Going in circles

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Going in circles
a
v
Why is circular motion cool?
you get accelerated! (due to change in direction)
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Circular Motion in Our Daily Lives

• Driving around curves banks
• Amusement park rides (loops circles)
• Weather patterns (jet streams, coriolis effect)

3
Horizontal Circles (Rotor)
Friction between Bart and wall
wall pushing in on Bart
Barts weight
The inward wall force keeps Bart in the
circle. Friction keeps him from falling down.
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vertical circles

• Track provides centripetal force
• You feel heavier at bottom since larger
  centripetal force needed to battle gravity
• You feel lighter on top since gravity helps the
  track push you down

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Angled turns wings provide centripetal
forcefeel heavier if go faster in a tighter turn
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• Earth rotates in a tilted circle
• -high speed (800 mph), but small
  acceleration
• (adds
  .1 extra gravity)
• -west
  to east motion
• curves
  south
  (Coriolis effect- )

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Uniform circular motion
The speed stays constant, but the direction
changes
a
v
R
The acceleration in this case is
called centripetal acceleration, pointed toward
the center!
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Uniform Circular Motion Period
The time it takes to travel one cycle is the
period .

• Distance circumference 2pr
• Velocity distance / time
• Period time for one circle

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Centripetal acceleration

• centripetal acceleration
• V is the tangential velocity(constant number
  with changing direction)
• F ma is now F mv2/r

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Wide turns and tight turns
little R
big R
for the same speed, the tighter turn requires
more acceleration
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Example

• What is the tension in a string used to twirl a
  0.3 kg ball at a speed of 2 m/s in a circle of 1
  meter radius?
• Force mass x acceleration m ? aC
• acceleration aC v2 / R (2 m/s)2/ 1 m
• 4 m/s2
• force m aC 0.3 ? 4 1.2 N
• If the string is not strong enough to handle this
  tension it will break and the ball goes off in a
  straight line.

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Applying Newtons 2nd Law
Always points toward center of circle. (Always
changing direction!)
Centripetal force is the magnitude of the force
required to maintain uniform circular motion.
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Examples of centripetal force

• Tension- ball on a string
• Gravity- planet motion
• Friction- cars
• Normal Force- coasters banked cars

Centripetal force is NOT a new force. It is
simply a way of quantifying the magnitude of the
force required to maintain a certain speed around
a circular path of a certain radius.
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Whats this Centrifugal force ? ?

• The red object will make the turn only if there
  is enough friction on it
• otherwise it goes straight
• the apparent outward force is called the
  centrifugal force
• it is NOT A REAL force!
• an object will not move in a circle until
  something makes it!

object on the dashboard
straight line object naturally follows
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Work Done by the Centripetal Force

• Since the centripetal force on an object is
  always perpendicular to the objects velocity,
  the centripetal force never does work on the
  object - no energy is transformed.
• W Fd cos(90)0

Fcent
v
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Direction of Centripetal Force, Acceleration and
Velocity
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Tension Can Yield a Centripetal Acceleration
If the person doubles the speed of the airplane,
what happens to the tension in the cable? F
Tension mv2/r
Doubling the speed, quadruples the force (i.e.
tension) to keep the plane in uniform circular
motion.
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Friction Can Yield a Centripetal Acceleration
F friction umg mv2/r
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Gravity Can Yield a Centripetal Acceleration
Hubble Space Telescope orbits at an altitude of
598 km (height above Earths surface). What is
its orbital speed?
F mMG/r2 mv2/r
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Banked Curves
Why exit ramps in highways are banked?
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Artifical Gravity
F Normal force mv2/r If v2/r 9.8, seems
like earth!
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horizontal Circular Motion(normal force always
same)
F Normal force mv2/r (doesnt matter
where) Like center of a vertical circle
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Vertical Circular Motion(normal force varies)
Top mg normal mv2/r (normal
smallest, v same) side normal mv2/r
(weight not centripetal, v same) bottom
normal - mg mv2/r (normal largest, v
same)
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Relationship Between Variables of Uniform
Circular Motion

• Suppose two identical objects go around in
  horizontal circles of identical diameter but one
  object goes around the circle twice as fast as
  the other. The force required to keep the faster
  object on the circular path is
• the same as
• one fourth of
• half of
• twice
• four times
• the force required to keep the slower object on
  the path.

The answer is E. As the velocity increases the
centripetal force required to maintain the circle
increases as the square of the speed.
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Relationship Between Variables of Uniform
Circular Motion

• Suppose two identical objects go around in
  horizontal circles with the same speed. The
  diameter of one circle is half of the diameter of
  the other. The force required to keep the object
  on the smaller circular path is
• the same as
• one fourth of
• half of
• twice
• four times
• the force required to keep the object on the
  larger path.

The answer is D. The centripetal force needed to
maintain the circular motion of an object is
inversely proportional to the radius of the
circle. Everybody knows that it is harder to
navigate a sharp turn than a wide turn.
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Relationship Between Variables of Uniform
Circular Motion

• Suppose two identical objects go around in
  horizontal circles of identical diameter and
  speed but one object has twice the mass of the
  other. The force required to keep the more
  massive object on the circular path is
• the same as
• one fourth of
• half of
• twice
• four times

Answer D.The mass is directly proportional to
centripetal force.
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The Apple the Moon

• Isaac Newton realized that the motion of a
  falling apple and the motion of the Moon were
  both actually the same motion, caused by the same
  force - the gravitational force.

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Universal Gravitation

• Newtons idea was that gravity was a universal
  force acting between any two objects.

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At the Earths Surface

• Newton knew that the gravitational force on the
  apple equals the apples weight, mg, where g
  9.8 m/s2.

W mg
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Weight of the Moon

• Newton reasoned that the centripetal force on the
  moon was also supplied by the Earths
  gravitational force.

?
Fc mg
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Law of Universal Gravitation

• In symbols, Newtons Law of Universal Gravitation
  is
• Fgrav ma G
• Where G is a constant of proportionality.
• G 6.67 x 10-11 N m2/kg2

Mm
r 2
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An Inverse-Square Force
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Gravitational Field Strength(acceleration)

• Near the surface of the Earth, g F/m 9.8
  N/kg 9.8 m/s2.
• In general, g GM/r2, where M is the mass of the
  object creating the field, r is the distance from
  the objects center, and G 6.67 x10-11 Nm2/kg2.

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Gravitational Force

• If g is the strength of the gravitational field
  at some point, then the gravitational force on an
  object of mass m at that point is Fgrav mg.
• If g is the gravitational field strength at some
  point (in N/kg), then the free fall acceleration
  at that point is also g (in m/s2).

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Gravitational Field Inside a Planet

• The blue-shaded partof the planet pulls
  youtoward point C.
• The grey-shaded partof the planet does not pull
  you at all.

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Black Holes

• When a very massive star gets old and runs out of
  fusionable material, gravitational forces may
  cause it to collapse to a mathematical point - a
  singularity. All normal matter is crushed out of
  existence. This is a black hole.

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Earths Tides

• 2 high tides and 2 low tides per day.
• The tides follow the Moon.
• Differences due to sun not signficant

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Why Two Tides?

• Tides due to stretching of a planet.
• Stretching due to difference in forces on the
  two sides of an object.
• Since gravitational force depends on distance,
  there is more gravitational force on the side of
  Earth closest to the Moon and less gravitational
  force on the side of Earth farther from the Moon.
  Not much difference from the Sun since its much
  further awayI

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Why Two Tides?

• Remember that