On an outdoor research range, a stationary laser produces a narrow, horizontal beam directed due east. A square flat mirror is positioned 120 m east of the laser so that the laser beam strikes the centre of the mirror. The mirror can rotate only about a vertical axis through its centre. A light-sensing instrument is located 300 m due north of the mirror.
(a) Derive an expression that relates the azimuthal angle θ through which the mirror must be rotated (measured from its initial orientation in which its surface faces south) to the azimuthal angle ϕ at which the reflected beam leaves the mirror relative to due east.
(b) Using your expression from part (a), determine the precise rotation angle θ required so that the reflected beam reaches the instrument to the north.
(c) The instrument’s active area subtends an angle of only 0.10∘ as viewed from the mirror. Calculate the maximum permissible error in the mirror’s rotation angle θ if the beam is still to strike the instrument. Comment on the level of mechanical precision required.