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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
r
q
r
s
Angle = arc length / radius
q
from one complete rotation = 360o = 2p rad
45o = p/4 rad
90o = p/2 rad
180o = p rad
1 rad ≈ 57.30o
Phys 250 Ch5 p1
s
r
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.
Phys 250 Ch5 p2
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.
v
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.
Phys 250 Ch5 p3
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
v2
ac 
r
mv 2
Fc  mac 
r
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
f=1/T
Phys 250 Ch5 p4
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?
Phys 250 Ch5 p5
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.
Phys 250 Ch5 p6
Angular velocity
an object which rotates an angle q in a time t has an average angular velocity w :
Dq
w 
Dt
usually rad/s but sometime rpm, rps
For a particle traveling in uniform circular motion
w
v
r
or v  w r
and
w  2pf
so
a c  w 2r
Phys 250 Ch5 p7
r
q
s
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?
Phys 250 Ch5 p8
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 earth’s 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?
Phys 250 Ch5 p9
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?
Phys 250 Ch5 p10
Gravitation
History: Kepler’s Laws
T2
k
ellipses, equal areas and period-distance
R3
pick units for T: earth years
pick units for R: Astronomical Units (AU) = Earth’s orbit radius about the sun
k = 1
Gravitation as a force
A fundamental force of nature
(electromagnetism, weak nuclear force, strong nuclear force)
Newton’s 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
Fgrav  G
m A mB
r2
G  6.67  10
11
Nm 2
kg 2
The force between two 1 kg masses separated by 1m is 6.67x10-11 N ~ 1.5x10-11 lb
Phys 250 Ch5 p11
Example: What is the force of gravity exerted by a 1 kg object on the surface of the earth?
Earth’s mass: 5.98E24 kg
Earth’s radius: 6.38E6m
Example: Show that Kepler’s third law follows from Universal GravitationC
Phys 250 Ch5 p12
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 earth’s surface?
Example: What would be the radius of a satellite in geosynchronous orbit?
Phys 250 Ch5 p13