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) If the reflected laser light makes a spot on the floor $1.40 \mathrm{cm}$ away from the point vertically below the laser, find the angle $\phi$.

Key Concepts

Law of Reflection

This principle states that the angle of incidence is equal to the angle of reflection. It is fundamental in optics, defining how light behaves when it encounters a reflective surface and is essential for analyzing how a beam of light bounces off a mirror.

Mirror Rotation and Deflection

When a mirror rotates by an angle ? from its original position, the normal to the mirror also rotates by ? relative to the vertical. Due to the law of reflection, this results in the reflected beam deviating by twice the angle of rotation (2?) from its original path. This doubling effect is a crucial concept when assessing how small rotations impact the path of reflected light.

Geometric Analysis of Reflection

Using geometric relations helps in visualizing and proving how rotations of a reflective surface affect the beam path. By analyzing angles between the incident beam, the rotated normal, and the reflected beam, one can derive the relationship between the mirror’s tilt and the displacement of the reflected beam.

Trigonometric Application in Optical Measurements

In many optical problems, trigonometry is used to relate angular deviations to spatial displacements. By applying trigonometric functions in the context of a known distance (such as the height of a ceiling), one can quantitatively determine small angular deviations (like ?) from measured lateral shifts of the light spot.

Transcript

00:01 So we have this mirror that is 7 .2 meters above the ground.

00:16 And it's rotated by some angle phi, some angle phi.

00:32 And the first thing we're asked is if it's rotated by this angle of phi, we need to show that this angle that is made by someone shining a light from the floor up is also phi.

00:46 So how are we going to prove that? well, if we were to draw a right triangle that comes down like this, right, i'm going to do this in red.

00:57 So we have a, we're going to make a right triangle, and it's going to go in right here.

01:12 So clearly this angle is going to be, this angle right here is going to be 90 minus phi, which means that this angle.

01:25 Angle right here is five because if this is a right angle then and this is 90 minus five this has to be five so this is the same angle as this guy now we know that the angle between this line and this line is a 90 degree angle so that means that this angle is 90 minus phi we also know that the angle i'm going to draw this in blue because is getting a little cluttered, the angle between the incident light and the horizontal is also 90.

02:06 So that means that this angle right here is going to be just phi, because 90 minus 90 minus phi is just phi.

02:18 So we have this angle equal to phi.

02:21 So now we have established that this is equal to phi.

02:24 We want to prove, you know what, i was going to leave that there, we want to prove that the reflected light is going to be at an angle of two -fi to the incident light.

02:55 So we know that when light shines on any surface and it's completely reflected on a smooth surface, 5 -theta prime is going to be equal to theta.

03:09 So this light that's being reflected is going to have the same angle with respect to the normal line as this does.

03:22 So if this angle is 5, then this angle is also going to be 5, which means that as a whole, the angle between these 2 is going to be 2 .5.

03:33 So now we've proven a, we've proven b.

03:37 And the next step is two, is that we are told that the reflected laser light makes a spot on the floor 1 .4 centimeters away from the point vertically below the laser.

03:54 So we have our ceiling, we have our mirror, which is at an angle thigh, and we have our light, which i'm going to be.

04:09 Drawing in red, shining upon that, and then being reflected as shown...

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