Download Rotation Lecture Notes B

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 or a = v/t
t f - ti
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
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 m/s)2/(12.5 m) = 0.297 m/s2 or 2.97 x 10-1 m/s2
Example
The earth orbits the sun at a distance of 1.5 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 = 42(1.5 x 1011 m)/( 3.15 x 107 s)2
a = 5.92 x 1012 m/9.92 x 1014 s2
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.