Motion Characteristics for Circular Motion
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Teacher's Guide
The Uniform Circular Motion activity from the Shockwave Studios is an excellent accompaniment for this reading.
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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 that 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 that 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 could be described as experiencing uniform circular motion. Uniform circular motion is the motion of an object in a circle with a constant or uniform speed.
Calculation of the Average 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. With a uniform speed of 5 m/s, a car could make a complete cycle around a circle that had a circumference of 5 meters. 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
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 an
algebraic recipe for problem solving. 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. To illustrate, consider a strand of four
LED lights positioned at various locations along the strand.
The strand is held at one end and spun rapidly in a circle.
Each LED light traverses a circle of different radius. Yet
since they are connected to the same wire, their period of
rotation is the same. Subsequently, the LEDs that are
further from the center of the circle are traveling faster
in order to sweep out the circumference of the larger circle
in the same amount of time. If the room lights are turned
off, the LEDs created an arc that could be perceived to be
longer for those LEDs that were traveling faster - the LEDs
with the greatest radius. This is illustrated in the diagram
at the right.
The Direction of the Velocity Vector
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 the
instantaneous speed of the object. The direction of the
velocity
vector is directed in the same direction that 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 that touches a
circle at one point but does not intersect it.) The diagram
at the right shows the direction of the velocity vector at
four different points 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, its
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.
Check Your Understanding
1.
A tube is been placed upon the table and shaped into a
three-quarters circle. A golf ball is pushed into the tube
at one end at high speed. The ball rolls through the tube
and exits at the opposite end. Describe the path of the golf
ball as it exits the tube.