Circular Motion Lecturer: Professor Stephen T. Thornton

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Circular Motion Lecturer Professor Stephen T.
Thornton
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Reading Quiz When a car is moving around a
banked curve, which of the following statements
is most true?

• The friction is always up the slope of the banked
  road.
• The friction is always down the slope of the
  banked road.
• The friction can either be up or down the slope
  of the banked road.
• It sounds as if the cars tires are too bald to
  matter.

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Answer C

• Friction can be up or down the road. This
  is in many textbooks and is worked out. There is
  even one speed where there is no friction.

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Last Time

• Review
• Friction

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Today

• Circular motion
• Centripetal motion, force
• Motion on banked curves

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(No Transcript)
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Try this with a block of wood. This is similar
to the car moving on banked curve.
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Conceptual Quiz

• Below you see two cases a father pulling or
  pushing a sled with a force F that is applied at
  an angle q. In which case is the normal force
  greater?

A) case 1 B) case 2 C) its the same for
both D) depends on the magnitude of the force
F E) depends on the ice surface
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Conceptual Quiz

• Below you see two cases a father pulling or
  pushing a sled with a force F that is applied at
  an angle q. In which case is the normal force
  greater?

A) case 1 B) case 2 C) its the same for
both D) depends on the magnitude of the force
F E) depends on the ice surface
In case 1, the force F is pushing down (in
addition to mg), so the normal force needs to be
larger. In case 2, the force F is pulling up,
against gravity, so the normal force is lessened.
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Conceptual Quiz

• Two blocks of masses 2m and m are in contact
  on a horizontal frictionless surface. If a force
  F is applied to mass 2m, what is the force on
  mass m ? The blocks accelerate.

A) 2F B) F C) F D) F E) F
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Conceptual Quiz

• Two blocks of masses 2m and m are in contact
  on a horizontal frictionless surface. If a force
  F is applied to mass 2m, what is the force on
  mass m ? The blocks accelerate.

A) 2F B) F C) F D) F E) F
The force F leads to a specific acceleration of
the entire system. In order for mass m to
accelerate at the same rate, the force on it must
be smaller! How small?? Lets see...
F 3ma a F/3m, F1maF/3
Follow-up What is the acceleration of each mass?
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Conceptual Quiz
A) the force from the rushing air pushed it
off B) the force of friction pushed it off C)
no net force acted on the box D) truck went into
reverse by accident E) none of the above

• A box sits in a pickup truck on a frictionless
  truck bed. When the truck accelerates forward,
  the box slides off the back of the truck because

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Conceptual Quiz
A) the force from the rushing air pushed it
off B) the force of friction pushed it off C)
no net force acted on the box D) truck went into
reverse by accident E) none of the above

• A box sits in a pickup truck on a frictionless
  truck bed. When the truck accelerates forward,
  the box slides off the back of the truck because

Generally, the reason that the box in the truck
bed would move with the truck is due to friction
between the box and the bed. If there is no
friction, there is no force to push the box
along, and it remains at rest. The truck
accelerated away, essentially leaving the box
behind!!
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• Mass Moving Up Ramp. A small block of mass m is
  given an initial speed up a ramp inclined at
  angle to the horizontal. It travels a
  distance d up the ramp and comes to rest. (a)
  Determine a formula for the coefficient of
  kinetic friction between block and ramp. (b) What
  can you say about the value of the coefficient of
  static friction?

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Circular motion
Do demo with string and ball.
Note that the direction of the velocity is
changing. The ball is accelerating!
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Notice that tends to point towards the
center of the circle. As ?? becomes smaller and
smaller, points directly to center.
Therefore the acceleration points towards the
center of the circle.
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Centripetal acceleration

• Centripetal means center seeking.
• Look at your textbook for a derivation of the
  magnitude of the centripetal acceleration acp
• where r is the radius and v is the speed.

r
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Dynamics of Uniform Circular Motion
We can see that the force must be inward by
thinking about a ball on a string. Strings only
pull they never push.
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For an object to be in uniform circular motion,
there is an acceleration, and, therefore, a net
force acting on it.
We already know the acceleration, so we can
immediately write the force
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Centripetal force

• Where in the world did this centripetal force
  come from?
• There has to be a force to keep the object moving
  in a circle. In the case of the ball and string,
  it is the tension in the string. The tension
  always points towards the center!
• The direction of the centripetal force must also
  be towards the center!

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The moon rotates around the Earth in a circle.
What is the centripetal force that causes
this?If you drive around in a circle with a
bicycle or even with a car, what is the
centripetal force?In a simple atomic model of
the hydrogen atom, the electron rotates around
the proton in a circle. What is the centripetal
force?
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How can we make a bowling ball go around in a
circle?

• How are you going to do it?
• What is the centripetal force?

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Circular motion

• Results for circular motion
• Consider an object moving in a circle of radius r
  with a constant speed v.
• A centripetal acceleration of magnitude v2/r must
  be present.
• There must be a centripetal force Fcp of value

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Conceptual Quiz What other forces are exerted
on the ball besides mg? A) Friction B)
Tension C) A normal force perpendicular
to mg. D) A normal force perpendicular to
the surface of the cone at the ball.
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Answer D

• The only other possible force is the normal
  force, and it must be perpendicular to the
  surface that the ball is rolling upon.

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Conceptual Quiz What is the direction of the net
force? A) towards the center of the
dashed circle at the ball (radially). B)
away from the center of the circle at the
ball. C) up at the ball. D) down at the
ball. E) cannot tell with information
given.
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Answer A

• Because the ball is moving at constant speed in a
  circle, the net force must be along the radial
  direction, towards the center of the circle.
  This is the centripetal force.

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Look at ball moving in vertical circle

• In this case we do not have uniform circular
  motion. The tension always points towards the
  center, but gravity points down.
• Look at Examples in textbook.

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Turning corners

• What do we notice when we turn corners at high
  speed?
• Good chance of falling
• over or skidding.
• When skiing, we lean
• over and tilt our skis.
• Interstate highways
• are banked.
• Motorcycles tilt.

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Highway Curves Banked and Unbanked
When a car goes around a curve, there must be a
net force toward the center of the circle of
which the curve is an arc. If the road is flat,
that force is supplied by friction.
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If the frictional force is insufficient, the car
will tend to move more nearly in a straight line,
as the skid marks show.
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You really do not want your tires to slip!!

• As long as the tires do not slip, the friction is
  static. If the tires do start to slip, the
  friction is kinetic, which is bad in two ways
• The kinetic frictional force is smaller than the
  static.
• The static frictional force can point toward the
  center of the circle, but the kinetic frictional
  force opposes the direction of motion, making it
  very difficult to regain control of the car and
  continue around the curve.

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Banking the curve can help keep cars from
skidding. In fact, for every banked curve, there
is one speed at which the entire centripetal
force is supplied by the horizontal component of
the normal force, and no friction is required.
This occurs when
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• Whirling Bucket. A bucket of mass 2.00 kg is
  whirled in a vertical circle of radius 1.10 m. At
  the lowest point of its motion the tension in the
  rope supporting the bucket is 25.0 N. (a) Find
  the speed of the bucket. (b) How fast must the
  bucket move at the top of the circle so that the
  rope does not go slack?

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Conceptual Quiz

• Antilock brakes keep the car wheels from locking
  and skidding during a sudden stop. Why does this
  help slow the car down?

A) mk gt ms so sliding friction is better B) mk
gt ms so static friction is better C) ms gt mk so
sliding friction is better D) ms gt mk so static
friction is better E) none of the above
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Conceptual Quiz

• Antilock brakes keep the car wheels from locking
  and skidding during a sudden stop. Why does this
  help slow the car down?

A) mk gt ms so sliding friction is better B) mk
gt ms so static friction is better C) ms gt mk so
sliding friction is better D) ms gt mk so static
friction is better E) none of the above
Static friction is greater than sliding
friction, so by keeping the wheels from skidding,
the static friction force will help slow the car
down more efficiently than the sliding friction
that occurs during a skid.
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Conceptual Quiz
A) component of the gravity force parallel to
the plane increased B) coefficient of static
friction decreased C) normal force exerted by
the board decreased D) both A and C E) all of
A, B, and C

• A box sits on a flat board. You lift one end of
  the board, making an angle with the floor. As
  you increase the angle, the box will eventually
  begin to slide down. Why?

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Conceptual Quiz
A) component of the gravity force parallel to
the plane increased B) coefficient of static
friction decreased C) normal force exerted by
the board decreased D) both A and C E) all of
A, B, and C

• A box sits on a flat board. You lift one end of
  the board, making an angle with the floor. As
  you increase the angle, the box will eventually
  begin to slide down. Why?

As the angle increases, the component of weight
parallel to the plane increases and the component
perpendicular to the plane decreases (and so does
the normal force). Because friction depends on
normal force, we see that the friction force gets
smaller and the force pulling the box down the
plane gets bigger.
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Conceptual Quiz

• A mass m is placed on an inclined plane (m gt 0)
  and slides down the plane with constant speed.
  If a similar block (same m) of mass 2m were
  placed on the same incline, it would

A) not move at all B) slide a bit, slow down,
then stop C) accelerate down the incline D)
slide down at constant speed E) slide up at
constant speed
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Conceptual Quiz

• A mass m is placed on an inclined plane (m gt 0)
  and slides down the plane with constant speed.
  If a similar block (same m) of mass 2m were
  placed on the same incline, it would

A) not move at all B) slide a bit, slow down,
then stop C) accelerate down the incline D)
slide down at constant speed E) slide up at
constant speed
The component of gravity acting down the plane
is double for 2m. However, the normal force (and
hence the friction force) is also double (the
same factor!). This means the two forces still
cancel to give a net force of zero.