Solucionario Tippens Cap.34

Physics, 6th Edition

Chapter 34. Reflection and Mirrors

Chapter 34. Reflection and Mirrors Reflection from Plane Mirrors 34-1. A man man 1.8 m tall tall stands 1.! m from a lar"e lar"e plane plane mirror mirror.. #o$ tall tall is his his ima"e% ima"e% #o$ far is he from his ima"e% ( Image symmetric, reversed.) Image distance q = object distance p: q = p = 1.! m&

1.! m

1.8 m

y’ = y ' 1.8 m

33-!. (hat is is the shorte shortest st mirror mirror len"th len"th re)*ir re)*ired ed to ena+le ena+le a 1.68 m $oman $oman to see her her entire entire ima"e%  Virtual image is behind the mirror . ays sho! that only hal" length is needed. needed. #irror #irror height = ($%&) object height h =

 ft, 8 in. !

1.68 m

h = ! ft, 1 in.

It doesn’t doesn’t matter matter !here she stands.

/33-3. A plane mirror mo0es at a speed of 3 m2h a$ay from a stationary person. #o$ fast fast does this persons ima"e appear to +e mo0in" in the opposite direction% 'ach time mirror moves, image image also moves, so that speed is doubled: vi = 6 m2h

/33-4. he optical le0er is a sensiti0e sensiti0e meas*rin" de0ice that *tilies min*te rotations of a plane mirror to meas*re small deflections. he de0ice is ill*strated in 5i". 34-1. (hen the mirror is in position $, the li"ht ray follo$s the path IV path IV$. 7f the mirror is rotated thro*"h an an"le θ to position &, the ray $ill follo$ the path IV path IV&. ho$ that the reflected reflected +eam t*rns thro*"h an an"le !θ , $hich is t$ice the an"le thro*"h $hich the mirror itself t*rns. emember the basic principles o" ray tracing and that the angle o" incidence al!ays equals the angle o" re"lection. re"lection. e"er to the "igure "igure dra!n on the "ollo!ing "ollo!ing page.

!19


Chapter 34. Reflection and Mirrors /34-4. (ont.)

ho! that the deviated ray turns through t!ice the angle turned by mirror.

In hori*ontal initial position, θi ' θr . +o! "rom "igure: 7 θ i $ = θ

 θ ;

θ r ! ' θ r 1 : θ

i$

θ i ! ' θ r ! ' θ

1

!

R 1

θ

R !

!θ

: θ : θ r 1 : θ

!

i1

1

∠R 1R ! ' ! θ

θ

Images Formed by Spherical Mirrors 4-. A li"ht +*l+ 3 cm hi"h is placed ! cm in front of a conca0e mirror $ith a radi*s of c*r0at*re of 1 cm. ;etermine the nat*re, sie, and location of the ima"e formed. etch the ray-tracin" dia"ram. < y ' 3 cm, p ' ! cm = q=

p" p − "

# =

y > y

=

! cm9. cm

=

! cm - 9. cm

−q

p

& y> =

− qy

p

=

&

q = 1! cm, real

−1! cm3 cm

! cm

&

y’ = -1.8 cm

q = 1!. cm, y? ' - 1.8 cm& eal, inverted, and diminished.

34-6.

A spherical conca0e mirror has a focal len"th of ! cm. (hat are the nat*re sie and location of the ima"e formed $hen a 6 cm tall o+@ect is located 1 cm from this mirror% q=

p" p − "

=

1 cm! cm 1 cm - ! cm

&

C 5

q = -6 cm, virtual and enlarged # =

y > y

=

−q

p

& y> =

−qy

p

=

− −6 cm6 cm

! cm

&

y’ = :18 cm

q = -6 cm, y? ' 18. cm& virtual, erect, and enlarged.

!18

virtual image



34-9. A 8-cm pencil is placed 1 cm from a di0er"in" mirror of radi*s 3 cm. ;etermine the nat*re, sie, and location of the ima"e formed. etch the ray-tracin" dia"ram. virtual image

-or diverging mirror B " = (%&) = 1 cm q=

p" p − "

# =

y >

=

=

y

1 cm-1 cm 1 cm - -1 cm −q

p

& y> =

−qy

p

& q = -6. cm, virtual

=

− −6 cm8 cm

1 cm

&

C

5

y’ = :16 cm

q = -6. cm, y? ' 4.8 cm& virtual, erect, and diminished.

34-8. A spherical con0e mirror has a focal len"th ! cm. (hat are the nat*re, sie, and location of the ima"e formed of a -cm tall o+@ect located 3 cm from the mirror% virtual image

-or diverging mirror B " = ! cm q=

p" p − "

# =

y >

=

y

3 cm-! cm

=

3 cm - -! cm −q

p

& y> =

−qy

p

=

& q = -13.6 cm, virtual − −13.6 cm cm

3 cm

&

C

5

y’ = :!.!9 cm

q = -13.6 cm, y? ' !.!9 cm& virtual, erect, and diminished.

34-. An o+@ect  cm tall is place half$ay +et$een the focal point and the center of c*r0at*re of a conca0e spherical mirror of radi*s 3 cm. ;etermine the location and ma"nification the of the ima"e% " = (%&) = 1 cm. p ' !!. cm q=

p" p − "

# =

y > y

=

=

−q

p

!!. cm1 cm !!. cm - 1 cm & # =

C &

−4 cm

!!. cm

q = 4 cm, real 5

&

# = !.

q = 4. cm, M ' - !.& eal, inverted, and enlarged.

!1



34-1. A 4-cm hi"h so*rce of li"ht is placed in front of a spherical conca0e mirror $hose radi*s is 4 cm. ;etermine the nat*re sie and location of the ima"es formed for the follo$in" o+@ect distancesB a 6 cm, + 4 cm, c 3 cm, d ! cm, and e 1 cm. ;ra$ the appropriate ray-tracin" dia"rams.  In the interest o" space, diagrams are not dra!n) /ivens: " = 4 cm2! = ! cm& y ' 4 cm, p ' 6, 4, 3, !, and 1 cm. q=

a q =

+ q =

c q =

d q =

e q =

6 cm! cm 6 cm - ! cm 4 cm! cm 4 cm - ! cm 3 cm! cm 3 cm - ! cm ! cm! cm ! cm - ! cm 1 cm! cm 1 cm - ! cm

p" p − "

&

#

=

y >

=

y

−

q

p

&

= 3 cm, real

y > =

= 4 cm, real

y > =

= 6 cm, real

y > =

= ∞, no image

y > =

= -! cm, virtual

y > =

y>=

−

qy p

−3 cm4 cm

6 cm −4 cm4 cm

4 cm −6 cm4 cm

3 cm −∞4 cm

! cm

' -! cm, inverted

' -4 cm, inverted

' -8 cm, inverted

' ∞, no ima"e

−−! cm4 cm

1 cm

' 8 cm, erect

34-11. At $hat distance from a conca0e spherical mirror of radi*s 3 cm m*st an o+@ect +e placed to form an enlar"ed, in0erted ima"e located 6 cm from the mirror% 'nlarged and inverted means object bet!een - and . C

and the image is beyond the radius.

5 " = (%&) = 3 cm2!&

" = :1 cm

q = :6 cm , positive since image is real. p =

q" q − "

=

6 cm1 cm 6 cm - 1 cm

&

!!

p = ! cm



Magnification 34-1!. (hat is the ma"nification of an o+@ect if it is located 1 cm from a mirror and its ima"e is erect and seems to +e located 4 cm +ehind the mirror% 7s this mirror di0er"in" or con0er"in"%  'rect image means diverging mirror.) C

q = -4 cm , p = 1 cm # =

−q

p

=

−−4 cm

1 cm

virtual image

5

&

# = -4.

34-13. A Christmas tree ornament has a sil0ered s*rface and a diameter of 3 in. (hat is the ma"nification of an o+@ect placed 6 in. from the s*rface of this ornament% < = 02! = " =

 !

=

q =

−1. in.

!

6 in.-.9 in. 6 in. - -.9 in. # =

virtual image

& " = -.9 in. (1) since converging

−q

p

=

C 5 R ' 3 in.

& q = -.669 in., virtual

− −.669 in.

6. in.

&

# = :.111

34-14. (hat type of mirror is re)*ired to form an ima"e on a screen ! m a$ay from the mirror $hen an o+@ect is placed 1! cm in front of the mirror% (hat is the ma"nification% It must be a converging mirror, since image is real . 2lso, "rom position and si*e, it must be enlarged and inverted. 3hus, ) ' :!. m, p ' 1! cm. # =

" =

pq p + q

−q

p =

=

−! m

.1! m

&

.1! m! m .1! m : ! m

C 5

# = -16.9

&

" = :11.3 cm 4ositive means it is

!!1

converging.



/34-1. A conca0e sha0in" mirror has a focal len"th of ! mm. #o$ far a$ay from it sho*ld an o+@ect +e placed for the ima"e to +e erect and t$ice its act*al sie% # =

−q

p

−! p =

= +!&

p" p − "

q = −! p&

&

p"

q=

p − "

&

C

− ! p − "  = " &

! p + ! "

−

=

"&

virtual image

5

p=

" !

=

! mm !

&

p ' !6 mm

/34-16. 7f a ma"nification of :3 is desired, ho$ far sho*ld the mirror of Pro+lem 34-1 +e placed from the face% # =

−q

p

= +3&

q = −3 p&

−3 p + 3 " = " &

p=

p"

q=

! " 3

p − "

& −3 p =

!! mm

=

3

&

p" p − "

&

− 3 p − "  = " &

p ' 39mm

/34-19. An o+@ect is placed 1! cm from the s*rface of a spherical mirror. 7f an erect ima"e is formed that is one-third the sie of the o+@ect, $hat is the radi*s of the mirror. 7s it con0er"in" or di0er"in"%  It is diverging since image is erect and diminished.) # =

" =

−q

p

pq p + q

=

=+

1 3

& q=

−p

3

=

1! cm-4 cm 1! cm : -4 cm

−1! cm

3

&

&

q = −4 cm

" = 6. cm, diverging

/34-18. A conca0e spherical mirror has a radi*s of 3 cm and forms an in0erted ima"e on a $all  cm a$ay. (hat is the ma"nification% < " = 3 cm2! ' 1 cm& p =

, cm,1 cm  cm - 1 cm

=

18 cm&

# =

!!!

−

 cm

18 cm

&

q ' : cm =

# = -.



Challenge Problems 34-1. (hat are the nat*re, sie, and location of the ima"e formed $hen a 6-cm tall o+@ect is located 1 cm from a spherical conca0e mirror of focal len"th ! cm% q =

1 cm! cm 1 cm - ! cm

y > =

C

= -6 cm, virtual

− −6 cm6 cm

1 cm

5

virtual image

' !4 cm, erect

q = -6. cm, y? ' !4. cm& virtual, erect, and enlarged.

34-!. An erect ima"e has a ma"nification of :.6. 7s the mirror di0er"in" or con0er"in"% (hat is the o+@ect distance if the ima"e distance is D1! cm% 3he mirror is diverging since image is diminished, erect, and virtual. # =

−q

p

= .6& p =

−q

.6

=

− −1! cm

.6

& p = ! cm

34-!1. An o+@ect is located  cm from a con0er"in" mirror $hose radi*s is 4 cm. (hat is the ima"e distance and the ma"nification% q=

p" p − "

# =

y > y

=

=

 cm! cm

−q

p

 cm - ! cm =

−33.3 cm

 cm

&

&

q = 33.3 cm, real

# = -.669

q = 33.3 cm, M ' - .669& eal, inverted, and diminished.

34-!!. (hat is the focal len"th of a di0er"in" mirror if the ima"e of an o+@ect located ! mm from the mirror appears to +e a distance of 1! mm +ehind the mirror% " =

pq p + q

=

! mm−1! mm ! mm - -1! mm

!!3

&

" ' -9. mm



34-!3. A sil0er +all is 4. cm in diameter. ocate the ima"e of a 6-cm o+@ect located  cm from the s*rface of the +all. (hat is the ma"nification% <  ' 4 cm2! ' ! cm. = " = (%&)=-! cm2! ' -1 cm& p '  cm& q=

p" p − " # =

 cm-1 cm

=

 cm - -1 cm

−q

−−. cm

=

p

. cm

&

virtual image

q = . cm C 5 R ' 4 cm

&

# = .1

34-!4. An o+@ect 8 mm tall is placed 4 mm in front of a di0er"in" mirror of radi*s D6 mm. ;etermine the nat*re sie and location of the ima"e

virtual image

" = (%&)=-6 mm2! ' -3 mm q=

p" p − "

4 mm-3 mm

=

4 mm - -3 mm

# =

y > y

=

−q

& y> =

p

&

C 5 R ' 3 in.

q = 191 cm

− −191 mm8 mm

4 mm

&

y? ' :34.3 mm

/34-!. An o+@ect 1 cm tall is located ! cm from a spherical mirror. 7f an erect ima"e -cm tall is formed, $hat is the focal len"th of the mirror% #

=

y > y

=

 cm 1 cm

.&

= +

" =

pq p + q

#

=

=

−

q

p

&

q = − #p = −,.,! cm&

! cm-1 cm ! cm : -1 cm

&

q = −1 cm

" = -! cm

/34-!6. (hat is the ma"nification if the ima"e of an o+@ect is located 1 cm from a di0er"in" mirror of focal len"th D! cm% < q = 1 cm& " ' -! cm = p =

q" q − "

=

,−1 cm,-! cm 1 cm

−

, ! cm

=

6 cm& # =

− −

!!4

−q

p

=

− −1 cm

6 cm

&

# ' :.!



/34-!9. An o+@ect is placed ! mm from the 0erte of a con0e spherical mirror $hose radi*s is 4 mm. (hat is the ma"nification of the mirror% virtual image " = %& = 4 mm2! ' -! mm& q=

pq p − q

=

! mm-! mm ! mm - -! mm

#agni"ication: # =

−q

=

p

p = ! mm

p C

&

q = - 1 mm

−−1 mm

! mm

&

5

R ' 4 mm.

# = 5

/34-!8. A con0e spherical mirror has a radi*s of D6 cm. #o$ far a$ay sho*ld an o+@ect +e held if the ima"e is to +e one-third the sie of the o+@ect% virtual image " = %& = -6m2! ' -3 mm& # = − p

3

=

−q

p

p" p − "

=

&

1 3

q=−

&

p 3

# = $%6 q=

&

 −1 p − "  = 3 " &

p"

C 5 R ' -6 cm

p − " − p + " − 3 " =  &

p = &" = -!-3 cm )7

p = 6 cm

/34-!. (hat sho*ld +e the radi*s of c*r0at*re of a con0e spherical mirror to prod*ce an ima"e one-fo*rth as lar"e as the o+@ect $hich is located 4 in. from the mirror% # =

−q

p

=

1 4

&

q=

-4 in. : " = 4 "

−p

4

=

" =

p" p− "

-1

&

−4 in.

3

4 &

=

" 4 in. − "

&

" ' -!6.9 in.

/34-3. A con0e mirror has a focal len"th of D mm. 7f an o+@ect is placed 4 mm from the 0erte, $hat is the ma"nification% q=

4 mm- mm 4 mm - - mm

= −!!! mm&

# =

!!

−q

p

=

− −!!! mm

4 mm

&

# = .6



/34-31. A spherical mirror forms a real ima"e 18 cm from the s*rface. he ima"e is t$ice as lar"e as the o+@ect. 5ind the location of the o+@ect and the focal len"th of the mirror. ince the image is real and enlarged, !e dra! as sho!n: +ote: q = 18 cm, # = ! (inverted) # =

" =

−q

p

= −!&

pq p + q

=

p=

q !

=

−−18 cm

!

 cm-18 cm  cm :-18 cm

&

&

C 5

p = . cm

" = 6. cm

/34-3!. A certain mirror placed ! m from an o+@ect prod*ces an erect ima"e enlar"ed three times. 7s the mirror di0er"in" or con0er"in"% (hat is the radi*s of the mirror% 2n erect, enlarged image is consistent

C

only "or converging mirrors. # =

−q

p

= +3&

" =

q = −3 p = −3! m&

pq p + q

=

virtual image

5

! cm-6 cm ! cm :-6 cm

q = -6. m

&

" = 3. cm, con0er"in"

/34-33. he ma"nification of a mirror is D.333. (here is the o+ @ect located if its ima"e is formed on a card 4 mm from the mirror% (hat is the focal len"th% # =

−q

p

" =

= −.333&

pq p + q

=

p=

q .333

=

4 mm

1.6! m.4 m 1.6! m : .4 m

!!6

.333 &

&

p = 1.6! m

" = :4 mm



/34-34. (hat sho*ld +e the radi*s of c*r0at*re of a conca0e mirror to prod*ce an ima"e onefo*rth as lar"e as an o+@ect  cm a$ay from the mirror% 3he magni"ication must be negative i" image is diminished by concave mirror: # =

−q

p

" =

=−

pq p + q

1 4

&

=

q=

p 4

=

 cm 4

 cm1!. cm  cm : 1!. cm

&

&

q = 1!. cm

" = :1 cm

/34-3. A spherical sha0in" mirror as a ma"nification of :!. $hen an o+@ect is located 1 cm from the s*rface. (hat is the focal len"th of the mirror% # =

−q

p

= +!.&

C

q = −!.1 cm&

5

virtual image

q = -39. cm " =

pq p + q

=

1 cm-39. cm 1 cm :-39. cm

&

" = !. cm, con0er"in"

Critical Thinking Problems 34-36. A +ase+all player is 6 ft tall and stands 3 ft in front of a plane mirror. he distance from the top of his cap to his eyes is 8 in. ;ra$ a dia"ram sho$in" location of the ima"es formed of his feet and of the top of his cap% (hat is the minim*m len"th of mirror re)*ired for him to see his entire ima"e. 7f he $als 1 ft closer to the mirror $hat is the ne$ separation of o+@ect and ima"e% -rom the dra!ing o" re"lected rays, you see that: y = F8 in. : F9! in. D 8 in. &

ymin ' 36 in.

ince the image moves closer by the same amount that the ballplayer does a ne! object distance o" ! ft means a separation o" !! ft ' 4 ft

!!9



/34-39. he diameter of the moon is 348 m and it is 3.84  18 m a$ay from the earth. A telescope on the earth *tilies a spherical mirror, $hose radi*s is 8. m to form an ima"e of the moon. (hat is the diameter of the ima"e formed. (hat is the ma"nification of the mirror%  y ' 3.48  1-6 m, p ' 3.84  18 m.  Moon q=

6

p" p − "

# =

y > y

=

=

3.84  1 m4 m

−q

p

3.84  18 m - 4 m & # =

= 4. m (at -)

−4 m

&

8

3.84  1 m

# = 1.4  1-8

-8

6

y > = #y =  −1.4  1 3.48  1 m&

y = 3.6! cm

/34-38. An ima"e 6 mm lon" is formed on a $all located !.3 m a$ay from a so*rce of li"ht ! mm hi"h. (hat is the focal len"th of this mirror% 7s it di0er"in" or con0er"in"% (hat is the ma"nification%  3he image in inverted so that # is negative, image is inverted.) q = p  !.3 m& p  !.3 m ' 3 p7 " =

pq p + q

=

# =

−q

p

=

− p + !.3 m

p

= −3

p C

p ' 1.1 m& q ' p : !.3 m ' 3.4 m

1.1 m3.4 m 1.1 m : 3.4 m

!.3 m &

" = .86! m

3he positive "ocal length indicates that the mirror is converging. # =

5

−q

p

=

−3.4 m

1.1 m

!!8

&

# = -3.



/34-3. ;eri0e an epression for calc*latin" the focal len"th of a mirror in terms of the o+@ect distance p and the ma"nification M. Apply it to Pro+lem 34-3. ;eri0e a similar relation for calc*latin" the ima"e distance q in terms of M and p. Apply it to Pro+lem 34-33. # =

−q

p

&

#(p 1 ") = "7

q = − #p&

#p  #" = "7

# = !., p = 1 cm7 " =

# =

q=

−q

p

(q 1 ") =# "7

&

p=

−q

#

 # − 1 p=

q  " = #"

# = -.333, q = 4 mm7 " =

&

p− "

− #p =

=

!.1 cm !. − 1

q" q− "

&

−q

#

=

&

 # − 1

!!

=

p − " #p , # − 1

" ' :! cm q"

q − "

#" 1 " = q −q

p"

" =

#" 1 " = #p

#p

&

p"

−4 mm −.333 − 1

" =

−

q

, # − 1

" = :4 mm



/34-4. A conca0e mirror of radi*s 8 mm is placed 6 mm from a plane mirror that faces it. A so*rce of li"ht placed mid$ay +et$een the mirrors is shielded so that the li"ht is first reflected from the conca0e s*rface. (hat are the position and ma"nification of the ima"e formed after reflection from the plane mirror% #intB treat the ima"e formed +y the first p

mirror as the o+@ect for the second mirror. p =

.6 m !

= .3 m&

" =

 !

C

q

= .4 m

5 6 mm

virtual image

-irst "ind q "ormed by spherical mirror. q=

p" p − "

=

.3 m.4 m .3 m - .4 m

&

q = −1.! m 7

(behind mirror)

ince plane mirror is .6 m in le"t o" spherical mirror, the "inal image q’= p’ is: q’ = 1.! m : .6 m ' 1.8 m& # 1 =

−q

p

=

− −1.! m

.3 m

= +4&

q’ ' 1.8 m to left or +ehind the plane mirror. # ! = +1& # 1  # ! =  +4 +1&

!3

# ' :4

Solucionario Tippens Cap.34

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