Lesson 1: Motion
Characteristics for Circular Motion
Speed and Velocity
Any moving object can be described using the kinematic
concepts discussed in Unit
1 of The Physics Classroom. The motion of a moving
object can be explained using either Newton's Laws
(Unit 2 of The Physics
Classroom) and vector principles (Unit
3 of The Physics Classroom) or by means of the
Work-Energy Theorem (Unit
5 of The Physics Classroom). The same concepts and
principles used to describe and explain the motion of an
object can be used to describe and explain the parabolic
motion of a projectile. In this unit, we will see that these
same concepts and principles can also be used to describe
and explain the motion of objects which either move in
circles or can be approximated to be moving in circles.
Kinematic concepts and motion principles will be applied to
the motion of objects in circles and then extended to
analyze the motion of such objects as roller coaster cars, a
football player making a circular turn, and a planet
orbiting the sun. We will see that the beauty and power of
physics lies in the fact that a few simple concepts and
principles can be used to explain the mechanics of the
entire universe. Lesson 1 of this study will begin with the
development of kinematic and dynamic ideas can be used to
describe and explain the motion of objects in circles.
Suppose that you were driving a car with
the steering wheel turned in such a manner that your car
followed the path of a perfect circle with a constant
radius. And suppose that as you drove, your speedometer
maintained a constant reading of 10 mi/hr. In such a
situation as this, the motion of your car would be described
to be experiencing uniform circular motion.
Uniform circular motion
is the motion of an object in a circle with a constant or
uniform speed.
Uniform circular motion - circular motion
at a constant speed - is one of many forms of circular
motion. An object moving in uniform circular motion would
cover the same linear distance in each second of time. When
moving in a circle, an object traverses a distance around
the perimeter of the circle. So if your car were to move in
a circle with a constant speed of 5 m/s, then the car would
travel 5 meters along the perimeter of the circle in each
second of time. The distance of one complete cycle around
the perimeter of a circle is known as the
circumference. At a
uniform speed of 5 m/s, if the circle had a circumference of
5 meters, then it would take the car 1 second to make a
complete cycle around the circle. At this uniform speed of 5
m/s, each cycle around the 5-m circumference circle would
require 1 second. At 5 m/s, a circle with a circumference of
20 meters could be made in 4 seconds; and at this uniform
speed, every cycle around the 20-m circumference of the
circle would take the same time period of 4 seconds. This
relationship between the circumference of a circle, the time
to complete one cycle around the circle, and the speed of
the object is merely an extension of the average speed
equation stated in Unit 1 of
The Physics Classroom.
The circumference of any circle can be computed using
from the radius according to the equation
Circumference =
2*pi*Radius
Combining these two equations above will lead to a new
equation relating the speed of an object moving in uniform
circular motion to the radius of the circle and the time to
make one cycle around the circle
(period).
where R represents
the radius of the circle and
T represents the period.
This equation, like all equations, can be used as a
algebraic recipe for problem solving. Yet it also can be
used to guide our thinking about the variables in the
equation relate to each other. For instance, the equation
suggests that for objects moving around circles of different
radius in the same period, the object traversing the circle
of larger radius must be traveling with the greatest speed.
In fact, the average speed and the radius of the circle are
directly proportional. A twofold increase in radius
corresponds to a twofold
increase in speed; a threefold increase in radius
corresponds to a three--fold increase in speed; and so on.
This principle was best illustrated in an classroom
demonstration using a series of LCD lights. The LCD lights
were positioned along an electrical wire at varying
locations from the end. The end of the wire was held and
spun rapidly in a circle. Each LCD light traversed a circle
of different radius. Yet since they were connected to the
same wire, their period of rotation was the same.
Subsequently, the LCDs which were further from the center of
the circle were traveling faster in order to sweep out the
circumference of the larger circle in the same amount of
time. With the room lights turned off, the LCDs created an
arc which could be perceived to be longer for those LCDs
which were traveling faster - the LCDs with the greatest
radius. This is illustrated in the diagram above.
Objects moving in uniform
circular motion will have a constant speed. But does this
mean that they will have a constant velocity? Recall from
Unit 1 of The Physics
Classroom that speed and velocity refer to two
distinctly different quantities. Speed is a scalar
quantity and velocity is a vector
quantity. Velocity, being a vector, has both a magnitude
and a direction. The magnitude of the velocity vector is
merely the instantaneous speed of the object; the direction
of the velocity
vector is directed in the same direction which the object
moves. Since an object is moving in a circle, its direction
is continuously changing. At one moment, the object is
moving northward such that the velocity vector is directed
northward. One quarter of a cycle later, the object would be
moving eastward such that the velocity vector is directed
eastward. As the object rounds the circle, the
direction of the velocity vector is different than it was
the instant before. So while the magnitude of the velocity
vector may be constant, the direction of the velocity vector
is changing. The best word that can be used to describe the
direction of the velocity vector is the word
tangential. The
direction of the velocity vector at any instant is in the
direction of a tangent line drawn to the circle at the
object's location. (A tangent line is a line which touches
the circle at one point but does not intersect it.) The
diagram at the right shows the direction of the velocity
vector at four different point for an object moving in a
clockwise direction around a circle. While the actual
direction of the object (and thus, of the velocity vector)
is changing, it's direction is always tangent to the
circle.
To summarize, an object moving in uniform
circular motion is moving around the perimeter of the circle
with a constant speed. While the speed of the object is
constant, its velocity is changing. Velocity, being a
vector, has a constant magnitude but a changing direction.
The direction is always directed tangent to the circle and
as the object turns the circle, the tangent line is always
pointing in a new direction. As we proceed through this
unit, we will see that these same principles will have a
similar extension to noncircular motion.
Check
Your Understanding
1. A spiraled tube lies fixed in
its horizontal position (i.e., it has been placed upon its
side upon a table). When a marble is rolled through it it
curves around the tube, draw the path of the marble after it
exits the tube.
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