In the previous part
of this lesson, the use of ray diagrams were introduced and
illustrated. Ray diagrams can be used to determine where a person must sight
along a mirror in order to see an image of him/herself. As such, ray diagrams
can be used to determine what portion of a plane mirror must be used in order
to view an image. The diagram below depicts a 6-foot tall man standing in front
of a plane mirror. To see the image of his feet, he must sight along a line
towards his feet; and to see the image of the top of his head, he must sight along
a line towards the top of his head. The ray diagram depicts these lines of
sight and the complete path of light from his extremities to the mirror and to the eye. In order
to view his image, the man must look as low as point Y (to see his feet) and as
high as point X (to see the tip of his head). The man only needs the portion of
mirror extending between points X and Y in order to view his entire image. All
other portions of the mirror are useless to the task of this man viewing his
own image.
The diagram
depicts some important information about plane mirrors. Using a cm-ruler,
measure the height of the man (the vertical arrow) on the computer screen and
measure the distance between points X and Y. What do you notice? The man is
twice as tall as the distance between points X and Y. In other words, to view
an image of yourself in a plane mirror, you will need an amount of mirror equal
to one-half of your height. A 6-foot tall man needs 3-feet of mirror
(positioned properly) in order to view his entire image.
But what if the
man stood a different distance from the mirror? Wouldn't that cause the man to
need a different amount of mirror to view his image? Maybe less mirror would be
required in such an instance? These questions can be explored with the help of
another ray diagram. The diagram below depicts a man standing different
distances from a plane mirror. Ray diagrams for each situation (standing close
and standing far away) are drawn. To assist in distinguishing between the two
ray diagrams, they have been color-coded. Red and blue light rays have been
used for the situation in which the man is standing far away. Green and purple
light rays have been used for the situation in which the man is standing close
to the mirror.
The two ray
diagrams above demonstrate that the distance that a person stands from the
mirror will notaffect the
amount of mirror that the person needs to see their image. Indeed in the
diagram, the man's line of sight crosses the mirror at the same locations. A
6-foot tall man needs 3-feet of mirror to view his whole image regardless of
where he is standing. In fact, the man needs the exact same 3-feet of mirror.
A common Physics
lab involves using a tall plane mirror to explore the relationship between
object height and the portion of mirror needed to view an image. A student
stands a few meters from a planer mirror and views her image. With the student
standing upright and still and staring at her feet, the lab partner moves a
marker up and down the mirror until the sight location on the mirror is
identified. The partner then marks this location on the mirror with an erasable
marker. The process is repeated for the student staring at the tip of her head.
Of course, being a lab, the procedure is subject to a variety of procedural and
measurement error that may yield less than ideal results. The mirrors are occasionally
mounted on a wall that is not perfectly vertical. Or a student will lean
forward a slight amount, thus reducing his/her effective height. Or the mirror
warps over the years leading to one that is concave or convex rather than
planar. Despite these potential complications, the 1:2 ratio between the amount
of mirror required to view the image and the height of the object is often
observed.
1. Ben Phooled is 6-feet tall. He is the tallest person in
his family. It just so happens that Ben learned the important principle of the
2:1 relationship just prior to his family's decision to purchase a mirror that
was to be used by the entire family. Enthused about the recent physics lesson,
Ben decided to put it to gooduse. Ben convinced his parents that it would be
a waste of money to buy a mirror longer than 3 feet. "After all," Ben
argued, "I'm the tallest person in the family and only three feet of mirror
would be required to view my image." Ben's parents conceded and they
purchased a 3-foot tall mirror and mounted it on the bathroom wall.
Comment on the
wisdom behind the Phooled family decision.
The Phooled family has been fooled. Unfortunately, the
3-foot mirror can be mounted in the perfect position for Ben to view his entire
image. Suzie, who is 4 feet tall, may only need 2 feet of mirror to view her
image. Yet the two foot section which Suzie needs extends to positions on the
wall below the 3 foot section which Ben needs. Suzie's eyeball position is
lower and thus she must sight at a lower position on the mirror in order to
view her feet.