If 10 objects, tinsels are placed between the mirror of a Kaleidoscope with an inclination of 30º, how many images are formed?

Answer Details

Two plane mirrors inclined at an angle \(\theta\) produce multiple images by repeated reflection. For a single object the number of images is

\[n = \frac{360^{\circ}}{\theta} - 1 \quad \text{when } \frac{360^{\circ}}{\theta} \text{ is a whole even number.}\]

The reason for subtracting one is that the \(360^{\circ}/\theta\) positions found by successive reflection include a pair that coincide on the line bisecting the angle behind the mirrors, so one of the counted images is not separate from another.

Here \(\theta = 30^{\circ}\), so

\[\frac{360^{\circ}}{30^{\circ}} = 12, \qquad n = 12 - 1 = 11.\]

Each of the tinsels acts as an independent object, and each is imaged by the same mirror system, so the total number of images is

\[N = 10 \times 11 = 110.\]

The frequent mistake is to use \(360^{\circ}/\theta\) itself, which gives \(12\) per object and \(120\) in total, or to forget to multiply by the number of objects and answer \(11\). Note also the rule for the other case: if \(360^{\circ}/\theta\) turns out to be an odd whole number, the object on the bisector still gives \(360^{\circ}/\theta - 1\) images, but an object placed off the bisector gives \(360^{\circ}/\theta\). In the examination, always evaluate \(360^{\circ}/\theta\) first, check whether it is even, apply the subtraction once, and only then multiply by the number of objects.