Physics 106 Lesson #25
Geometrical Optics:
Image Formation
Dr. Andrew Tomasch
2405 Randall Lab
atomasch@umich.edu
Review: Simple Harmonic Motion (SHM)
• For a mass-and-spring
oscillator, the oscillation
frequency f is governed by
inertia (mass) and the
restoring force (spring
constant)
– A bigger mass produces a
smaller f (slower oscillation
→ longer period)
– A bigger restoring force
(bigger spring constant)
produces a bigger f (faster
oscillation→shorter period)
Hooke’s Law: Fspring= -kx
1
f 
2
k
m
1
m
T   2
f
k
f is the natural frequency of a mass-and-spring oscillator.
Review: The
Simple Pendulum
•The frequency and period are
independent of the mass of the
pendulum bob
• Assuming g is fixed, the only way to
change the period is to change the
length of the pendulum
•A pendulum with a fixed length can
be used to measure g → different f, T
on a planet with a different g or at
different places on Earth
1
f 
2
g
1
L
T   2
L
f
g
f is the natural
frequency of a
simple pendulum.
Review:
Damping
• Damping forces are often put into a system
deliberately to prevent it from oscillating
indefinitely
• Example: the shock absorbers in your car
– Too little damping  too “bouncy” a ride
– Too much damping  too “stiff” a ride
• Q: How can you tell if a system is
underdamped or overdamped?
A: Disturb it. If it oscillates for a while it’s
underdamped. If it returns to the
equilibrium position without oscillating it’s
overdamped .
Review: Resonance
Demo: Mass and
Spring on Finger
• Natural Frequency: a
frequency determined by
the physical properties
of a vibrating object
f driving  f natural
• Driving an object at its
resonant frequency
(≡natural frequency)
produces high-amplitude
oscillations
f driving  f natural
• Objects driven at their
natural frequency can be
damaged or destroyed
Propagation of Light Waves
•
Light waves arrive at
objects and interact
with them in three
basic ways. They can:
1. Reflect (bounce off)
2. Refract (go through)
3. Be absorbed (stop)
•
Not exclusive, all three
may occur
Demonstration
Reflection
• Specular Reflection
Smooth mirror-like
surface → rays
reflected parallel
• Diffuse Reflection
Rough surface →
rays reflected in
various directions
The Law of Reflection
• The incident ray,
reflected ray and
the normal to the
surface are all in
the same plane.
• The angle of
incidence equals
the angle of
reflection.
i   r
How Do We See Objects?
• Light comes in straight lines from the
object to you
• Light from different points on the
object strikes your retina in different
places
• An image is a location at which there
is a one-to-one correspondence
between points on the object and
points on the image.
Concept Test #1
Why do you see images only in
mirrors, and not, for example, in the
wall?
A. Light is not reflected from the wall
B. The law of reflection does not apply to the
wall
C. Light does not pass through the wall
D. The wall isn’t flat enough
Plane Mirrors
• A ray of light from the top of the chess piece reflects
from the mirror
• To the eye, the ray seems to come from behind the
mirror
• Because none of the rays actually emanate from the
image, it is called a virtual image
Real Images From a Pinhole
•The oldest known camera
•Used by renaissance artists
for one-point perspective
drawings
•Called the Camera Obscura
in classical times
Refraction
• As light passes from one medium to another it
changes direction at the interface between the
two media
• This change of direction is known as refraction
The Index of Refraction
• Light travels
through materials at
a speed less than its
speed in a vacuum
c
c
n v
v
n
INDEX OF
REFRACTION
Indices of Refraction
Vacuum
1 (exactly)
Air
1.0003
Water
1.333
Ice
1.309
Glass
1.523
Diamond
2.419
Refraction at Surface of Water
http://www.opticalres.com/gentsupp_f.html
Concept Test #2
While boating on the Amazon, you decide to go
spear fishing. You look into the water and see
where a fish appears to be. Where should you
aim your spear?
1) Beyond where the fish appears
2) In front of where the fish appears
3) Directly where the fish appears
What you see
Where the fish really is
Where the fish really is
The Secret of the Archer Fish
• Shot with a powerful jet of water, the prey falls into the water
and the Archer Fish swims to the surface to retrieve its meal.
• The secret to the Archer Fish's success is that it lines up its
aim from a position directly underneath the prey.
Thin Lenses
• Lenses refract light in such a way that
an image of the light source is formed
• Two types of lenses: converging
(convex) and diverging (concave)
Converging Lens
Diverging Lens
A Thin Converging Lens Produces a Real, Inverted
Image for Objects Outside the Focal Length
A Thin Converging Lens Has a Positive
Focal Length-a Real Image can be
Produced on the Side Opposite the Object.
A Thin Converging Lens Produces a Virtual,
Upright Image for Objects Inside the Focal Length
A Thin Diverging Lens Produces a Virtual, Upright
Image for Objects Outside the Focal Length
A Thin Diverging Lens Has a Negative
Focal Length—the Image Appears on the
Same Side as the Object. A Diverging
Lens Cannot Form a Real Image
The Thin Lens Equation
•Relates the Image Distance (i), the Object
Distance (o) and the Focal Length (f)
•Works for both Converging and Diverging lenses
provided the focal length for a Diverging lens is
defined to be negative.
1 1 1
 
i o f
A Typical Camera
Lens
f/3 Lens
f/22 Lens
Object
Object
Film
f
f
Two Cameras With Different f/numbers
Film
Dispersion
• The index of refraction for a given material will vary
with the wavelength of the light passing through it
• This means that different colors of light will be
refracted through different angles when passing
through the same medium.
• This is called dispersion and can be demonstrated
with a prism or by observing a rainbow
The Spectrum of a Prism
White light is a
combination of all
the visible colors
nr< no < ny < ng < nb < ni <nv
Rainbows
• Rainbows are a spectacular example of
dispersion by droplets of water in the air when
the the sun is at your back
• Rainbows can be observed when sunlight is
refracted and then reflected off the back
surface of water droplets
The Secondary Rainbow
• A secondary rainbow is caused by light
which reflects twice inside the drop
• The colors in the secondary rainbow
are reversed!
Primary Rainbow
Secondary Rainbow
A Double Rainbow…