Greenwood press

14
CIRCLES
A keystone at the center of an arc above a
doorway maintains its structure and support. 

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can also be used for aesthetic design purposes, such as the arches seen over some
doorways. Roman engineers mastered the use of the circular arch in buildings,
bridges, and aqueducts. A keystone, the stone placed at the top of the arch, is the
essential component that keeps the structure of the arch together. Without a key-
stone, the arch may crumble if it is not cemented properly.
All materials are not designed to include circles, however, because a circle
does not necessarily serve all functions. For example, a book is shaped like a rec-
tangular prism instead of a cylinder, because it may be easier to store on a shelf
and retrieved easily with its visible binding.
Circular, or angular, motion has several useful applications. It affects the lin-
ear speed and performance of many objects. For example, circular disks spin in
an automobile engine to move its timing belts. The size of the disks can vary,
allowing the engine to distribute its power in different ways. In order to move a
belt, larger wheels do not need to spin as fast as smaller wheels, because they
cover a greater distance in a smaller amount of time. (See Variation.)
Another way to think about the connection between angular and linear speed
is to envision the motion of an ice skater. The spinning rate of the skater will
change with the movement of the radius of his or her arms from the body. To
move faster, the skater will pull his or her arms in towards the body; conversely,
to spin more slowly, the skater will gradually pull his or her arms away from the
body. As an equation, the linear speed, s, is the product of the radius, r, and angu-
lar speed, 
ω, written as s = rω. Suppose the skater has a constant linear speed of
500 cm/sec. If his or her arm radius is 100 cm, then the skater will be spinning
at a rate of 5 radians/sec, or less than 1 revolution in a second. If he or she pulls
the arms in so that they are 25 cm from the body, then the skater’s angular speed
picks up to 20 radians/sec, about 3
1
/
2
revolutions in 1 second. 
If the angular speed is held constant, then an object can have different linear
velocities depending on its position on the circular object. For example, a spin-
ning object on a playground or at an amusement park, such as a merry-go-round,
typically has a constant angular speed. Therefore linear velocity increases as the
radius increases. This means that you would feel like you were moving faster if
you stood further away from the center. If you like rides that make you feel dizzy,
then make sure you stand near the outside of a circular wheel when it is in
motion.