Question

A concave mirror has a focal length of 32.2 cm. The distance between an object and its image is 54.7 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.

Answer 1

Mirror equation is given by:

1/f = 1/u + 1/v

f = focal length of mirror = 32.2 cm

u = object distance from mirror

v = image distance from mirror

Part A

when object lies beyond the center of curvature than image for concave mirror will be between the focus and center of curvature, So

u - v = 54.7 cm

Also we have

1/u + 1/v = 1/32.2 cm

(u + v)/uv = 1/32.2 cm

from above equation, v = u - 54.7

So,

(u + (u - 54.7))/(u*(u - 54.7)) = 1/32.2

2*u*32.2 - 32.2*54.7 = u^2 - 54.7*u

u^2 - 119.1u + 32.2*54.7 = 0

Solving above quadratic equation:

u = [119.1 +/- sqrt (119.1^2 - 4*32.2*54.7)]/(2*1)

by taking +ve sign

u = 101.8 cm

v = 101.8 - 54.7 = 47.1 cm

(a) Object distance = 101.8 cm

(b) Image distance = 47.1 cm

Now when object lies between the focal point and the mirror, then in this case image will be behind the mirror, So now

we just have to use 2nd solution of above quadratic equation, which is

by taking negative sign

u = 17.3 cm

in that case

v = 17.3 - 54.7 = -37.4 cm

(a) Object distance = 17.3 cm

(b) Image distance = -37.4 cm

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