Transcript Tangential speed

Chapter Ten Notes:
Circular Motion
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There are two types of circular motion,
rotation and revolution.
When an object
turns about an internal axis, the motion is
called rotation, or spin. When an object turns
about an external axis, the motion is called
revolution. Ea: The earth revolves around
the sun once every 365½ days, and it rotates
around it’s axis every 24 hours!
An object moving in a circle is accelerating.
Accelerating objects are objects which are
changing their velocity - either the speed
(i.e., magnitude of the velocity vector) or the
direction. An object undergoing uniform
circular motion is moving with a constant
speed. Nonetheless, it is accelerating due to
its change in direction. The direction of the
acceleration is inwards. The animation at the
right depicts this by means of a vector arrow.
The final motion characteristic
for an object undergoing
uniform circular motion is the
net force. The net force acting
upon such an object is
directed towards the center of
the circle. The net force is
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said to be an inward or centripetal force.
Without such an inward force, an object
would continue in a straight line, never
deviating from its direction. Yet, with the
inward net force directed perpendicular to
the velocity vector, the object is always
changing its direction and undergoing an
inward acceleration.
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Types of Speed:
◦ Linear Speed – A point on the outside of a turntable
moves a greater distance than a spot near the
middle, in the same time. The speed of something
moving along a circular path is called tangential
speed because the direction of motion is always
tangent to the circle.
◦ Rotational speed – (Sometimes called angular
speed) is the number of rotations per unit of time.
It is common to express rotational speed in
revolutions per minute (RPM).
Ea: phonograph
records commonly rotate at 331/3 RPM
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Tangential and Rotational Speed:
These are related to each other. If you are on the
outside of a giant rotating platform, the faster it
turns, the faster your tangential speed.
 Tangential speed ~ radial distance x rotational speed
 In symbol form
 v ~ rω
 Where v is tangential speed and
 ω (pronounced oh MAY guh) is rotational speed .
◦ Tangential speed depends on rotational speed
and the distance you are from the axis of
rotation!
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Railroad train wheels:
Why does a moving freight train stay on the
tracks. Most people assume it is because of the
flanges at the edge of the wheel. However, these
are only for emergency situations or when they
follow slots that switch the train from one set of
tracks to another. They stay on the tracks because
their rims are slightly tapered. See figures 10.4
and 10.5 on page 173 of your book for two of the
reasons for tapered wheels. Also read pages 173
to174 for a complete discussion of this process.
FIGURE 10.6 
The tapered shape of railroad train wheels
(shown exaggerated here) is essential on
the curves of railroad tracks.
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Recall that on slides 3 & 4 when we said: “The
final
motion characteristic for an object undergoing uniform circular
motion is the net force. The net force acting upon such an object is
directed towards the center of the circle. The net force is said to be
an inward or centripetal force. Without such an inward force, an
object would continue in a straight line, never deviating from its
direction. Yet, with the inward net force directed perpendicular to
the velocity vector, the object is always changing its direction and
undergoing an inward acceleration.”
Acceleration : As mentioned earlier, an object
moving in uniform circular motion is moving in a
circle with a uniform or constant speed. The
velocity vector is constant in magnitude but
changing in direction.
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Because the speed is constant for such a motion, many
students have the misconception that there is no acceleration.
"After all," they might say, "if I were driving a car in a circle at
a constant speed of 20 mi/hr, then the speed is neither
decreasing nor increasing; therefore there must not be an
acceleration." At the center of this common student
misconception is the wrong belief that acceleration has to do
with speed and not with velocity. But the fact is that an
accelerating object is an object which is changing its velocity.
And since velocity is a vector which has both magnitude and
direction, a change in either the magnitude or the direction
constitutes a change in the velocity. For this reason, it can be
safely concluded that an object moving in a circle at constant
speed is indeed accelerating. It is accelerating because the
direction of the velocity vector is changing.
To understand this at a deeper level, we will have to combine
the definition of acceleration with a review of some basic
vector principles.
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Recall from previous chapters, that acceleration as
a quantity was defined as the rate at which the
velocity of an object changes. As such, it is
calculated using the following equation:
where vi represents the initial velocity and vf
represents the final velocity after some time of t.
The numerator of the equation is found by
subtracting one vector (vi) from a second vector
(vf). But the addition and subtraction of vectors
from each other is done in a manner much
different than the addition and subtraction of
scalar quantities.
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Consider the case of an object moving in a circle about point
C as shown in the diagram below. In a time of t seconds, the
object has moved from point A to point B. In this time, the
velocity has changed from vi to vf. The process of subtracting
vi from vf is shown in the vector diagram; this process yields
the change in velocity.
Direction of the Acceleration Vector: Note in the diagram
above that there is a velocity change for an object moving in a
circle with a constant speed. A careful inspection of the
velocity change vector in the above diagram shows that it
points down and to the left.
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At the midpoint along the arc connecting points A and B,
the velocity change is directed towards point C - the center of
the circle. The acceleration of the object is dependent upon
this velocity change and is in the same direction as this
velocity change. The acceleration of the object is in the same
direction as the velocity change vector; the acceleration is
directed towards point C as well - the center of the circle.
Objects moving in circles at a constant speed accelerate
towards the center of the circle.
The acceleration of an object is often measured using a
device known as an accelerometer. A simple
accelerometer consists of an object immersed in a fluid
such as water. Consider a sealed jar which is filled with
water. A cork attached to the lid by a string can serve as
an accelerometer. To test the direction of acceleration for
an object moving in a circle, the jar can be inverted and
attached to the end of a short section of a wooden 2x4. A
second accelerometer constructed in the same manner
can be attached to the opposite end of the 2x4. If the 2x4
and accelerometers are clamped to a rotating platform and
spun in a circle, the direction of the acceleration can be
clearly seen by the direction of lean of the corks.
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Calculating Centripetal Force:
The centripetal
force on an object depends on the object’s
tangential speed, its mass, and the radius of its
circular path. In equation form,
mass x speed2
 Centripetal force =
radius of curvature
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Fc = mv2/r
Centripetal force, Fc , is measured in newtons (N)
when m is expressed in kilograms (kg), v in
meters/second (m/s), and r in meters (m).
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Adding Force Vectors:
Figure 10.11 is a sketch of a conical pendulum – a bob held
in a circular path by a string attached above. Only two forces
act on the bob: mg, the force due to gravity, and tension T in
the string. Both are vectors. Figure 10.12 shows vector T
resolved into two perpendicular components, Tx (horizontal)
and Ty (vertical).
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Since the bob doesn’t accelerate vertically, the net force in
the vertical direction must be zero. Therefore: Ty = -mg
Now, what do we know about Tx ?
That’s the net force on the bob, centripetal force! It’s
magnitude is mv2/r. Note that this lies along the radius of
the circle swept out.
Another example is shown below. There are two forces
acting on the car, gravity mg and the normal force n. Gravity
mg and ny balance out, and nx is the centripetal force.
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Inertia, Force and Acceleration for an Automobile
Passenger The idea expressed by Newton's law of inertia should not be
surprising to us. We experience this phenomenon of inertia nearly everyday
when we drive our automobile. For example, imagine that you are a
passenger in a car at a traffic light. The light turns green and the driver
accelerates from rest. The car begins to accelerate forward, yet relative to the
seat which you are on your body begins to lean backwards. Your body being
at rest tends to stay at rest. This is one aspect of the law of inertia - "objects
at rest tend to stay at rest." As the wheels of the car spin to generate a
forward force upon the car and cause a forward acceleration, your body
tends to stay in place. It certainly might seem to you as though your body
were experiencing a backwards force causing it to accelerate backwards. Yet
you would have a difficult time identifying such a backwards force on your
body. Indeed there isn't one. The feeling of being thrown backwards is
merely the tendency of your body to resist the acceleration and to remain in
its state of rest. The car is accelerating out from under your body, leaving
you with the false feeling of being pushed backwards.
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Now imagine that you are in the same car moving along at a constant speed
approaching a stoplight. The driver applies the brakes, the wheels of the car
lock, and the car begins to skid to a stop. There is a backwards force upon
the forward moving car and subsequently a backwards acceleration on the
car. However, your body, being in motion, tends to continue in motion while
the car is skidding to a stop. It certainly might seem to you as though your
body were experiencing a forwards force causing it to accelerate forwards.
Yet you would once more have a difficult time identifying such a forwards
force on your body. Indeed there is no physical object accelerating you
forwards. The feeling of being thrown forwards is merely the tendency of
your body to resist the deceleration and to remain in its state of forward
motion. This is the second aspect of Newton's law of inertia - "an object in
motion tends to stay in motion with the same speed and in the same
direction... ." The unbalanced force acting upon the car causes the car to
slow down while your body continues in its forward motion. You are once
more left with the false feeling of being pushed in a direction which is
opposite your acceleration.
These two driving scenarios are summarized by the following graphic.
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Suppose that on the next part of your travels the driver of the car makes a
sharp turn to the left at constant speed. During the turn, the car travels in a
circular-type path. That is, the car sweeps out one-quarter of a circle. The
friction force acting upon the turned wheels of the car cause an unbalanced
force upon the car and a subsequent acceleration. The unbalanced force and
the acceleration are both directed towards the center of the circle about
which the car is turning. Your body however is in motion and tends to stay in
motion. It is the inertia of your body - the tendency to resist acceleration which causes it to continue in its forward motion. While the car is
accelerating inward, you continue in a straight line. If you are sitting on the
passenger side of the car, then eventually the outside door of the car will hit
you as the car turns inward. This phenomenon might cause you to think that
you are being accelerated outwards away from the center of the circle. In
reality, you are continuing in your straight-line inertial path tangent to the
circle while the car is accelerating out from under you. The sensation of an
outward force and an outward acceleration is a false sensation. There is no
physical object capable of pushing you outwards. You are merely
experiencing the tendency of your body to continue in its path tangent to the
circular path along which the car is turning. You are once more left with the
false feeling of being pushed in a direction which is opposite your
acceleration.
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This apparent (fictitious) outward force on a rotating or
revolving body is called centrifugal force. Centrifugal means
“center-fleeing,” or “away from the center.”
Now suppose there is a ladybug inside the whirling can, as
shown in figure 10.16. The can presses against the bug’s
feet and provides the centripetal force that holds it in a
circular path. The ladybug, in turn presses against the floor
of the can.
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Neglecting gravity, the only force exerted on the ladybug is
the force on the can on its feet. From our outside stationary
frame of reference, we see that there is no centrifugal force
exerted on the ladybug. The centrifugal-force effect is
attributed not to any real force but to inertia – the tendency
of the moving object to follow a straight-line path.
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Our view of nature depends upon the frame of reference from
which we view it.
Recall the ladybug in the previous slide. We can see that
there is no centrifugal force acting on her. However, we do
see centripetal force acting on the can and the ladybug,
producing circular motion.
But nature seen from the rotating frame of reference (the
can), is different. To the ladybug, the centrifugal force
appears in its own right, as real as the pull of gravity.
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Centrifugal force is an effect of rotation. It is not
part of an interaction and therefore it cannot be a
true force.
For this reason, physicists refer to centrifugal force
as a fictitious force, unlike gravitational,
electromagnetic, and nuclear forces. Nevertheless,
to observers who are in a rotating system,
centrifugal force is very real, just as gravity is ever
present at Earth’s surface, centrifugal force is ever
present within a rotating system.
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Even learned physics types would admit that circular motion
leaves the moving person with the sensation of being thrown
outward from the center of the circle. But before drawing
hasty conclusions, ask yourself three probing questions:
◦ Does the sensation of being thrown outward from the center of a circle
mean that there was definitely an outward force?
◦ If there is such an outward force on my body as I make a left-hand turn in
an automobile, then what physical object is supplying the outward push or
pull?
◦ And finally, could that sensation be explained in other ways which are
more consistent with our growing understanding of Newton's laws?
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If you can answer the first of these questions with "No" then
you have a chance.
Key Terms:
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Axis
Rotation
Revolution
Linear Speed
Tangential Speed
Rotational Speed
Centripetal force
Centrifugal force