Download Motion Characteristics for Circular Motion

Rotation Lecture Notes A
Motion Characteristics for Circular Motion
Speed and Velocity
Any moving object can be described using the
kinematic concepts discussed in unit one of this
course. The motion of a moving object can be
explained using either Newton's Laws and vector
principles or by work-energy concepts. 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. As we
will see, the beauty of physics is that a few simple
concepts and principles can be used to explain the
mechanics of much of what we see in the universe.
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
20 km/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.
Some other important terminology; a cycle can
be defined as one complete motion around the
circle. The period (T) of the motion can be defined
as the time required to complete once cycle of the
motion. The unit of the period is the second (s). The
frequency (f) of motion can be defined as the
number of complete cycles per second. The unit of
frequency is the hertz (Hz). One hertz is one cycle
per second. The period of motion and the frequency
of motion are inverses of each other.
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 an extension of the
average speed equation that we have already used:
The circumference of any circle can be computed
using 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.
Note that since the period of motion and the
frequency are inverses of each other, another
method to determine speed could be
v = 2πrf
These equations, like all equations, can be used
as an algebraic recipe for problem solving. Yet it
also can be used to guide our thinking about the
variables in the equation and how they relate to
each other. For instance, the equation suggests
that for objects moving around circles of different
radii in the same period, the object traversing the
circle of a 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.
Objects moving in uniform circular motion will
have a constant speed. But does this mean that they
will have a constant velocity? Recall 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 below 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.
The 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.
Example
Suppose that a model airplane is moving in a circle
of radius 10 m at 30 revolutions per minute.
Determine:
a) the frequency
b) the period
c) the speed of the airplane
Solution
a) frequency = cycles/time frequency = 30
rev./60 s = 0.5 Hz
b) period = 1/f = 1/0.5 = 2 s
c) v = 2πr/T = 2π(10 m)/2 s = 31.4 m/s
v = 2πrf = 2π(10 m)(0.5) = 31.4 m/s
Rotation Lecture B
Centripetal Acceleration
In this lesson we examine acceleration and
force in circular motion. In doing so, we will learn
that circular motion is unique in how the velocity
and acceleration vectors are related to each other.
The direction of the acceleration in circular motion
may not be immediately obvious. The application of
this work can range from the motion of jet planes
and race cars to planets and moons moving in
circular orbits to a centrifuge in a medical
laboratory.
We saw in the previous lesson that the
velocity of an object moving in a circle is always
tangent to the circle (or perpendicular to the
radius). The diagram below shows an object at
position X at a time ti moving in a clockwise fashion.
The velocity vector at this point is drawn and
labeled vi. At a later time, tf, the object has moved
to a new position Y. The velocity vector at this new
position is also drawn and labeled vf.
Recall from our study of kinematics that the
definition of acceleration is the change in velocity
divided by the difference in time.
a = (vf – vi)/tf - ti or a = ∆v/∆t
The diagram below connects the two velocity
vectors to show the difference between them.
The diagram shows that the difference in velocity
vectors, ∆v, is pointing somewhere towards the interior of
the circle. If the time interval between vf and vi is very
small, then it would become more obvious that the direction
of the difference in velocity, and therefore the
acceleration, is perpendicular to the velocity vector and
hence along the radius towards the center of the circle. In
fact the object accelerates towards the center of the circle
at every moment. This acceleration is called centripetal
acceleration because the word centripetal means “centerseeking.”
It can be shown mathematically (although not
discussed here) that the magnitude ac of the centripetal
acceleration is given by
The centripetal acceleration vector always points
toward the center of the circle and continually changes
direction as the object moves.
Lastly, by combining
with
centripetal acceleration can also be written as
the
.
Example
A 25 kg child moves with a speed of 1.93 m/s when
sitting 12.5 m from the center of a merry-go-round.
Calculate the centripetal acceleration.
Solution
a = (1.93)2/(12.5)
a = 0.297 m/s2 or 2.97 x 10-1 m/s2
Example
The earth orbits the sun at a distance of 1.50 x 1011
m. What is the centripetal acceleration of the earth in its
orbit?
Solution
We first must calculate the earth’s period in seconds.
T = 365 days x 24 hours x 60 minutes x 60 seconds
T = 3.15 x 107 s
a = 4π2(1.5 x 1011)/( 3.15 x 107)2
a = 5.92 x 1012/9.92 x 1014
a = 0.00596 m/s2 or 5.96 x 10-3 m/s2
Centripetal Force
From Newton’s second law, F = ma, we know that
whenever an object accelerates, there must be a net force
causing the acceleration. It follows that in uniform circular
motion, there must be a net force producing the centripetal
acceleration. According to the second law, the net force Fc
is equal to the product of the object’s mass and
acceleration. This net force is called the centripetal force
and points in the same direction as the centripetal
acceleration, that is, toward the center of the circle.
In the previous lesson we learned that
,
and if we combine this with F = ma, the centripetal force can
be written as
We can describe the centripetal force in a formal way
as follows:
The centripetal force is the name given to the net
force required to keep an object of mass m, moving at
a speed v, on a circular path of radius r that has a
magnitude of
The centripetal force always points toward the center
of the circle and continually changes direction as the object
moves.
The term “centripetal” force is not a new and
separate force. It is simply a label given to the net force
pointing toward the center of the circular path, and this net
force is the vector sum of all the force components that
point along the radial direction.
Sources of Centripetal Force
(none of the following formulas will be tested in this unit,
but you should know the names of all of the different
sources)
Tension
In some cases, it is easy to identify the source of
centripetal force. When a model airplane on a guideline flies
in a horizontal circle, the only force pulling the plane inward
is the tension in the line, so this force is the centripetal
force.
As a formula, one could use
Friction
When a car moves at a steady speed around an
unbanked curve, the centripetal force keeping the car on the
curve comes from the static friction between the road and
the tires. It is static rather than kinetic friction because
the tires are not slipping on the road surface. If the static
frictional force is not sufficient, given the radius of the
turn and the speed, the car will skid off the road. For the
car to make the turn, the centripetal force is equal to the
force of friction.
As a formula, one could use
Gravity
Today there are many satellites in orbit about the
earth. The ones in circular orbits are examples of uniform
circular motion. Like a model airplane on a guideline, each
satellite is kept on its circular path by a centripetal force.
The gravitational pull of the earth provides the centripetal
force and acts like a guideline for the satellite.
As a formula, one could use
Centrifugal Force
There is a common misconception that an object
moving in a circle has an outward force acting on it, a socalled centrifugal (center-fleeing) force. Consider for
example a person swinging a ball on the end of a string. If
you have ever done this yourself, you know that you feel a
force pulling outward on your hand. This misconception
arises when this pull is interpreted as an outward
"centrifugal" force pulling on the ball that is transmitted
along the string to the hand. But this is not what is
happening at all. To keep the ball moving in a circle, the
person pulls inwardly on the ball. The ball then exerts an
equal and opposite force on the hand (Newton`s third law)
due to the fact that a) the ball has inertia, and b) this
inertia is in motion. The ball in and of itself is not pulling on
your hand at all, but due to its inertia and motion, you
perceive that it is. Hence, the term centrifugal force is
really a misnomer.
For even more convincing evidence that a centrifugal
force does not act on the ball, consider what happens when
you let go of the string. If a centrifugal force were acting,
the ball would fly straight out along the radius of motion. Of
course this does not happen. The ball flies off tangentially in
the direction of the velocity it had at the moment it was
released because the inward force no longer acts.