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Math Help - mirror equation...hate them!!

  1. #1
    Newbie
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    Unhappy mirror equation...hate them!!

    Please help me!!

    A mirror of focal length 26.6 cm creates an image with magnification −0.350. What is the image distance?

    if know what the answer is... can you explain it to me. thnxs!!!
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  2. #2
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
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    539
    You haven't mentioned whether it is a concave or convex mirror. So I am taking cases.

    Using the mirror equation,

    \frac{1}{u} + \frac{1}{v} = \frac{1}{f}
    where:
    u - object distance
    v - image distance
    f - focal length of the mirror
    (NOTE: The equation is formulated defining the pole as origin and the quantities u,v\ \mbox{and}\ f denote the x-coordinates)

    CASE I(For concave mirror):
    We have,
    f = - 26.6 cm (Considering focus towards left of pole as it is a concave mirror)

    So using mirror equation,
    \frac{1}{u} + \frac{1}{v} = -\frac{1}{26.6}

    \frac{u + v}{uv} = -\frac{1}{26.6} ---------

    We know that,

    Magnification = \frac{\mbox{Height of image}}{\mbox{Height of object}} = -\frac{v}{u}

    (The equation is formulated taking proper signs)

    Thus,
    - 0.35 = -\frac{v}{u}

    \implies v = 0.35u --------

    Using the above relation in ■,
    \frac{u(0.35 + 1)}{0.35u^2} = -\frac{1}{26.6}


    \implies (1.35\times 26.6)u = -0.35u^2


    \implies 35.91u + 0.35u^2 = 0


    \implies u(35.91 + 0.35u) = 0


    \implies u = 0 (rejected as image cannot be formed at the pole)
    or u = -\frac{35.91}{0.35}\ \mbox{cm} = - 102.6\ \mbox{cm}


    Using ●,
    v = 0.35\times -102.6 = -35.91\ \mbox{cm}


    Thus, the image is formed at a distance of \boxed{35.91\ \mbox{cm}} towards the left of the concave mirror. (The negative sign implies negative x-axis which is towards the left of the origin i.e the pole)

    Similarly,
    CASE II(For Convex mirror):

    We have,
    f = + 26.6 cm [Considering focus to the right of the pole]

    Repeating the rest of the process,
    v = +35.91\ \mbox{cm}

    Thus, the image is formed at a distance of \boxed{35.91\ \mbox{cm}} towards the right of the convex mirror.
    (The positive sign implies positive x-axis which is towards the right of the origin i.e. the pole)
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