# Mini Mirror Project

## Introduction

This semester, I've been in a number of classes that have required me to walk through the lower level of the Math Tower at Stony Brook University. Each time I entered through a large set of double doors, I had to pass by a rather large exhibit that demonstrated the behavior of light and sound. It was a simple one: a large, plastic reflecting dish. When walking past it, a friend of mine and I would always approach it head on, placing our heads at what we approximated to be in line with the center of the mirror. Upon approach, we hummed or whistled and observed when the sound changed and our images became inverted.

My first instinct was that the moment the sound of my voice changed (in that it appeared as though I was speaking back at myself) coincided with the moment that my image flipped, which I immediately assumed to be at the focal point of the mirror. I brought this up to Dr. Noe, and we discussed the idea for a while. It became apparent that I was wrong with my assumption, and that sound emitted from a source located at the center of curvature, not the focal point, would be reflected back at itself. But, in order to convince myself of this property, I investigated further.

## Calculations

In order to appropriately observe this phenomena , I needed a way to find the focal point of a curve. However, this presented itself as an issue, as I only had a portion of the reflecting dish to work with. To begin, I assumed the reflecting dish to be a portion of a sphere. Essentially, the dish is formed by a great arc that spans less than have the full circumference and rotating it about its radius. Thus, to find the radius of the dish, we simply need to find the radius of one of these arcs.

From the dish, I was able to measure two distances: a chord length that spans the great-arc and the sagitta of that arc. To get the chord length, I took four pieces of string to perpendicularly bisect two arbitrary chords on the on the very outer rim of the mirror. One of these could have found the radius, but by taking their average length, I reduced the error slightly. Also, since their intersection marked the center of the rim, the shortest distance between their intersection and the mirror was the sagitta.

I will refer to the two chord lengths as c_1 and c_2, their average being c. The length of the sagitta is stated as s. So, with c_1 = 41.5 in, c_2 = 41.0 in, and s = 23.5, I could calculate r by the formula

determined by examining the accompanying diagram.

From this, I determined the radius of curvature to be roughly 48 in.

## Testing

Having calculated a value for r, it was possible for me to test to see if my revised ideas were correct. I went with another length of string, measured to be the same length as the mirror's radius of curvature, and a bit of tape in hand to the basement of the Math Tower to make some observations.

Initially, I found that my calculations proved the optical component of the idea, as I found the image of my friend flipped as he passed the 24 inch mark, or the focal point. Additionally, the prediction of acoustical behavior was accurate. After placing my head at a location 48 inches away from the center of the mirror and speaking, I could tell my voice was the loudest, in that it was being reflected back at me.