P1252109402JXTfC

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Uniform Circular Motion and Gravitation
Rotational Motion in close analogy with linear
motion (distance/displacement, velocity,
acceleration) Angular measure in natural
units Angles and Rotation in radians
from one complete rotation 360o 2p rad 45o
p/4 rad 90o p/2 rad 180o p rad 1 rad 57.30o
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Example The ancient Greek Eratosthenes new that
when the sun was directly overhead in Syene, the
sun was about 7 degrees from overhead in
Alexandria. (see astro 03f14.jpg) Using the
known distance between the cities, he was able to
determine the radius of the Earth. Using the
distance of 770 km between these cities,
calculate the radius of the earth.
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v
A curved path requires an inward force Center
seeking Centripetal Centripetal force is the
force perpendicular to the velocity of an object
moving along a curved path. The centripetal force
is directed toward the center of curvature of the
path.
Dv
Dv
Dv aDt (F/m)dt
examples ball on a string, car rounding a
corner. Centrifugal Effect the fictitious
force felt by an object when the frame of
reference moves along (and therefore accelerates)
along a curved path. This effect is simply
inertia. Stop the force and the object will
undergo straight line motion.
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Uniform Circular Motion motion in a circle at
constant speed centripetal force Fc and
centripetal acceleration ac is always directed
towards the center centripetal force and
acceleration have constant magnitudes
the period T of the motion is the time to make
one orbit the frequency f is the number of
complete revolutions per unit time f1/T
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Example A bicycle racer rides with a constant
speed around a circular track 25 m in diameter.
What is the centripetal acceleration of the
bicycle if its speed is 6.0 m/s?
Example A grinding wheel with a 25.4 cm diameter
spins at a rate of 1910 revolutions per minute.
What is the linear speed of a poihnt on the rim?
What is the acceleration of a point on the rim?
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Example What is the centripetal acceleration of
the moon as it circles the earth? Its orbital
period is 27.3 days and its orbital radius is
3.84E8 m.
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Angular velocity an object which rotates an angle
q in a time t has an average angular velocity w
usually rad/s but sometime rpm, rps For a
particle traveling in uniform circular motion
s
r
q
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Example An amusement park ride carries
passengers in a circular path 7.70 m in radius.
The ride makes a complete rotation every 4.00 s.
What is the angular velocity of the passengers?
What is the angular acceleration of the
passengers?
Example A student ties a 0,060 kg lead fishing
weight to the end of a string and whirls it
around his head in a horizontal circle. if the
radius of the circle is 0.30 m and the object
moves with a speed of 2.0 m/s, what is the
horizontal component of the force that keeps the
string in circular motion? What is the tension
in the string?
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Example A space station is to consist of a torus
with an outside diameter of 1.5 km. What period
of rotation must the space station have in order
to simulate earths gravity?
Example A centrifuge separates blood cells from
from blood plasma by rotating a tube at 55
rotations per second. What is the acceleration
at the center of a centrifuge tube 8.0 cm from
the axis of rotation?
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Example A race track designed for average speeds
of 240 km/hr (66.7 m/s) is to have a turn with a
radius of 975 m. To what angle must the track be
banked so that cars traveling at the design speed
have no tendency to slip sideways?
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Gravitation History Keplers Laws ellipses,
equal areas and period-distance pick units for T
earth years pick units for R Astronomical Units
(AU) Earths orbit radius about the sun ?k
1 Gravitation as a force A fundamental force of
nature (electromagnetism, weak nuclear force,
strong nuclear force) Newtons law of universal
gravitation All objects interact by virtue of
having mass Force is proportional to each
mass Force is inversely proportional to the
square of the distance
The force between two 1 kg masses separated by 1m
is 6.67x10-11 N 1.5x10-11 lb
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Example What is the force of gravity exerted by
a 1 kg object on the surface of the earth?
Earths mass 5.98E24 kg Earths radius 6.38E6m
Example Show that Keplers third law follows
from Universal GravitationC
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Example omitted We can use our knowledge about g
and G and Universal Gravitation to determine the
mass of the earth, and from there its average
density
Example What would be the period of an
artificial satellite orbiting just above the
earths surface?
Example What would be the radius of a satellite
in geosynchronous orbit?