A dance hall is built without pillars and with a horizontal...

A dance hall is built without pillars and with a horizontal ceiling 7.20 $\mathrm{m}$ above the floor. A mirror is fastened flat against one section of the ceiling. Following an earthquake, the mirror is in place and unbroken. An engineer makes a quick check of whether the ceiling is sagging by directing a vertical beam of laser light up at the mirror and observing its reflection on the floor. (a) Show that if the mirror has rotated to make an angle $\phi$ with the horizontal, the normal to the mirror makes an angle $\phi$ with the vertical. (b) Show that the reflected laser light makes an angle 2$\phi$ with the vertical. (c) Assume the reflected laser light makes a spot floor 1.40 $\mathrm{cm}$ away from the point vertically below the laser. Find the angle $\phi$ .

Key Concepts

Geometrical Relation in Reflection

Due to the law of reflection combined with the mirror's rotation, the reflected beam deviates by an angle of 2? from its original vertical direction. This doubling effect arises from the symmetrical nature of reflection angles, and it enables the use of simple trigonometric relationships to connect angular deviations with measurable displacement, such as the distance between the point of reflection and the laser’s spot on the floor.

Law of Reflection

The law of reflection states that the angle of incidence (the angle between the incoming beam and the normal to the surface) is equal to the angle of reflection (the angle between the reflected beam and the normal). This principle is fundamental in understanding how the directions of light beams change upon striking a surface, and it underpins the reasoning for the reflected light deviating by twice the mirror's tilt angle.

Mirror Normal and Rotation

The geometry of mirror rotation indicates that when a mirror is rotated by an angle ? from its original horizontal position, its normal (a line perpendicular to its surface) rotates by the same angle relative to the vertical. This insight is crucial because the direction of reflection is defined with respect to the normal, making the tilt of the mirror directly influence the reflection angle.

Transcript

00:01 This question is about this slightly angled ceiling with a mirror attached to it.

00:04 Now we're going to focus just in the mirror and i made this quick diagram here where the blue line represents the mirror and the dotted line is the normal to the mirror.

00:12 First things first, we are told that the height of the mirror is 7 .20 meters.

00:19 And the first part of the question asks us to prove that the angle between the mirror and the horizontal which is phi is equal to the angle between the vertical and the normal.

00:30 So if you first draw a diagram, it becomes very intuitive that these two angles are the same.

00:37 But we can write it out.

00:39 So since the normal, by definition, is going to be perpendicular to the mirror and vertical is also going to be perpendicular to the horizontal, we can conclude that the angle between the normal and the vertical is always a very horizontal.

01:11 Going to be equal to the angle between the mirror and the horizontal.

01:24 Now this angle then equals the phi, so we can conclude that the angle between the normal and the vertical equals to phi.

01:34 Next, we have to prove that the reflected light makes an angle to phi.

01:38 So let me quickly just draw the path of the light in red.

01:42 Something like that, right? so, we can see that the incident angle is 5, right? it's the angle between the light and the normal.

01:55 And since we know, as a rule, that the incident angle is always equal to the reflected angle, you know that this angle over here is also equal to 5...

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