chapter_7 - Astronauts F. Story Musgrave and Jeffrey A....

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189 7 CHAPTER Rotational Motion and the Law of Gravity 7.1 Angular Speed and Angular Acceleration 7.2 Rotational Motion under Constant Angular Acceleration 7.3 Relations between Angular and Linear Quantities 7.4 Centripetal Acceleration 7.5 Newtonian Gravitation 7.6 Kepler’s Laws Courtesy NASA Rotational motion is an important part of everyday life. The rotation of the Earth creates the cycle of day and night, the rotation of wheels enables easy vehicular motion, and modern technology depends on circular motion in a variety of contexts, from the tiny gears in a Swiss watch to the operation of lathes and other machinery. The concepts of angular speed, angu- lar acceleration , and centripetal acceleration are central to understanding the motions of a di- verse range of phenomena, from a car moving around a circular race track to clusters of galaxies orbiting a common center. Rotational motion, when combined with Newton’s law of universal gravitation and his laws of motion, can also explain certain facts about space travel and satellite motion, such as where to place a satellite so it will remain Fxed in position over the same spot on the Earth. The generalization of gravitational potential energy and energy conservation offers an easy route to such results as planetary escape speed. ±inally, we present Kepler’s three laws of planetary motion, which formed the foundation of Newton’s approach to gravity. 7.1 ANGULAR SPEED AND ANGULAR ACCELERATION In the study of linear motion, the important concepts are displacement D x , velocity v , and acceleration a . Each of these concepts has its analog in rotational motion: angu- lar displacement D u , angular velocity v , and angular acceleration a . The radian , a unit of angular measure, is essential to the understanding of these concepts. Recall that the distance s around a circle is given by s ± 2 p r , where r is Astronauts F. Story Musgrave and Jeffrey A. Hoffman, along with the Hubble Space Telescope and the Space Shuttle Endeavor, are all “falling” around Earth.
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190 Chapter 7 Rotational Motion and the Law of Gravity the radius of the circle. Dividing both sides by r results in s/r ± 2 p . This quantity is dimensionless, because both s and r have dimensions of length, but the value 2 corresponds to a displacement around a circle. A half circle would give an answer of , a quarter circle an answer of /2. The numbers 2 , , and /2 corre- spond to angles of 360 8 , 180 8 , and 90 8 , respectively, so a new unit of angular measure, the radian , can be deFned as the arc length s along a circle divided by the radius r : [7.1] ±igure 7.1 illustrates the size of 1 radian, which is approximately 53 8 . ±or conver- sions, we use the fact that 360 2 radians (or 180 radians). ±or example, 45 8 (2 rad/360 8 ) ± ( /4) rad. Generally, angular quantities in physics must be expressed in radians. Be sure to set your calculator to radian mode; neglecting to do this is a common error.
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This note was uploaded on 09/14/2009 for the course PHY 303K taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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chapter_7 - Astronauts F. Story Musgrave and Jeffrey A....

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