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 Rotational Motion
 and
 The Law of Gravity

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 Show an understanding of the centripetal acceleration of objects in
circular motion and be able to apply Newton’s laws to such motion.
 Show an understanding of angular speed and angular acceleration.
 List Kepler’s Laws of Motion and understand them.
 Recognize how Kepler’s Laws resulted in Newton’s Law of Gravitation.
 Be able to calculate periods and velocities of orbiting objects.

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 Understand that gravitational force is proportional to both masses and
the inverse square of the distance between the centers of the masses.

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 Axis of rotation is the center of the disk
 Need a fixed reference line
 During time t, the reference line moves through angle θ

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 Every point on the object undergoes circular motion about the point O
 Angles generally need to be measured in radians
 s is the length of arc and r is the radius

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 The angular displacement is defined as the angle the object rotates
through during some time interval
 Every point on the disc undergoes the same angular displacement in any
given time interval

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 Comparing degrees and radians
 Converting from degrees to radians

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 So, we can say that angle theta is equal to the arc length s divided by
the radius r.
 If the arc length is the entire circumference of the circle, then the
angle in radians is 2pi or 360^{o}.

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 In magnitude, the angular velocity is equal to the linear velocity
divided by the radius of rotation,
 Angular velocity = v/r

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 The average angular speed, ω, of a rotating rigid object is the
ratio of the angular displacement to the time interval

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 The instantaneous angular speed is defined as the limit of the average
speed as the time interval approaches zero
 Units of angular speed are radians/sec
 Speed will be positive if θ is increasing (counterclockwise)
 Speed will be negative if θ is decreasing (clockwise)

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 The average angular acceleration,
,
 of an object is defined as the
ratio of the change in the angular speed to the time it takes for the
object to undergo the change:

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 Units of angular acceleration are rad/s˛
 When a rigid object rotates about a fixed axis, every portion of the
object has the same angular speed and the same angular acceleration

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 Similar to the techniques used in linear motion problems
 With constant angular acceleration, the techniques are much like those
with constant linear acceleration
 There are some differences to keep in mind
 For rotational motion, define a rotational axis
 The object keeps returning to its original orientation, so you can find
the number of revolutions made by the body

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 Displacements
 Speeds
 Accelerations
 Every point on the rotating object has the same angular motion
 Every point on the rotating object does not have the same linear motion

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 An object traveling in a circle, even though it moves with a constant
speed, will have an acceleration
 The centripetal acceleration is due to the change in the direction of
the velocity

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 Centripetal refers to “centerseeking”
 The direction of the velocity changes
 The acceleration is directed toward the center of the circle of motion

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 The angular velocity and the linear velocity are related (v = ωr)
 The centripetal acceleration can also be related to the angular velocity

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 The tangential component of the acceleration is due to changing speed
 The centripetal component of the acceleration is due to changing
direction
 Total acceleration can be found from these components

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 Assign a positive or negative direction in the problem
 A more complete way is by using the right hand rule
 Grasp the axis of rotation with your right hand
 Wrap your fingers in the direction of rotation
 Your thumb points in the direction of ω

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 Newton’s Second Law says that the centripetal acceleration is
accompanied by a force
 F = ma_{C}
 F stands for any force that keeps an object following a circular path
 Tension in a string
 Gravity
 Force of friction

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 Draw a free body diagram, showing and labeling all the forces acting on
the object(s)
 Choose a coordinate system that has one axis perpendicular to the
circular path and the other axis tangent to the circular path

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 Find the net force toward the center of the circular path (this is the
force that causes the centripetal acceleration)
 Solve as in Newton’s second law problems
 The directions will be radial and tangential
 The acceleration will be the centripetal acceleration

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 Many specific situations will use forces that cause centripetal
acceleration
 Level curves
 Banked curves
 Horizontal circles
 Vertical circles

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 Friction is the force that produces the centripetal acceleration
 Can find the frictional force, µ, v

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 A component of the normal force adds to the frictional force to allow
higher speeds

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 The horizontal component of the tension causes the centripetal
acceleration

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 Look at the forces at the top of the circle
 The minimum speed at the top of the circle can be found

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 Distinguish real forces from fictitious forces
 Centrifugal force is a fictitious force
 Real forces always represent interactions between objects

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 Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of the masses and
inversely proportional to the square of the distance between them.

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 G is the constant of universal gravitational
 G = 6.673 x 10^{11} N m˛ /kg˛
 This is an example of an inverse square law

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 Determined experimentally
 Henry Cavendish
 The light beam and mirror serve to amplify the motion

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 Mass of the earth
 Use an example of an object close to the surface of the earth

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 Acceleration due to gravity
 g will vary with altitude

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 PE = mgh is valid only near the
earth’s surface
 For objects high above the earth’s surface, an alternate expression is
needed
 Zero reference level is infinitely far from the earth

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 The escape speed is the speed needed for an object to soar off into
space and not return
 For the earth, v_{esc }is about 11.2 km/s
 Note, v is independent of the mass of the object

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 Johannes Kepler, with his laws of planetary motion greatly influenced
modern physics and astronomy.

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 Johannes Kepler was born on December 27, 1571 in Weil der Stadt,
Germany.
 He was a convinced Platonist, which is to say that he looked for
mathematical relationships in the observable universe.
 Kepler originally intended to become a priest, but was persuaded to take
up a post teaching mathematics at Graz.

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 Kepler attended the University of Tübingen, where his mathematical
ability was noticed by his astronomy teacher Michael Maestlin.
Officially Maestlin taught geocentric (Ptolemaic) astronomy, but able
pupils, among them Kepler, were also introduced to the heliocentric
astronomy of Copernicus (published in 1543).

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 In 1600, Kepler went to Prague, as assistant to Tycho Brahe. Brahe died
in 1601, but Kepler went on to use his observations to calculate
planetary orbits of unprecedented accuracy.
 Kepler used Tycho’s data to formulate his three laws of planetary
motion.

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 1^{st} Law. The orbits of planets are ellipses with the Sun at
one foci.
 2^{nd} Law. A planet moves faster as it gets closer to the Sun.
 3^{rd} Law. Relates the orbit period to the distance from the
Sun: p^{2} (in earth years) = a^{3 } (in astronomical units).
 One astronomical unit= 9.3 x 10^{7 }miles

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 Kepler believed this revealed the harmonies present in the construction
of the universe.
 He noticed the radii of the orbits of planets could be argued to form a
pattern like the five regular solids.

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 Newton made corrections to Kepler’s 1st and 3rd law.
 So Kepler’s 1st law becomes: The orbit of a planet around the Sun is an
ellipse, with the common center of mass of the planet and the Sun at one
focus.

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