Question

A concave mirror has a focal length of 37.4 cm. The distance between an object and its image is 52.8 cm. Find (a) the object and (b) image distances, assuming that the object lies beyond the center of curvature and (c) the object and (d) image distances, assuming that the object lies between the focal point and the mirror. (a) Number Units (b) Number i Units Units (c) Number i < (d) Number Units

Answer 1

let

focal length = f

object distance = d_o

image distance =d_i

we know that

1/f= 1/d_o+1/d_i

given focal length = f= 37.4 cm

the radius of curvature is R = 2 f = (2)( 37.4 cm)=74.8 cm

given that

distance between the object and its image is = 52.8 cm

d_o+d_i = 52.8 cm

d_i = d_o-52.8 cm

1/f= 1/d_o+1/d_i

becomes

1/37.4= 1/d_o+1/ d_o-52.8

1/37.4 =( (d_o-52.8)+d_o) / (d_o)( d_o-52.8)

d_o²-52.8d_o= 74.8d_o-1974.82

d_o²-127.6d_o+1974.82 =0

by solving we get

d_o=109.578 cm or

d_o=18.0221 cm

case 1)

when the object lies beyond the center of curvature then

object at d_o =109.578 cm

image at d_i =d_o-52.8cm = 56.778 cm

a) d_o =109.578 cm       

approximately

d_o=109.58 cm

b) d_i = 56.778 cm

approximately

d_i= 56.78 cm

case 2)

when the object is lies between the focal point and the mirror

then

c)

object is at d_o = 18.0221 cm

d)

image at d_i = d_o-52.8 cm = 18.0221 cm--52.8 cm = -34.7779 cm = -37.78 cm

approximately

d_i = -37.78 cm

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