10_1425_web_Lec_18_CircularMotion

10_1425_web_Lec_18_CircularMotion - Circular Motion Physics...

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Circular Motion Physics 1425 Lecture 18 Michael Fowler, UVa
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How Far is it Around a Circle? A regular hexagon (6 sides) can be made by putting together 6 equilateral triangles (all sides equal). The radius of the circle = 1. The distance all the way round the hexagon ( red path ) = 6. The distance all the way round the circle ( green path ) is a little more: in fact, it’s 2 π r = 6.283… a
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Arcs Subtending Angles: the Radian It’s 360° all the way round the circle, that’s 60° from each of the equilateral triangles. We say that the arc of circle between A and B “subtends” an angle of 60° at the center of the circle. One radian is defined as the angle subtended by an arc equal in length to the radius of the circle. a A B 60°
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Clicker Question One radian is: A. 60° B. 120° C. A bit less that 60° D. A bit more than 60° E. None of the above
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Full Circle For a circular path of radius r , if you walk a distance r along the path , you have gone around an angle of one radian relative to the center. If you walk all the way around the path, you have of course gone through 360°. BUT you’ve walked a total distance 2 π r , and therefore around an angle of 2 π radians. Conclusion: 360° = 2 π radians
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Radians and Trig Measuring the angle θ in radians , = (length green arc )/ r and sin = (length red line )/ r so for small angles sin a r
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Units for Angular Velocity How fast is something rotating? Car engine: units rpm , revs per minute, redlines around 6,000 rpm or 100 revs/sec. 1 Hertz, written 1Hz , means one cycle/sec, used for electrical generators, circuits. (Often called the rotational frequency , and written f .) Second hand on watch turns at 1 rpm, or 6°/sec.
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10_1425_web_Lec_18_CircularMotion - Circular Motion Physics...

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