Conceptual Physics

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Conceptual Physics

• Chapter Ten Notes
• Circular Motion

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10.1 Rotation and Revolution

• 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 its axis every 24
  hours!
• An object moving in a circle is accelerating.
  Accelerating objects are objects which are
  changing their velocity - either the speed

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• (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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10.2 Rotational Speed

• 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 r?
• Where v is tangential speed and
• ? (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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10.3 Centripetal Force

• 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 objects
  tangential speed, its mass, and the radius of its
  circular path. In equation form,
• 
  mass x speed2 .
• Centripetal force radius of curvature
• 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 doesnt accelerate vertically, the
  net force in the vertical direction must be zero.
  Therefore Ty -mg
• Now, what do we know about Tx ?
• Thats the net force on the bob, centripetal
  force! Its 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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10.4 Centripetal and Centrifugal
Forces

• 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 bugs 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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10.5 Centrifugal Force in a Rotating
Reference Frame

• 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 Earths
  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?
• If you can answer the first of these questions
  with "No" then you have a chance.
• Key Terms
• Axis Rotational
  Speed
• Rotation Centripetal force
• Revolution Centrifugal force
• Linear Speed
• Tangential Speed