Chapter 6: Uniform Circular Motion

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Chapter 6 Uniform Circular Motion

• Introductory Formulas

v w r w 2pf
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Introductory Formulas(2)

• s arc length (m)
• r radius (m)
• q angular displacement (radians rad)
• w angular velocity (rad/s)
• v linear (tangential) velocity (m/s)
• f frequency (Hertz Hz rev/s s-1)
• T orbital period (s)
• NOTE rad m/m and therefore rad are
  dimensionless

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Derivation of Centripetal Force
r r (coswt i sinwt j ) v dr/dt wr
(-sinwt i coswt j ) a dv/dt -w2r (coswt i
sinwt j ) a -w2r
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Centripetal Force(2)

• Therefore from previous slide

Note This acceleration vector always points
toward the center of the circle and so is called
centripetal acceleration . If we multiple
this centripetal accel. by mass , we get
centripetal force, by Newtons 2nd Law .
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Centripetal Force(3)

• To solve circular motion problems, always
• realize that we must set some physical
• force (like gravity or friction) equal to the
• centripetal force.
• F (centripetal) F (physical)
• See active Figure 6.2 at pse6.com

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Example centripetal force friction
Fc f
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Vertical Circle
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Vertical Circle (2)
See Active Figure 6.8 at pse6.com
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Vertical Circle (3)

• PROBLEM A PAIL OF WATER IS SPUN AROUND ON A
  STRING OF LENGTH 1 METER. CALC. THE MINIMUM
  VELOCITY AT THE TOP SO WATER DOESNT SPILL.

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Rotating Space Station Artificial Gravity

• In 1974, Gerard ONeill, a physicist, wrote a
  detailed engineering plan for a series of
  rotating space stations where thousands of people
  could live. NASA actually built the current
  space station that doesnt rotate. The advantage
  of rotation is that then people can experience
  artificial gravity due to the centripetal
  acceleration
• g w2r

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Student Problem

• A CHILD m 30 kg RIDES ON A FERRIS WHEEL AT
  VELOCITY v 6 m/s . THE WHEEL HAS RADIUS r 20
  m. CALC. THE NORMAL FORCE EXERTED BY THE SEAT AT
  THE TOP OF THE WHEEL, IN NEWTONS.

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Newtons Law of Universal Gravitation
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Example gravity force

• W mg
• Calc. weight 50 kg person
• W 50X9.8 490N

NOTE THEY ARE THE SAME!
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Acceleration due to gravity on a planet
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Orbital velocity
Is above speed concerning the first satellite,
Sputnik, correct?
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Explorer I, first USA satellite

• Celebrating Explorer I (January 31, 1958)
• This is a photo of (from left to right) William
  Pickering of Jet Propulsion Laboratory, James Van
  Allen of the State University of Iowa, and Wehner
  Von Braun proclaiming the launch of Explorer I.
• Explorer I discovers Van Allen radiation belts in
  1958

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Keplers 3rd Law
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Example geosynchronous comsat
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P13.56 Vanguard I
This is the oldest artificial satellite still in
orbit launched March 17, 1958. Perigee
4400 miles Apogee 6500 miles Semi-major axis
5500 miles Orbital period T 130 min. 2 hr.