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 *tilies 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
Physics, 6th Edition
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, sie, and location of the ima"e formed. etch the ray-tracin" dia"ram. < y ' 3 cm, p ' ! cm = q=
p" p − "
# =
y > y
=
! cm9. cm
=
! cm - 9. cm
−q
p
& y> =
− qy
p
=
&
q = 1! cm, real
−1! cm3 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 sie and location of the ima"e formed $hen a 6 cm tall o+@ect is located 1 cm from this mirror% q=
p" p − "
=
1 cm! cm 1 cm - ! cm
&
C 5
q = -6 cm, virtual and enlarged # =
y > y
=
−q
p
& y> =
−qy
p
=
− −6 cm6 cm
! cm
&
y’ = :18 cm
q = -6 cm, y? ' 18. cm& virtual, erect, and enlarged.
!18
virtual image
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
34-9. A 8-cm pencil is placed 1 cm from a di0er"in" mirror of radi*s 3 cm. ;etermine the nat*re, sie, and location of the ima"e formed. etch the ray-tracin" dia"ram. virtual image
-or diverging mirror B " = (%&) = 1 cm q=
p" p − "
# =
y >
=
=
y
1 cm-1 cm 1 cm - -1 cm −q
p
& y> =
−qy
p
& q = -6. cm, virtual
=
− −6 cm8 cm
1 cm
&
C
5
y’ = :16 cm
q = -6. cm, y? ' 4.8 cm& virtual, erect, and diminished.
34-8. A spherical con0e mirror has a focal len"th ! cm. (hat are the nat*re, sie, and location of the ima"e formed of a -cm tall o+@ect located 3 cm from the mirror% virtual image
-or diverging mirror B " = ! cm q=
p" p − "
# =
y >
=
y
3 cm-! cm
=
3 cm - -! cm −q
p
& y> =
−qy
p
=
& q = -13.6 cm, virtual − −13.6 cm cm
3 cm
&
C
5
y’ = :!.!9 cm
q = -13.6 cm, y? ' !.!9 cm& virtual, erect, and diminished.
34-. An o+@ect cm tall is place half$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
!!. cm1 cm !!. cm - 1 cm & # =
C &
−4 cm
!!. cm
q = 4 cm, real 5
&
# = !.
q = 4. cm, M ' - !.& eal, inverted, and enlarged.
!1
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
34-1. A 4-cm hi"h so*rce of li"ht is placed in front of a spherical conca0e mirror $hose radi*s is 4 cm. ;etermine the nat*re sie and location of the ima"es formed for the follo$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 cm4 cm
6 cm −4 cm4 cm
4 cm −6 cm4 cm
3 cm −∞4 cm
! cm
' -! cm, inverted
' -4 cm, inverted
' -8 cm, inverted
' ∞, no ima"e
−−! cm4 cm
1 cm
' 8 cm, erect
34-11. At $hat distance from a conca0e spherical mirror of radi*s 3 cm m*st an o+@ect +e placed to form an enlar"ed, in0erted ima"e located 6 cm from the mirror% 'nlarged and inverted means object bet!een - and . C
and the image is beyond the radius.
5 " = (%&) = 3 cm2!&
" = :1 cm
q = :6 cm , positive since image is real. p =
q" q − "
=
6 cm1 cm 6 cm - 1 cm
&
!!
p = ! cm
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
Magnification 34-1!. (hat is the ma"nification of an o+@ect if it is located 1 cm from a mirror and its ima"e is erect and seems to +e located 4 cm +ehind the mirror% 7s this mirror di0er"in" or con0er"in"% 'rect image means diverging mirror.) C
q = -4 cm , p = 1 cm # =
−q
p
=
−−4 cm
1 cm
virtual image
5
&
# = -4.
34-13. A Christmas tree ornament has a sil0ered s*rface and a diameter of 3 in. (hat is the ma"nification of an o+@ect placed 6 in. from the s*rface of this ornament% < = 02! = " =
!
=
q =
−1. in.
!
6 in.-.9 in. 6 in. - -.9 in. # =
virtual image
& " = -.9 in. (1) since converging
−q
p
=
C 5 R ' 3 in.
& q = -.669 in., virtual
− −.669 in.
6. in.
&
# = :.111
34-14. (hat type of mirror is re)*ired to form an ima"e on a screen ! m a$ay from the mirror $hen an o+@ect is placed 1! cm in front of the mirror% (hat is the ma"nification% It must be a converging mirror, since image is real . 2lso, "rom position and si*e, it must be enlarged and inverted. 3hus, ) ' :!. m, p ' 1! cm. # =
" =
pq p + q
−q
p =
=
−! m
.1! m
&
.1! m! m .1! m : ! m
C 5
# = -16.9
&
" = :11.3 cm 4ositive means it is
!!1
converging.
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/34-1. A conca0e sha0in" mirror has a focal len"th of ! mm. #o$ 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 sie% # =
−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 ' 39mm
/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 sie 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 =
# = -.
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
Challenge Problems 34-1. (hat are the nat*re, sie, 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 cm6 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
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
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 sie 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 mm8 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
&
# ' :.!
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/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 sie of the o+@ect% virtual image " = %& = -6m2! ' -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, $hat is the ma"nification% q=
4 mm- mm 4 mm - - mm
= −!!! mm&
# =
!!
−q
p
=
− −!!! mm
4 mm
&
# = .6
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/34-31. A spherical mirror forms a real ima"e 18 cm from the s*rface. he ima"e is t$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
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/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
cm1!. 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 $als 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 = F8 in. : F9! 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
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/34-39. he diameter of the moon is 348 m and it is 3.84 18 m a$ay from the earth. A telescope on the earth *tilies 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 18 m. Moon q=
6
p" p − "
# =
y > y
=
=
3.84 1 m4 m
−q
p
3.84 18 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 m3.4 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.
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/34-3. ;eri0e an epression 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
Physics, 6th Edition
Chapter 34. Reflection and Mirrors
/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