Rotation Lecture Notes A
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 = 2rf
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 = 2r/T = 2(10 m)/2 s = 31.4 m/s
v = 2rf = 2(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 center-seeking.
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 the centripetal acceleration can also
be written as
.
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 earths period in seconds.
T = 365 days x 24 hours x 60 minutes x 60 seconds
T = 3.15 x 107 s
a = 42(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 Newtons 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 objects 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 so-called 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.
