Transcript circular motion monster review

Going in circles
a
v
Why is circular motion cool?
you get accelerated! (due to change in direction)
Circular Motion in Our Daily Lives
Driving around curves & banks
Amusement park rides (loops & circles)
Weather patterns
(jet streams, coriolis effect)
Horizontal Circles (Rotor)
Friction between
Bart and wall
wall pushing
in on Bart
Bart’s
weight
The inward wall force keeps Bart in the circle.
Friction keeps him from falling down.
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
Spring 2008
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Angled turns –
wings provide
centripetal force
feel heavier if go faster in a
tighter turn
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- )
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!
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
2
v
aC =
R
• V is the tangential velocity
(constant number with changing direction)
• F= ma is now…… F = mv2/r
Wide turns and tight turns
little R
big R
for the same
speed, the tighter
turn requires more
acceleration
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.
Applying Newton’s 2nd Law:
F  ma
mv
F
r
2
Centripetal Force
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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What’s this Centrifugal force ? ?
object on
the dashboard
straight line
object naturally
follows
• 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!
Work Done by the Centripetal
Force
• Since the centripetal force on an object is
always perpendicular to the object’s
velocity, the centripetal force never does
work on the object - no energy is
transformed.
• W= Fd cos(90)=0
Fcent
v
Direction of Centripetal Force,
Acceleration and Velocity
With a centripetal
force, an object in
motion continues along
a straight-line path.
Without a centripetal
force, an object in
motion continues along
a straight-line path.
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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 = u*mg = mv2/r
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Gravity Can Yield a Centripetal Acceleration:
Hubble Space Telescope
orbits at an altitude of 598 km
(height above Earth’s 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
(doesn’t matter where)
Like center of a vertical circle
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Vertical Circular Motion
(normal force varies)
Top: mg + normal = mv2/r
side: normal = mv2/r
(normal smallest, v same)
(weight not centripetal, v same)
bottom: normal - mg = mv2/r
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(normal largest, v same)
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 answer is E. As the
A. the same as
velocity increases the
B. one fourth of
centripetal force required to
maintain the circle increases
C. half of
as the square of the speed.
D. twice
E. four times
the force required to keep the slower object on the path.24
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
A. the same as
The answer is D. The centripetal force needed
B. one fourth of to maintain the circular motion of an object is
inversely proportional to the radius of the circle.
C. half of
Everybody knows that it is harder to navigate a
D. twice
sharp turn than a wide turn.
E. four times
the force required to keep the object on the larger
path.
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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
A. the same as
B. one fourth of
Answer: D.The mass is directly
C. half of
proportional to centripetal force.
D. twice
E. four times
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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.
Universal Gravitation
 Newton’s
idea was that gravity was a
universal force acting between any
two objects.
At the Earth’s Surface
 Newton
knew that the gravitational
force on the apple equals the apple’s
weight, mg, where g = 9.8 m/s2.
W = mg
Weight of the Moon
 Newton
reasoned that the centripetal
force on the moon was also supplied by
the Earth’s gravitational force.
?
Fc = mg
Law of Universal Gravitation
symbols, Newton’s Law of
Universal Gravitation is:
 In
 Fgrav
= ma = G Mm
 Where
r2
G is a constant of proportionality.
 G = 6.67 x 10-11 N m2/kg2
An Inverse-Square Force
Gravitational Field Strength
(acceleration)
 Near
the surface of the Earth,
g = F/m = 9.8 N/kg = 9.8 m/s2.
general, g = GM/r2, where M is the
mass of the object creating the field, r is
the distance from the object’s center,
and G = 6.67 x10-11 Nm2/kg2.
 In
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).
Gravitational Field Inside a
Planet
 The
blue-shaded part
of the planet pulls you
toward point C.
 The grey-shaded part
of the planet does
not pull you at all.
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.
Earth’s Tides
2
high tides and 2 low tides per day.
 The tides follow the Moon.
 Differences due to sun not signficant
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 it’s much further away
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Why Two Tides?
 Remember
that