin f hw a real image is frmed Reflectin frm Plane Mirrrs Undubtedly, as lng as there have been humans, they have seen their faces reflecten the quiet water f lakes and pnds. When yu lk at the surface f a bdy f water, hwever, yu dn t always see a clear reflectin. Smetimes, the wind causes ripples in the water, and passing bats create waves. Disturbances n the surface f the water prevent the light frm reflecting in a manner such that a clear reflectin is visible. Almst 4000 years ag, Egyptians understd that reflectin requires smth surfaces. They used plished metal mirrrs t view their images. Sharp, well-defined, reflectemages were nt pssible until 1857, hwever, when Jean ucault, a rench scientist, develped a methd f cating glass with silver. Mdern mirrrs are prduced using ever-increasing precisin. They are made with greater reflecting ability by the evapratin f aluminum r silver nt highly plished glass. The quality f reflecting surfaces is even mre imprtant in applicatins such as lasers and telescpes. Mre than ever befre, clear reflectins in mdern, ptical instruments require smth surfaces. Objectives Explain the law f reflectin. Distinguish between specular and diffuse reflectin. Lcate the images frmed by plane mirrrs. Vcabulary specular reflectin diffuse reflectin plane mirrr bject image virtual image Sectin 17.1 Reflectin frm Plane Mirrrs 457 Hrizns Cmpanies

3 The Law f Reflectin Reflected light Nrmal r Incident light i Surface Plane f travel r i igure 17-1 The incident ray and the reflected ray are in the same plane f travel. What happens t the light that is striking this bk? When yu hld the bk up t the light, yu will see that n light passes thrugh it. Recall frm Chapter 16 that an bject like this is called paque. Part f the light is absrbed, and part is reflected. The absrbed light spreads ut as thermal energy. The behavir f the reflected light depends n the type f surface and the angle at which the light strikes the surface. Recall frm Chapter 14 that when a wave traveling in tw dimensins encunters a barrier, the angle f incidence is equal t the angle f reflectin f the wave. This same tw-dimensinal reflectin relatinship applies t light waves. Cnsider what happens when yu bunce-pass a basketball. The ball bunces in a straight line, as viewed frm abve, t the ther player. Light reflects in the same way as a basketball. igure 17-1 shws a ray f light striking a reflecting surface. The nrmal is an imaginary line that is perpendicular t a surface at the lcatin where light strikes the surface. The reflected ray, the incident ray, and the nrmal t the surface always will be in the same plane. Althugh the light is traveling in three dimensins, the reflectin f the light is planar (tw-dimensinal). The planar and angle relatinships are knwn tgether as the law f reflectin. Law f Reflectin r i The angle that a reflected ray makes as measured frm the nrmal t a reflective surface equals the angle that the incident ray makes as measured frm the same nrmal. Light rays and wave frnts are red. Mirrrs are light blue. This law can be explainen terms f the wave mdel f light. igure 17-2 shws a wave frnt f light appraching a reflective surface. As each pint alng the wave frnt reaches the surface, it reflects ff at the same angle as the preceding pint. Because all pints are traveling at the same speed, they all travel the same ttal distance in the same time. Thus, the wave frnt as a whle leaves the surface at an angle equal t its incident angle. Nte that the wavelength f the light des nt affect this prcess. Red, green, and blue light all fllw this law. igure 17-2 A wave frnt f light appraches a reflective surface. Pint P n the wave frnt strikes the surface first (a). Pint Q strikes the surface after pint P reflects at an angle equal t the incident angle (b). The prcess cntinues with all pints reflecting ff at angles equal t their incident angles, resulting in a reflected wave frnt (c). a Nrmal b c Nrmal Nrmal Wave frnt Wave frnt Wave frnt P Q i P Reflective surface 458 Chapter 17 Reflectin and Mirrrs P i r Q i i r Q Reflective surface Reflective surface i

4 a b igure 17-3 When a beam f light strikes a mirrred surface (a), the parallel rays in the beam reflect in parallel and maintain the light as a beam (b). When the light beam strikes a rugh surface (c), the parallel rays that make up the beam are reflected frm different micrscpic surfaces, thereby diffusing the beam (d). c d Smth and rugh surfaces Cnsider the beam f light shwn in igure 17-3a. All f the rays in this beam f light reflect ff the surface parallel t ne anther, as shwn in igure 17-3b. This ccurs nly if the reflecting surface is nt rugh n the scale f the wavelength f light. Such a surface is cnsidered t be smth relative t the light. A smth surface, such as a mirrr, causes specular reflectin, in which parallel light rays are reflecten parallel. What happens when light strikes a surface that appears t be smth, but actually is rugh n the scale f the wavelength f light, such as the page f this textbk r a white wall? Is light reflected? Hw culd yu demnstrate this? igure 17-3c shws a beam f light reflecting ff a sheet f paper, which has a rugh surface. All f the light rays that make up the beam are parallel befre striking the surface, but the reflected rays are nt parallel, as shwn in igure 17-3d. This is diffuse reflectin, the scattering f light ff a rugh surface. The law f reflectin applies t bth smth and rugh surfaces. r a rugh surface, the angle that each incident ray makes with the nrmal equals the angle that its reflected ray makes with the nrmal. Hwever, n a micrscpic scale, the nrmals t the surface lcatins where the rays strike are nt parallel. Thus, the reflected rays cannt be parallel. The rugh surface prevents them frm being parallel. In this case, a reflected beam cannt be seen because the reflected rays are scatteren different directins. With specular reflectin, as with a mirrr, yu can see yur face. But n matter hw much light is reflected ff a wall r a sheet f paper, yu will never be able t use them as mirrrs. Sectin 17.1 Reflectin frm Plane Mirrrs 459 Henry Leap/James Lehman

5 Changing the Angle f Incidence A light ray strikes a plane mirrr at an angle f 52.0 t the nrmal. The mirrr then rtates 35.0 arund the pint where the beam strikes the mirrr s that the angle f incidence f the light ray decreases. The axis f rtatin is perpendicular t the plane f the incident and the reflected rays. What is the angle f rtatin f the reflected ray? Analyze and Sketch the Prblem Sketch the situatin befre the rtatin f the mirrr. Draw anther sketch with the angle f rtatin applied t the mirrr. Knwn: Unknwn: i, initial 52.0 r? mirrr 35.0 Slve fr the Unknwn r the angle f incidence t reduce, rtate clckwise. θ i, final θ i, initial θ mirrr Substitute θ i, initial 52.0, θ mirrr clckwise frm the new nrmal Apply the law f reflectin. θ r, final θ i, final 17.0 cunterclckwise Substitute θ i, final 17.0 frm the new nrmal Using the tw sketches, determine the angle thrugh which the reflected ray has rtated. θ r clckwise frm the riginal angle Initial nrmal i, final r, final mirrr i, initial 52.0 r, initial 52.0 Math Handbk Operatins with Significant Digits pages Evaluate the Answer Is the magnitude realistic? Cmparing the final sketch with the initial sketch shws that the angle the light ray makes with the nrmal decreases as the mirrr rtates clckwise tward the light ray. It makes sense, then, that the reflected ray als rtates clckwise. mirrr Explain why the reflectin f light ff grund glass changes frm diffuse t specular if yu spill water n it. 2. If the angle f incidence f a ray f light is 42, what is each f the fllwing? a. the angle f reflectin b. the angle the incident ray makes with the mirrr c. the angle between the incident ray and the reflected ray 3. If a light ray reflects ff a plane mirrr at an angle f 35 t the nrmal, what was the angle f incidence f the ray? 4. Light frm a laser strikes a plane mirrr at an angle f 38 t the nrmal. If the laser is mved s that the angle f incidence increases by 13, what is the new angle f reflectin? 5. Tw plane mirrrs are psitined at right angles t ne anther. A ray f light strikes ne mirrr at an angle f 30 t the nrmal. It then reflects tward the secnd mirrr. What is the angle f reflectin f the light ray ff the secnd mirrr? 460 Chapter 17 Reflectin and Mirrrs

6 Objects and Plane-Mirrr Images If yu lked at yurself in a mirrr this mrning yu saw yur reflectin in a plane mirrr. A plane mirrr is a flat, smth surface frm which light is reflected by specular reflectin. T understand reflectin frm a mirrr, yu must cnsider the bject f the reflectin and the type f image that is frmed. In Chapter 16, the wrd bject was used t refer t surces f light. In describing mirrrs, the wrd bject is usen the same way, but with a mre specific applicatin. An bject is a surce f light rays that are t be reflected by a mirrred surface. An bject can be a luminus surce, such as a lightbulb, r an illuminated surce, such as a girl, as shwn in igure Mst bjects that yu will wrk with in this chapter are a surce f light that diverges, r spreads ut frm the surce. Cnsider a single pint n the birn igure Light reflects diffusely frm the crest f the bird, the bject pint. What happens t the light? Sme f the light travels frm the bird t the mirrr and reflects. What des the by see? Sme f the reflected light frm the bird hits his eyes. Because his brain prcesses this infrmatin as if the light has traveled a straight path, it seems t the by as if the light had fllwed the dashed lines. The light seems t have cme frm a pint behind the mirrr, the image pint. The by in igure 17-5 sees rays f light that cme frm many pints n the bird. The cmbinatin f the image pints prduced by reflected light rays frms the image f the bird. It is a virtual image, which is a type f image frmed by diverging light rays. A virtual image is always n the ppsite side f the mirrr frm the bject. Images f real bjects prduced by plane mirrrs are always virtual images. igure 17-4 The lightbulb is a luminus surce that prduces diverging light by shining in all directins. The girl is an illuminated surce that prduces diverging light by the diffused reflectin frm her bdy f light that cmes frm the lightbulb. Prperties f Plane-Mirrr Images Lking at yurself in a mirrr, yu can see that yur image appears t be the same distance behind the mirrr as yu are in frnt f the mirrr. Hw culd yu test this? Place a ruler between yu and the mirrr. Where des the image tuch the ruler? Yu als see that yur image is riented as yu are, ant matches yur size. This is where the expressin mirrr image riginates. If yu mve tward the mirrr, yur image mves tward the mirrr. If yu mve away, yur image als mves away. Object pint Diverging rays Image pint Diverging rays Mirrr igure 17-5 The reflected rays that enter the eye appear t riginate at a pint behind the mirrr. Sectin 17.1 Reflectin frm Plane Mirrrs 461 Laura Sifferlin

7 O d Ray 1 P 1 I h Ray 2 h i i P 2 r igure 17-6 Light rays (tw are shwn) leave a pint n the bject. Sme strike the mirrr and are reflectent the eye. Sight lines, drawn as dashed lines, extend frm the lcatin n the mirrr where the rays are reflected back t where they cnverge. The image is lcated where the sight lines cnverge: d. Virtual Image Psitin Suppse yu are lking at yur image in a plane mirrr. Can yu measure the lcatin f the image? 1. Obtain a camera frm yur teacher with a fcusing ring that has distances marked n it. 2. Stand 1.0 m frm a mirrr and fcus n the edge f the mirrr. Check the reading n the fcusing ring. It shuld be 1.0 m. 3. Measure the psitin f the image by fcusing the camera n yur image. Check the reading n the fcusing ring. Analyze and Cnclude 4. Hw far is the image behind the mirrr? 5. Why is the camera able t fcus n a virtual image that is behind the mirrr even thugh there is n real bject behind the mirrr? Image psitin and height The gemetric mdel f igure 17-6 demnstrates why the distances are the same. Tw rays frm pint O at the tip f the candle strike the mirrr at pint P 1 and pint P 2, respectively. Bth rays are reflected accrding t the law f reflectin. The reflected rays are extended behind the mirrr as sight lines, cnverging at pint I, which is the image f pint O. Ray 1, which strikes the mirrr at an angle f incidence f 0, is reflected back n itself, s the sight line is at 90 t the mirrr, just as ray 1. Ray 2 is als reflected at an angle equal t the angle f incidence, s the sight line is at the same angle t the mirrr as ray 2. This gemetric mdel reveals that line segments O P 1 and IP 1 are crrespnding sides f tw cngruent triangles, OP 1 P 2 and IP 1 P 2. The psitin f the bject with respect t the mirrr, r the bject psitin, d, has a length equal t the length f O P 1. The apparent psitin f the image with respect t the mirrr, r the image psitin,, has a length equal t the length f IP 1. Using the cnventin that image psitin is negative t indicate that the image is virtual, the fllwing is true. Plane-Mirrr Image Psitin d With a plane mirrr, the image psitin is equal t the negative f the bject psitin. The negative sign indicates that the image is virtual. Yu can draw rays frm the bject t the mirrr t determine the size f the image. The sight lines f tw rays riginating frm the bttm f the candle in igure 17-6 will cnverge at the bttm f the image. Using the law f reflectin and cngruent-triangle gemetry, the fllwing is true f the bject height, h, anmage height, h i, and any ther dimensin f the bject anmage. Plane-Mirrr Image Height h i h With a plane mirrr, image height is equal t bject height. 462 Chapter 17 Reflectin and Mirrrs

8 igure 17-7 The image frmed in a plane mirrr is the same size as the bject ans the same distance behind the mirrr as the bject is in frnt. Hwever, when the by blinks his right eye, the left eye f his image blinks. Image rientatin A plane mirrr prduces an image with the same rientatin as the bject. If yu are standing n yur feet, a plane mirrr prduces an image f yu standing n yur feet. If yu are ding a headstand, the mirrr shws yu ding a headstand. Hwever, there is a difference between yu and the appearance f yur image in a mirrr. llw the sight lines in igure The ray that diverges frm the right hand f the by cnverges at what appears t be the left hand f his image. Left and right appear t be reversed by a plane mirrr. Why, then, are tp and bttm nt als reversed? This des nt happen because a plane mirrr des nt really reverse left and right. The mirrr in igure 17-7 nly reverses the by s image such that it is facing in the ppsite directin as the by, r, in ther wrds, it prduces a frnt-t-back reversal. Cnsider the image f the muntain in the pht at the beginning f the chapter. In this case, the image f the muntain can be described as upside dwn, but the image is actually a frnt-t-back reversal f yur view f the actual muntain. Because the mirrr (the lake surface) is hrizntal, rather than vertical, yur perspective, r angle f view, makes the image lk upside dwn. Turn yur bk 90 cunterclckwise and lk at igure 17-7 again. The actual by is nw facing dwn, and his image is facing up, upside dwn relative t the actual by, just like the image f the muntain. The nly thing that has changes yur perspective Sectin Review 6. Reflectin A light ray strikes a flat, smth, reflecting surface at an angle f 80 t the nrmal. What is the angle that the reflected ray makes with the surface f the mirrr? 7. Law f Reflectin Explain hw the law f reflectin applies t diffuse reflectin. 8. Reflecting Surfaces Categrize each f the fllwing as a specular r a diffuse reflecting surface: paper, plished metal, windw glass, rugh metal, plastic milk jug, smth water surface, and grund glass. 9. Image Prperties A 50-cm-tall dg stands 3 m frm a plane mirrr and lks at its image. What is the image psitin, height, and type? 10. Image Diagram A car is fllwing anther car dwn a straight rad. The first car has a rear windw tilted 45. Draw a ray diagram shwing the psitin f the Sun that wuld cause sunlight t reflect int the eyes f the driver f the secnd car. 11. Critical Thinking Explain hw diffuse reflectin f light ff an bject enables yu t see an bject frm any angle. physicspp.cm/self_check_quiz Sectin 17.1 Reflectin frm Plane Mirrrs 463

9 17.2 Curved Mirrrs Objectives Explain hw cncave and cnvex mirrrs frm images. Describe prperties and uses f spherical mirrrs. Determine the lcatins and sizes f spherical mirrr images. Vcabulary cncave mirrr principal axis fcal pint fcal length real image spherical aberratin magnificatin cnvex mirrr If yu lk at the surface f a shiny spn, yu will ntice that yur reflectin is different frm what yu see in a plane mirrr. The spn acts as a curved mirrr, with ne side curvenward and the ther curved utward. The prperties f curved mirrrs and the images that they frm depend n the shape f the mirrr and the bject s psitin. Cncave Mirrrs The inside surface f a shiny spn, the side that hlds fd, acts as a cncave mirrr. A cncave mirrr has a reflective surface, the edges f which curve tward the bserver. Prperties f a cncave mirrr depend n hw much it is curved. igure 17-8 shws hw a spherical cncave mirrr wrks. In a spherical cncave mirrr, the mirrr is shaped as if it were a sectin f a hllw sphere with an inner reflective surface. The mirrr has the same gemetric center, C, and radius f curvature, r, as a sphere f radius, r. The line that includes line segment CM is the principal axis, which is the straight line perpendicular t the surface f the mirrr that divides the mirrr in half. Pint M is the center f the mirrr where the principal axis intersects the mirrr. P i igure 17-8 The fcal pint f a spherical cncave mirrr is lcated halfway between the center f curvature and the mirrr surface. Rays entering parallel t the principal axis are reflected t cnverge at the fcal pint,. Principal axis C r f r Mirrr surface M When yu pint the principal axis f a cncave mirrr tward the Sun, all the rays are reflected thrugh a single pint. Yu can lcate this pint by mving a sheet f paper tward and away frm the mirrr until the smallest and sharpest spt f sunlight is fcused n the paper. This spt is called the fcal pint f the mirrr, the pint where incident light rays that are parallel t the principal axis cnverge after reflecting frm the mirrr. The Sun is a surce f parallel light rays because it is very far away. All f the light that cmes directly frm the Sun must fllw almst parallel paths t Earth, just as all f the arrws sht by an archer must fllw almst parallel paths t hit within the circle f a bull s-eye. When a ray strikes a mirrr, it is reflected accrding t the law f reflectin. igure 17-8 shws that a ray parallel t the principal axis is reflected and crsses the principal axis at pint, the fcal pint. is at the halfway pint between M and C. The fcal length, f, is the psitin f the fcal pint with respect t the mirrr alng the principal axis and can be expressed as f r/2. The fcal length is psitive fr a cncave mirrr. 464 Chapter 17 Reflectin and Mirrrs

10 a b c O Object O Object O Object Image 2 1 I C Mirrr 2 1 I C Mirrr Image 2 1 I C Mirrr Graphical Methd f inding the Image Yu have already drawn rays t fllw the path f light that reflects ff plane mirrrs. This meths even mre useful when applied t curved mirrrs. Nt nly can the lcatin f the image vary, but s can the rientatin and size f the image. Yu can use a ray diagram t determine prperties f an image frmed by a curved mirrr. igure 17-9 shws the frmatin f a real image, an image that is frmed by the cnverging f light rays. The image is inverted and larger than the bject. The rays actually cnverge at the pint where the image is lcated. The pint f intersectin, I, f the tw reflected rays determines the psitin f the image. Yu can see the image flating in space if yu place yur eye s that the rays that frm the image fall n yur eye, as in igure 17-9a. As igure 17-9b shws, hwever, yur eye must be riented s as t see the rays cming frm the image lcatin. Yu cannt lk at the image frm behind. If yu were t place a mvie screen at this pint, the image wuld appear n the screen, as shwn in igure 17-9c. Yu cannt d this with virtual images. T mre easily understand hw ray tracing wrks with curved mirrrs, yu can use simple, ne-dimensinal bjects, such as the arrw shwn in igure 17-10a. A spherical cncave mirrr prduces an inverted real image if the bject psitin, d, is greater than twice the fcal length, f. The bject is then beynd the center f curvature, C. If the bject is placed between the center f curvature and the fcal pint,, as shwn in igure 17-10b, the image is again real annverted. Hwever, the size f the image is nw greater than the size f the bject. a d b igure 17-9 The real image, as seen by the unaided eye (a). The unaided eye cannt see the real image if it is nt in a lcatin t catch the rays that frm the image (b). The real image as seen n a white paque screen (c). igure When the bject is farther frm the mirrr than C, the image is a real image that is inverted and smaller cmpared t the bject (a). When the bject is lcated between C and, the real image is inverted, larger than the bject, and lcated beynd C (b). O 1 Ray 1 I 1 Object Ray 2 C Image Image C Object Ray 2 Ray 1 I 1 O 1 d Principal plane Principal plane Sectin 17.2 Curved Mirrrs 465

11 igure A Gregrian telescpe prduces a real image that is upright. Secndary cncave mirrr Telescpe tube Primary cncave mirrr Astrnmy Cnnectin Hw can the inverted real image created by a cncave mirrr be turned right-side up? In 1663, Scttish astrnmer James Gregry develped the Gregrian telescpe, shwn in igure 17-11, t reslve this prblem. It is cmpsed f a large cncave mirrr and a small cncave mirrr arranged such that the smaller mirrr is utside f the fcal pint f the larger mirrr. Parallel rays f light frm distant bjects strike the larger mirrr and reflect tward the smaller mirrr. The rays then reflect ff the smaller mirrr and frm a real image that is riented exactly as the bject is. Using Ray Tracing t Lcate Images rmed by Spherical Mirrrs Use the fllwing strategies fr spherical-mirrr prblems. Refer t igure Using lined r graph paper, draw the principal axis f the mirrr as a hrizntal line frm the left side t the right side f yur paper, leaving six blank lines abve and six blank lines belw. 2. Place a pint and a label n the principal axis the bject, C, and, as fllws. a. If the mirrr is a cncave mirrr and the bject is beynd C, away frm the mirrr, place the mirrr at the right side f the page, place the bject at the left side f the page, and place C and t scale. b. If the mirrr is a cncave mirrr and the bject is between C and, place the mirrr at the right side f the page, place C at the center f the paper, halfway between the mirrr and C, and the bject t scale. c. r any ther situatin, place the mirrr in the center f the page. Place the bject r (whichever is the greatest distance frm the mirrr) at the left side f the page, and place the ther t scale. 3. T represent the mirrr, draw a vertical line at the mirrr pint that extends the full 12 lines f space. This is the principal plane. 4. Draw the bject as an arrw and label its tp O 1. r cncave mirrrs, bjects inside f C shuld nt be higher than three lines high. r all ther situatins, the bjects shuld be six lines high. The scale fr the height f the bject will be different frm the scale alng the principal axis. 5. Draw ray 1, the parallel ray. It is parallel t the principal axis and reflects ff the principal plane and passes thrugh. 6. Draw ray 2, the fcus ray. It passes thrugh, reflects ff the principal plane, and is reflected parallel t the principal axis. 7. The image is lcated where rays 1 and 2 (r their sight lines) crss after reflectin. Label the pint I 1. The image is an arrw perpendicular frm the principal axis t I Chapter 17 Reflectin and Mirrrs Lick Observatry

12 a C Mirrr diameter b igure A cncave spherical mirrr reflects sme rays, such that they cnverge at pints ther than the fcus (a). A parablic mirrr fcuses all parallel rays at a pint (b). Real image defects in cncave mirrrs In tracing rays, yu have reflected the rays frm the principal plane, which is a vertical line representing the mirrr. In reality, rays are reflected ff the mirrr itself, as shwn in igure 17-12a. Ntice that nly parallel rays that are clse t the principal axis, r paraxial rays, are reflected thrugh the fcal pint. Other rays cnverge at pints clser t the mirrr. The image frmed by parallel rays reflecting ff a spherical mirrr with a large mirrr diameter and a small radius f curvature is a disk, nt a pint. This effect, called spherical aberratin, makes an image lk fuzzy, nt sharp. A mirrr grund t the shape f a parabla, as in igure 17-12b, suffers n spherical aberratin. Because f the cst f manufacturing large, perfectly parablic mirrrs, many f the newest telescpes use spherical mirrrs and smaller, specially-designed secndary mirrrs r lenses t crrect fr spherical aberratin. Als, spherical aberratin is reduced as the rati f the mirrr s diameter, shwn in igure 17-12a, t its radius f curvature is reduced. Thus, lwer-cst spherical mirrrs can be usen lwer-precisin applicatins. Mathematical Methd f Lcating the Image The spherical mirrr mdel can be used t develp a simple equatin fr spherical mirrrs. Yu must use the paraxial ray apprximatin, which states that nly rays that are clse t and almst parallel with the principal axis are used t frm an image. Using this, in cmbinatin with the law f reflectin, leads t the mirrr equatin, relating the fcal length, f, bject psitin, d, anmage psitin,, f a spherical mirrr. Hubble Truble In 1990, NASA launched the Hubble Space Telescpe int rbit arund Earth. Hubble was expected t prvide clear images withut atmspheric distrtins. Hwever, sn after it was deplyed, Hubble was fund t have a spherical aberratin. In 1993, crrective ptics, called COSTAR, were installed n Hubble t enable it t prduce clear images. Mirrr Equatin 1 f 1 1 d d i The reciprcal f the fcal length f a spherical mirrr is equal t the sum f the reciprcals f the image psitin and the bject psitin. When using this equatin t slve prblems, it is imprtant t remember that it is nly apprximately crrect. It des nt predict spherical aberratin, because it uses the paraxial ray apprximatin. In reality, light cming frm an bject tward a mirrr is diverging, s nt all f the light is clse t r parallel t the axis. When the mirrr diameter is small relative t the radius f curvature t minimize spherical aberratin, this equatin predicts image prperties mre precisely. Sectin 17.2 Curved Mirrrs 467

13 Adding and Subtracting ractins When using the mirrr equatin, yu first use math t mve the fractin that cntains the quantity yu are seeking t the left-hand side f the equatin and everything else t the right. Then yu cmbine the tw fractins n the right-hand side using a cmmn denminatr that results frm multiplying the denminatrs. Math 1 x 1 y 1 z Physics 1 f 1 1 d d i 1 y 1 x 1 z 1 1 d f 1 d 1 y ( 1 x)( z z) ( 1 z)( x x) 1 d ( 1 f )( d d ) ( 1 d )( f f ) 1 y z x 1 d f xz d fd i i i xz y z x fd d f Using this apprach, the fllwing relatinships can be derived fr image psitin, bject psitin, and fcal length: fd d f fd d i di f di f d d Magnificatin Anther prperty f a spherical mirrr is magnificatin, m, which is hw much larger r smaller the image is relative t the bject. In practice, this is a simple rati f the image height t the bject height. Using similar-triangle gemetry, this rati can be written in terms f image psitn and bject psitin. Magnificatin hi m d i h d The magnificatin f an bject by a spherical mirrr, defined as the image height divided by the bject height, is equal t the negative f the image psitin, divided by the bject psitin. Image psitin is psitive fr a real image when using the preceding equatins. Thus, the magnificatin is negative, which means that the image is inverted cmpared t the bject. If the bject is beynd pint C, the abslute value f the magnificatin fr the real image is less than 1. This means that the image is smaller than the bject. If the bject is placed between pint C and pint, the abslute value f the magnificatin fr the real image is greater than 1. Thus, the image is larger than the bject. 468 Chapter 17 Reflectin and Mirrrs

14 Real Image rmatin by a Cncave Mirrr A cncave mirrr has a radius f 20.0 cm. A 2.0-cm-tall bject is 30.0 cm frm the mirrr. What is the image psitin anmage height? Analyze and Sketch the Prblem Draw a diagram with the bject and the mirrr. Draw tw principal rays t lcate the image in the diagram. Knwn: Unknwn: h 2.0 cm? d 30.0 cm h i? r 20.0 cm Slve fr the Unknwn cal length is half the radius f curvature. f 2 r 20.0 cm Substitute r 20.0 cm cm Use the mirrr equatin and slve fr image psitin. 1 f 1 1 di d fd d f (10.0 cm)(30.0 cm) Substitute f 10.0 cm, d 30.0 cm 30.0 cm 10.0 cm 15.0 cm (real image, in frnt f the mirrr) Use the magnificatin equatin and slve fr image height. hi m d i h d di h h i d (15.0 cm)(2.0 cm) Substitute 15.0 cm, h 2.0 cm, d 30.0 cm 30.0 cm 1.0 cm (inverted, smaller image) Evaluate the Answer Are the units crrect? All psitins are in centimeters. D the signs make sense? Psitive psitin and negative height agree with the drawing. h O 1 C h i d I 1 Math Handbk ractins page Use a ray diagram, drawn t scale, t slve Example Prblem An bject is 36.0 cm in frnt f a cncave mirrr with a 16.0-cm fcal length. Determine the image psitin. 14. A 3.0-cm-tall bject is 20.0 cm frm a 16.0-cm-radius cncave mirrr. Determine the image psitin anmage height. 15. A cncave mirrr has a 7.0-cm fcal length. A 2.4-cm-tall bject is 16.0 cm frm the mirrr. Determine the image height. 16. An bject is near a cncave mirrr f 10.0-cm fcal length. The image is 3.0 cm tall, inverted, and 16.0 cm frm the mirrr. What are the bject psitin and bject height? Sectin 17.2 Curved Mirrrs 469

15 a Ray 2 I 1 b O 1 Ray 1 Object Image d igure When an bject is lcated between the fcal pint and a spherical cncave mirrr, a virtual image that is upright and larger cmpared t the bject is frmed behind the mirrr (a), as shwn with the stack f blcks (b). What else d yu see in this picture? Virtual Images with Cncave Mirrrs Yu have seen that as an bject appraches the fcal pint,, f a cncave mirrr, the image mves farther away frm the mirrr. If the bject is at the fcal pint, all reflected rays are parallel. They never meet, therefre, and the image is said t be at infinity, s the bject culd never be seen. What happens if the bject is mved even clser t the mirrr? What d yu see when yu mve yur face clse t a cncave mirrr? The image f yur face is right-side up and behind the mirrr. A cncave mirrr prduces a virtual image if the bject is lcated between the mirrr and the fcal pint, as shwn in the ray diagram in igure 17-13a. Again, tw rays are drawn t lcate the image f a pint n an bject. As befre, ray 1 is drawn parallel t the principal axis and reflected thrugh the fcal pint. Ray 2 is drawn as a line frm the pint n the bject t the mirrr, alng a line defined by the fcal pint and the pint n the bject. At the mirrr, ray 2 is reflected parallel t the principal axis. Nte that ray 1 and ray 2 diverge as they leave the mirrr, s there cannt be a real image. Hwever, sight lines extended behind the mirrr cnverge, shwing that the virtual image frms behind the mirrr. When yu use the mirrr equatin t slve prblems invlving cncave mirrrs fr which an bject is between the mirrr and the fcal pint, yu will find that the image psitin is negative. The magnificatin equatin gives a psitive magnificatin greater than 1, which means that the image is upright and larger cmpared t the bject, like the image in igure 17-13b. An bject f height h is lcated at d relative t a cncave mirrr with fcal length f. 1. Draw and label a ray diagram shwing the fcal length and lcatin f the bject if the image is lcated twice as far frm the mirrr as the bject. Prve yur answer mathematically. Calculate the fcal length as a functin f bject psitin fr this placement. 2. Draw and label a ray diagram shwing the lcatin f the bject if the image is lcated twice as far frm the mirrr as the fcal pint. Prve yur answer mathematically. Calculate the image height as a functin f the bject height fr this placement. 3. Where shuld the bject be lcated s that n image is frmed? 470 Chapter 17 Reflectin and Mirrrs Hrizns Cmpanies

16 O 1 Ray 1 igure A cnvex mirrr always prduces virtual images that are upright and smaller cmpared t the bject. I 1 Object Ray 2 d Image Cnvex Mirrrs In the first part f this chapter, yu learned that the inner surface f a shiny spn acts as a cncave mirrr. If yu turn the spn arund, the uter surface acts as a cnvex mirrr, a reflective surface with edges that curve away frm the bserver. What d yu see when yu lk at the back f a spn? Yu see an upright, but smaller image f yurself. Prperties f a spherical cnvex mirrr are shwn in igure Rays reflected frm a cnvex mirrr always diverge. Thus, cnvex mirrrs frm virtual images. Pints and C are behind the mirrr. In the mirrr equatin, f and are negative numbers because they are bth behind the mirrr. The ray diagram in igure represents hw an image is frmed by a spherical cnvex mirrr. The figure uses tw rays, but remember that there are an infinite number f rays. Ray 1 appraches the mirrr parallel t the principal axis. The reflected ray is drawn alng a sight line frm thrugh the pint where ray 1 strikes the mirrr. Ray 2 appraches the mirrr n a path that, if extended behind the mirrr, wuld pass thrugh. The reflected part f ray 2 ants sight line are parallel t the principal axis. The tw reflected rays diverge, and the sight lines intersect behind the mirrr at the lcatin f the image. An image prduced by a single cnvex mirrr is a virtual image that is upright and smaller cmpared t the bject. The magnificatin equatin is useful fr determining the apparent dimensins f an bject as seen in a spherical cnvex mirrr. If yu knw the diameter f an bject, yu can multiply by the magnificatin fractin t see hw the diameter changes. Yu will find that the diameter is smaller, as are all ther dimensins. This is why the bjects appear t be farther away than they actually are fr cnvex mirrrs. ield f view It may seem that cnvex mirrrs wuld have little use because the images that they frm are smaller than the bjects. Hwever, this prperty f cnvex mirrrs des have practical uses. By frming smaller images, cnvex mirrrs enlarge the area, r field f view, that an bserver sees, as shwn in igure Als, the center f this field f view is visible frm any angle f an bserver ff the principal axis f the mirrr; thus, the field f view is visible frm a wide perspective. r this reasn, cnvex mirrrs ften are usen cars as passenger-side rearview mirrrs. igure Cnvex mirrrs prduce images that are smaller than the bjects. This increases the field f view fr bservers. Sectin 17.2 Curved Mirrrs 471 Stewart Weir/Eye Ubiquitus/CORBIS

17 Image in a Security Mirrr A cnvex security mirrr in a warehuse has a 0.50-m fcal length. A 2.0-m-tall frklift is 5.0 m frm the mirrr. What are the image psitin and image height? Analyze and Sketch the Prblem Draw a diagram with the mirrr and the bject. Draw tw principal rays t lcate the image in the diagram. Knwn: Unknwn: h 2.0 m? d 5.0 m h i? f 0.50 m Slve fr the Unknwn Use the mirrr equatin and slve fr image psitin. 1 f 1 1 di d fd d f ( 0.50 m) ( 5. 0 m) Substitute f 0.50 m, d 5.0 m m 5.0 m 0.45 m (virtual image, behind the mirrr) Use the magnificatin equatin and slve fr image height. hi m d i h d h i di h d ( 0.45 m)(2.0 m) Substitute 0.45 m, h 2.0 m, d 5.0 m (5.0 m) 0.18 m (upright, smaller image) Evaluate the Answer Are the units crrect? All psitins are in meters. D the signs make sense? A negative psitin indicates a virtual image; a psitive height indicates an image that is upright. These agree with the diagram. h O 1 d I 1 Math Handbk Islating a Variable page An bject is lcated 20.0 cm in frnt f a cnvex mirrr with a 15.0-cm fcal length. ind the image psitin using bth a scale diagram and the mirrr equatin. 18. A cnvex mirrr has a fcal length f 13.0 cm. A lightbulb with a diameter f 6.0 cm is placed 60.0 cm frm the mirrr. What is the lightbulb s image psitin and diameter? 19. A cnvex mirrr is needed t prduce an image that is three-furths the size f an bject and lcated 24 cm behind the mirrr. What fcal length shuld be specified? 20. A 7.6-cm-diameter ball is lcated 22.0 cm frm a cnvex mirrr with a radius f curvature f 60.0 cm. What are the ball s image psitin and diameter? 21. A 1.8-m-tall girl stands 2.4 m frm a stre s security mirrr. Her image appears t be 0.36 m tall. What is the fcal length f the mirrr? 472 Chapter 17 Reflectin and Mirrrs

18 Table 17-1 Single-Mirrr System Prperties Mirrr Type f d m Image Plane N/A d 0 d (negative) Same size Virtual d r r f Reduced, inverted Real Cncave r d f r Enlarged, inverted Real f d 0 d (negative) Enlarged Virtual Cnvex d 0 f 0 (negative) Reduced Virtual Mirrr Cmparisn Hw d the varius types f mirrrs cmpare? Table 17-1 cmpares the prperties f single-mirrr systems with bjects that are lcated n the principal axis f the mirrr. Virtual images are always behind the mirrr, which means that the image psitin is negative. When the abslute value f a magnificatin is between zer and ne, the image is smaller than the bject. A negative magnificatin means the image is inverted relative t the bject. Ntice that the single plane mirrr and cnvex mirrr prduce nly virtual images, whereas the cncave mirrr can prduce real images r virtual images. Plane mirrrs give simple reflectins, and cnvex mirrrs expand the field f view. A cncave mirrr acts as a magnifier when an bject is within the fcal length f the mirrr Sectin Review 22. Image Prperties If yu knw the fcal length f a cncave mirrr, where shuld yu place an bject s that its image is upright and larger cmpared t the bject? Will this prduce a real r virtual image? 23. Magnificatin An bject is placed 20.0 cm in frnt f a cncave mirrr with a fcal length f 9.0 cm. What is the magnificatin f the image? 24. Object Psitin The placement f an bject in frnt f a cncave mirrr with a fcal length f 12.0 cm frms a real image that is 22.3 cm frm the mirrr. What is the bject psitin? 25. Image Psitin and Height A 3.0-cm-tall bject is placed 22.0 cm in frnt f a cncave mirrr having a fcal length f 12.0 cm. ind the image psitin and height by drawing a ray diagram t scale. Verify yur answer using the mirrr and magnificatin equatins. 26. Ray Diagram A 4.0-cm-tall bject is lcated 14.0 cm frm a cnvex mirrr with a fcal length f 12.0 cm. Draw a scale ray diagram shwing the image psitin and height. Verify yur answer using the mirrr and magnificatin equatins. 27. Radius f Curvature A 6.0-cm-tall bject is placed 16.4 cm frm a cnvex mirrr. If the image f the bject is 2.8 cm tall, what is the radius f curvature f the mirrr? 28. cal Length A cnvex mirrr is used t prduce an image that is tw-thirds the size f an bject and lcated 12 cm behind the mirrr. What is the fcal length f the mirrr? 29. Critical Thinking Wuld spherical aberratin be less fr a mirrr whse height, cmpared t its radius f curvature, is small r large? Explain. physicspp.cm/self_check_quiz Sectin 17.2 Curved Mirrrs 473

19 Alternate CBL instructins can be fund n the Web site. physicspp.cm Cncave Mirrr Images A cncave mirrr reflects light rays that arrive parallel t the principal axis thrugh the fcal pint. Depending n the bject psitin, different types f images can be frmed. Real images can be prjected nt a screen while virtual images cannt. In this experiment yu will investigate hw changing the bject psitin affects the image lcatin and type. QUESTION What are the cnditins needed t prduce real and virtual images using a cncave mirrr? Objectives Cllect and rganize data f bject and image psitins. Observe real and virtual images. Summarize cnditins fr prductin f real and virtual images with a cncave mirrr. Safety Precautins D nt lk at the reflectin f the Sun in a mirrr r use a cncave mirrr t fcus sunlight. Materials cncave mirrr flashlight screen supprt mirrr hlder tw metersticks fur meterstick supprts screen lamp with a 15-W lightbulb Prcedure 1. Determine the fcal length f yur cncave mirrr by using the fllwing prcedure. CAUTION: D nt use the Sun t perfrm this prcedure. Reflect light frm a flashlight nt a screen and slwly mve the screen clser r farther away frm the mirrr until a sharp, bright image is visible. Measure the distance between the screen and the mirrr alng the principal axis. Recrd this value as the actual fcal length f the mirrr, f. 2. On the lab table, set up tw metersticks n supprts in a V rientatin. Place the zer measurement ends at the apex f the tw metersticks. 3. Place the mirrr in a mirrr hlder and place it at the apex f the tw metersticks. 4. Using the lamp as the bject f the reflectin, place it n ne meterstick at the ppsite end frm the apex. Place the mirrr and the screen, supprted by a screen supprt, n the ther meterstick at the ppsite end frm the apex. 5. Turn the rm lights ff. 6. Turn n the lamp. CAUTION: D nt tuch the ht lightbulb. Measure bject psitin, d, and recrd this as Trial 1. Measure the bject height, h, and recrt as Trial 1. This is measured as the actual height f the lightbulb, r glwing filament if the bulb is clear. 7. Adjust the mirrr r metersticks, as necessary, such that the reflected light shines n the screen. Slwly mve the screen back and frth alng the meterstick until a sharp image is seen. Measure image psitin,, and the image height, h i, and recrd these as Trial Hrizns Cmpanies

20 Data Table Trial d (cm) (cm) h (cm) h i (cm) Calculatin Table 1 Trial (cm 1 1 ) (cm ) (cm 1 ) f d d d d calc (cm) % errr i i 8. Mve the lamp clser t the mirrr s that d is twice the fcal length, f. Recrd this as Trial 2. Mve the screen until an image is btained n the screen. Measure and h i, and recrd these as Trial Mve the lamp clser t the mirrr s that d is a few centimeters larger than f. Recrd this as Trial 3. Mve the screen until an image is btained n the screen. Measure and h i, and recrd these as Trial Mve the lamp s that d is equal t f. Recrd this as Trial 4 data. Mve the screen back and frth and try t btain an image. What d yu bserve? 11. Mve the lamp s that d is less than f by a few centimeters. Recrd this as Trial 5. Mve the screen back and frth and try t btain an image. What d yu bserve? Analyze 1. Use Numbers Calculate 1/d and 1/ and enter the values in the calculatin table. 2. Use Numbers Calculate the sum f 1/d and 1/ and enter the values in the calculatin table. Calculate the reciprcal f this number and enter it in the calculatin table as f calc. 3. Errr Analysis Cmpare the experimental fcal length, f calc, with f, the accepted fcal length, by finding the percent errr. percent errr = f f calc 100 f Cnclude and Apply 1. Classify What type f image was bserven each f the trials? 2. Analyze What cnditins cause real images t be frmed? 3. Analyze What cnditins cause virtual images t be frmed? Ging urther 1. What are the cnditins needed fr the image t be larger than the bject? 2. Review the methds used fr data cllectin. Identify surces f errr and what might be dne t imprve accuracy. Real-Wrld Physics What advantage wuld there be in using a telescpe with a cncave mirrr? T find ut mre abut reflectin, visit the Web site: physicspp.cm 475

21 Adaptive Optical Systems Objects in space are difficult t bserve frm Earth because they twinkle. Our mving, unevenlyheated atmsphere refracts their light in a chatic manner. It s like trying t lk at a small bject thrugh the bttm f an empty, clear, glass jar while rtating the jar. Adaptive mirrr lexible Adaptive Mirrr An adaptive ptical system (AOS) cntinuusly cmpensates fr the distrtin f the atmsphere, remving the twinkle frm star images t allw astrnmers t view and phtgraph steady images f the mst distant bjects in the visible universe. An AOS directs the magnified image f the stars frm the telescpe nt a flexible adaptive mirrr made f thin glass. This mirrr is stretched acrss mvable pistns that can pke r pull the surface f the mirrr int many cmplicated shapes. Each pistn is driven by a fast, cmputer-cntrlled mtr. When the mirrr surface is shapent just the right pattern at just the right time, it will cmpensate fr the cnvective mvement f the atmsphere between the telescpe and the star, and reflect a clear image t the bserver r camera. Wave-rnt Sensr T detect the atmspheric distrtin at each instant f time, a wave-frnt sensr lks at a single star thrugh the telescpe. This device has an array f tiny lenses (lenslets) in several rws. Each lenslet frms an image f the star n a sensitive screen behint. The psitin f each image can be read by the cmputer. If each image is nt directly behints lenslet, then the cmputer knws that the star s light waves are being distrted by the atmsphere. A star is a distant pint surce f light, s it shuld prduce plane waves. Distrtemages f a star are nn-planar light waves, and these uneven waves cause the images f the star behind sme f the lenslets t be displaced. Cntrl system AOS cmpensates fr distrtin when viewing Titan, Saturn s largest mn. Cnventinal telescpe Light frm telescpe Distrted wave frnt Wave-frnt sensr Beam splitter Hubble Space Telescpe Crrected wave frnt High-reslutin camera Keck telescpe with AOS The cmputer lks at this errr and calculates hw the adaptive mirrr shuld be wrinkled t bring each f the lenslet images back int place. The star image reflected t the bserver then will be crrect, and a clear image f all ther bjects (like galaxies and planets) in the vicinity als will be seen clearly. The adaptive mirrr is re-shaped abut 1000 times per secnd. Ging urther 1. Research What is dne if there is nt a suitable star fr the wave-frnt sensr t analyze in a regin f space under bservatin? 2. Apply Research hw adaptive ptics will be usen the future t crrect visin. 476 uture Technlgy

22 17.1 Reflectin frm Plane Mirrrs Vcabulary specular reflectin (p. 459) diffuse reflectin (p. 459) plane mirrr (p. 461) bject (p. 461) image (p. 461) virtual image (p. 461) Key Cncepts Accrding t the law f reflectin, the angle that an incident ray makes with the nrmal equals the angle that the reflected ray makes with the nrmal. r i The law f reflectin applies t bth smth and rugh surfaces. A rugh surface, hwever, has nrmals that g in lts f different directins, which means that parallel incident rays will nt be reflecten parallel. A smth surface prduces specular reflectin. A rugh surface prduces diffuse reflectin. Specular reflectin results in the frmatin f images that appear t be behind plane mirrrs. An image prduced by a plane mirrr is always virtual, is the same size as the bject, has the same rientatin, ans the same distance frm the mirrr as the bject. d h i h 17.2 Curved Mirrrs Vcabulary cncave mirrr (p. 464) principal axis (p. 464) fcal pint (p. 464) fcal length (p. 464) real image (p. 465) spherical aberratin (p. 467) magnificatin (p. 468) cnvex mirrr (p. 471) Key Cncepts Yu can lcate the image created by a spherical mirrr by drawing tw rays frm a pint n the bject t the mirrr. The intersectin f the tw reflected rays is the image f the bject pint. The mirrr equatin gives the relatinship amng image psitin, bject psitin, and fcal length f a spherical mirrr. 1 f 1 1 d d The magnificatin f a mirrr image is given by equatins relating either the psitins r the heights f the image and the bject. i hi m d i h d A single cncave mirrr prduces a real image that is inverted when the bject psitin is greater than the fcal length. A single cncave mirrr prduces a virtual image that is upright when the bject psitin is less than the fcal length. A single cnvex mirrr always prduces a virtual image that is upright and smaller cmpared t the bject. By frming smaller images, cnvex mirrrs make images seem farther away and prduce a wide field f view. Mirrrs can be usen cmbinatins t prduce images f any size, rientatin, and lcatin desired. The mst cmmn use f cmbinatins f mirrrs is as telescpes. physicspp.cm/vcabulary_puzzlemaker 477

23 Cncept Mapping 30. Cmplete the fllwing cncept map using the fllwing terms: cnvex, upright, inverted, real, virtual. plane Mirrrs cncave Applying Cncepts 44. Wet Rad A dry ras mre f a diffuse reflectr than a wet rad. Based n igure 17-16, explain why a wet rad appears blacker t a driver than a dry rad des. Richard Megna/undamental Phtgraphs virtual virtual Wet asphalt upright upright Dry asphalt Mastering Cncepts 31. Hw des specular reflectin differ frm diffuse reflectin? (17.1) 32. What is meant by the phrase nrmal t the surface? (17.1) 33. Where is the image prduced by a plane mirrr lcated? (17.1) 34. Describe the prperties f a plane mirrr. (17.1) 35. A student believes that very sensitive phtgraphic film can detect a virtual image. The student puts phtgraphic film at the lcatin f a virtual image. Des this attempt succeed? Explain. (17.1) 36. Hw can yu prve t smene that an image is a real image? (17.1) 37. An bject prduces a virtual image in a cncave mirrr. Where is the bject lcated? (17.2) igure Bk Pages Why is it desirable that the pages f a bk be rugh rather than smth and glssy? 46. Lcate and describe the physical prperties f the image prduced by a cncave mirrr when the bject is lcated at the center f curvature. 47. An bject is lcated beynd the center f curvature f a spherical cncave mirrr. Lcate and describe the physical prperties f the image. 48. Telescpe Yu have t rder a large cncave mirrr fr a telescpe that prduces high-quality images. Shuld yu rder a spherical mirrr r a parablic mirrr? Explain. 49. Describe the prperties f the image seen in the single cnvex mirrr in igure What is the defect that all cncave spherical mirrrs have and what causes it? (17.2) 39. What is the equatin relating the fcal pint, bject psitin, anmage psitin? (17.2) 40. What is the relatinship between the center f curvature and the fcal length f a cncave mirrr? (17.2) 41. If yu knw the image psitin and bject psitin relative t a curved mirrr, hw can yu determine the mirrr s magnificatin? (17.2) 42. Why are cnvex mirrrs used as rearview mirrrs? (17.2) 43. Why is it impssible fr a cnvex mirrr t frm a real image? (17.2) igure Chapter 17 Reflectin and Mirrrs r mre prblems, g t Additinal Prblems, Appendix B.

24 50. List all the pssible arrangements in which yu culd use a spherical mirrr, either cncave r cnvex, t frm a real image. 51. List all pssible arrangements in which yu culd use a spherical mirrr, either cncave r cnvex, t frm an image that is smaller cmpared t the bject. 52. Rearview Mirrrs The utside rearview mirrrs f cars ften carry the warning Objects in the mirrr are clser than they appear. What kind f mirrrs are these and what advantage d they have? Mastering Prblems 17.1 Reflectin frm Plane Mirrrs 53. A ray f light strikes a mirrr at an angle f 38 t the nrmal. What is the angle that the reflected angle makes with the nrmal? 54. A ray f light strikes a mirrr at an angle f 53 t the nrmal. a. What is the angle f reflectin? b. What is the angle between the incident ray and the reflected ray? 55. A ray f light incident upn a mirrr makes an angle f 36 with the mirrr. What is the angle between the incident ray and the reflected ray? 56. Picture in a Mirrr Penny wishes t take a picture f her image in a plane mirrr, as shwn in igure If the camera is 1.2 m in frnt f the mirrr, at what distance shuld the camera lens be fcused? 57. Tw adjacent plane mirrrs frm a right angle, as shwn in igure A light ray is incident upn ne f the mirrrs at an angle f 30 t the nrmal. a. What is the angle at which the light ray is reflected frm the ther mirrr? b. A retrreflectr is a device that reflects incming light rays back in a directin ppsite t that f the incident rays. Draw a diagram shwing the angle f incidence n the first mirrr fr which the mirrr system acts as a retrreflectr. 30 igure Draw a ray diagram f a plane mirrr t shw that if yu want t see yurself frm yur feet t the tp f yur head, the mirrr must be at least half yur height. 59. Tw plane mirrrs are cnnected at their sides s that they frm a 45 angle between them. A light ray strikes ne mirrr at an angle f 30 t the nrmal and then reflects ff the secnd mirrr. Calculate the angle f reflectin f the light ray ff the secnd mirrr. 60. A ray f light strikes a mirrr at an angle f 60 t the nrmal. The mirrr is then rtated 18 clckwise, as shwn in igure What is the angle that the reflected ray makes with the mirrr? Incident light 18 Nrmal 60 Mirrr igure igure physicspp.cm/chapter_test Chapter 17 Assessment 479 Hutchings Phtgraphy

25 17.2 Curved Mirrrs 61. A cncave mirrr has a fcal length f 10.0 cm. What is its radius f curvature? 62. An bject lcated 18 cm frm a cnvex mirrr prduces a virtual image 9 cm frm the mirrr. What is the magnificatin f the image? 63. un Huse A by is standing near a cnvex mirrr in a fun huse at a fair. He ntices that his image appears t be 0.60 m tall. If the magnificatin f the mirrr is 1, what is the by s height? Describe the image prduced by the bject in igure as real r virtual, inverted r upright, and smaller r larger than the bject. C igure Star Image Light frm a star is cllected by a cncave mirrr. Hw far frm the mirrr is the image f the star if the radius f curvature is 150 cm? 66. ind the image psitin and height fr the bject shwn in igure An bject is 30.0 cm frm a cncave mirrr f 15.0 cm fcal length. The bject is 1.8 cm tall. Use the mirrr equatin t find the image psitin. What is the image height? 69. Dental Mirrr A dentist uses a small mirrr with a radius f 40 mm t lcate a cavity in a patient s tth. If the mirrr is cncave ans held 16 mm frm the tth, what is the magnificatin f the image? 70. A 3.0-cm-tall bject is 22.4 cm frm a cncave mirrr. If the mirrr has a radius f curvature f 34.0 cm, what are the image psitin and height? 71. Jeweler s Mirrr A jeweler inspects a watch with a diameter f 3.0 cm by placing it 8.0 cm in frnt f a cncave mirrr f 12.0-cm fcal length. a. Where will the image f the watch appear? b. What will be the diameter f the image? 72. Sunlight falls n a cncave mirrr and frms an image that is 3.0 cm frm the mirrr. An bject that is 24 mm tall is placed 12.0 cm frm the mirrr. a. Sketch the ray diagram t shw the lcatin f the image. b. Use the mirrr equatin t calculate the image psitin. c. Hw tall is the image? 73. Shiny spheres that are placed n pedestals n a lawn are cnvex mirrrs. One such sphere has a diameter f 40.0 cm. A 12-cm-tall rbin sits in a tree that is 1.5 m frm the sphere. Where is the image f the rbin and hw tall is the image? Mixed Review 74. A light ray strikes a plane mirrr at an angle f 28 t the nrmal. If the light surce is mved s that the angle f incidence increases by 34, what is the new angle f reflectin? 75. Cpy igure n a sheet f paper. Draw rays n the diagram t determine the height and lcatin f the image. 3.8 cm 31 cm 16 cm 3.0 cm igure cm 4.0 cm 67. Rearview Mirrr Hw far des the image f a car appear behind a cnvex mirrr, with a fcal length f 6.0 m, when the car is 10.0 m frm the mirrr? igure Chapter 17 Reflectin and Mirrrs r mre prblems, g t Additinal Prblems, Appendix B.

26 76. An bject is lcated 4.4 cm in frnt f a cncave mirrr with a 24.0-cm radius. Lcate the image using the mirrr equatin. 77. A cncave mirrr has a radius f curvature f 26.0 cm. An bject that is 2.4 cm tall is placed 30.0 cm frm the mirrr. a. Where is the image psitin? b. What is the image height? 78. What is the radius f curvature f a cncave mirrr that magnifies an bject by a factr f 3.2 when the bject is placed 20 cm frm the mirrr? 79. A cnvex mirrr is needed t prduce an image ne-half the size f an bject and lcated 36 cm behind the mirrr. What fcal length shuld the mirrr have? 80. Surveillance Mirrr A cnvenience stre uses a surveillance mirrr t mnitr the stre s aisles. Each mirrr has a radius f curvature f 3.8 m. a. What is the image psitin f a custmer wh stands 6.5 m in frnt f the mirrr? b. What is the image height f a custmer wh is 1.7 m tall? 84. A 1.6-m-tall girl stands 3.2 m frm a cnvex mirrr. What is the fcal length f the mirrr if her image appears t be 0.28 m tall? 85. Magic Trick A magician uses a cncave mirrr with a fcal length f 8.0 m t make a 3.0-m-tall hidden bject, lcated 18.0 m frm the mirrr, appear as a real image that is seen by his audience. Draw a scale ray diagram t find the height and lcatin f the image. 86. A 4.0-cm-tall bject is placed 12.0 cm frm a cnvex mirrr. If the image f the bject is 2.0 cm tall, and the image is lcated at 6.0 cm, what is the fcal length f the mirrr? Draw a ray diagram t answer the questin. Use the mirrr equatin and the magnificatin equatin t verify yur answer. Thinking Critically 87. Apply Cncepts The ball in igure slwly rlls tward the cncave mirrr n the right. Describe hw the size f the ball s image changes as it rlls alng. 81. Inspectin Mirrr A prductin-line inspectr wants a mirrr that prduces an image that is upright with a magnificatin f 7.5 when it is lcated 14.0 mm frm a machine part. a. What kind f mirrr wuld d this jb? b. What is its radius f curvature? 82. The bject in igure mves frm psitin 1 t psitin 2. Cpy the diagram nt a sheet f paper. Draw rays shwing hw the image changes. C igure C Analyze and Cnclude The bject in igure is lcated 22 cm frm a cncave mirrr. What is the fcal length f the mirrr? 1.0 m 1.5 m 2.0 m 2.5 m igure cm 83. A ball is psitined 22 cm in frnt f a spherical mirrr and frms a virtual image. If the spherical mirrr is replaced with a plane mirrr, the image appears 12 cm clser t the mirrr. What kind f spherical mirrr was used? igure physicspp.cm/chapter_test Chapter 17 Assessment 481 CORBIS

27 89. Use Equatins Shw that as the radius f curvature f a cncave mirrr increases t infinity, the mirrr equatin reduces t the relatinship between the bject psitin and the image psitin fr a plane mirrr. 90. Analyze and Cnclude An bject is lcated 6.0 cm frm a plane mirrr. If the plane mirrr is replaced with a cncave mirrr, the resulting image is 8.0 cm farther behind the mirrr. Assuming that the bject is lcated between the fcal pint and the cncave mirrr, what is the fcal length f the cncave mirrr? 91. Analyze and Cnclude The layut f the twmirrr system shwn in igure is that f a Gregrian telescpe. r this questin, the larger cncave mirrr has a radius f curvature f 1.0 m, and the smaller mirrr is lcated 0.75 m away. Why is the secndary mirrr cncave? 92. Analyze and Cnclude An ptical arrangement usen sme telescpes is the Cassegrain fcus, shwn in igure This telescpe uses a cnvex secndary mirrr that is psitined between the primary mirrr and the fcal pint f the primary mirrr. Cnvex secndary mirrr igure Telescpe tube a. A single cnvex mirrr prduces nly virtual images. Explain hw the cnvex mirrr in this telescpe functins within the system f mirrrs t prduce real images. b. Are the images prduced by the Cassegrain fcus upright r inverted? Hw des this relate t the number f times that the light crsses? Writing in Physics Cncave primary mirrr Eyepiece 93. Research a methd used fr grinding, plishing, and testing mirrrs usen reflecting telescpes. Yu may reprt either n methds used by amateur astrnmers wh make their wn telescpe ptics, r n a methd used by a prject at a natinal labratry. Prepare a ne-page reprt describing the methd, and present it t the class. 94. Mirrrs reflect light because f their metallic cating. Research and write a summary f ne f the fllwing: a. the different types f catings used and the advantages and disadvantages f each b. the precisin ptical plishing f aluminum t such a degree f smthness that n glass is needen the prcess f making a mirrr Cumulative Review 95. A child runs dwn the schl hallway and then slides n the newly waxed flr. He was running at 4.7 m/s befre he started sliding and he slid 6.2 m befre stpping. What was the cefficient f frictin f the waxed flr? (Chapter 11) 96. A 1.0 g piece f cpper falls frm a height f m frm an airplane t the grund. Because f air resistance it reaches the grund mving at a velcity f 70.0 m/s. Assuming that half f the energy lst by the piece was distributed as thermal energy t the cpper, hw much dit heat during the fall? (Chapter 12) 97. It is pssible t lift a persn wh is sitting n a pillw made frm a large sealed plastic garbage bag by blwing air int the bag thrugh a sda straw. Suppse that the crss-sectinal area f the persn sitting n the bag is 0.25 m 2 and the persn s weight is 600 N. The sda straw has a crss-sectinal area f m 2. With what pressure must yu blw int the straw t lift the persn that is sitting n the sealed garbage bag? (Chapter 13) 98. What wuld be the perid f a 2.0-m-lng pendulum n the Mn s surface? The Mn s mass is kg, ants radius is m. What is the perid f this pendulum n Earth? (Chapter 14) 99. Organ pipes An rgan builder must design a pipe rgan that will fit int a small space. (Chapter 15) a. Shuld he design the instrument t have pen pipes r clsed pipes? Explain. b. Will an rgan cnstructed with pen pipes sund the same as ne cnstructed with clsed pipes? Explain ilters are added t flashlights s that ne shines red light and the ther shines green light. The beams are crssed. Explain in terms f waves why the light frm bth flashlights is yellw where the beams crss, but revert back t their riginal clrs beynd the intersectin pint. (Chapter 16) 482 Chapter 17 Reflectin and Mirrrs r mre prblems, g t Additinal Prblems, Appendix B.

28 Multiple Chice 1. Where is the bject lcatef the image that is prduced by a cncave mirrr is smaller than the bject? at the mirrr s fcal pint between the mirrr and the fcal pint between the fcal pint and center f curvature past the center f curvature 2. What is the fcal length f a cncave mirrr that magnifies, by a factr f 3.2, an bject that is placed 30 cm frm the mirrr? 23 cm 44 cm 32 cm 46 cm 3. An bject is placed 21 cm in frnt f a cncave mirrr with a fcal length f 14 cm. What is the image psitin? 42 cm 8.4 cm 8.4 cm 42 cm 4. The light rays in the illustratin belw d nt prperly fcus at the fcal pint. This prblem ccurs with. all spherical mirrrs all parablic mirrrs nly defective spherical mirrrs nly defective parablic mirrrs 6. A cncave mirrr prduces an invertemage that is 8.5 cm tall, lcated 34.5 cm in frnt f the mirrr. If the fcal pint f the mirrr is 24.0 cm, then what is the height f the bject that is reflected? 2.3 cm 14 cm 3.5 cm 19 cm 7. A cncave mirrr with a fcal length f 16.0 cm prduces an image lcated 38.6 cm frm the mirrr. What is the distance f the bject frm the frnt f the mirrr? 2.4 cm 22.6 cm 11.3 cm 27.3 cm 8. A cnvex mirrr is used t prduce an image that is three-furths the size f an bject and lcated 8.4 cm behind the mirrr. What is the fcal length f the mirrr? 34 cm 11 cm 6.3 cm 4.8 cm 9. A cup sits 17 cm frm a cncave mirrr. The image f the bk appears 34 cm in frnt f the mirrr. What are the magnificatin and rientatin f the cup s image? 0.5, inverted 2.0, inverted 0.5, upright 2.0, upright Extended Answer 10. A 5.0-cm-tall bject is lcated 20.0 cm frm a cnvex mirrr with a fcal length f 14.0 cm. Draw a scale-ray diagram shwing the image height. 5. A ray f light strikes a plane mirrr at an angle f 23 t the nrmal. What is the angle between the reflected ray and the mirrr? Yur Answers Are Better Than the Test s When yu knw hw t slve a prblem, slve it befre lking at the answer chices. Often, mre than ne answer chice will lk gd, s d the calculatins first, and arm yurself with yur answer befre lking. physicspp.cm/standardized_test Chapter 17 Standardized Test Practice 483