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Unformatted text preview: 1 February 28, 2012 Physics for Scientists&Engineers 1 1 Physics for Scientists & Engineers 1 Physics for Scientists & Engineers 1 Spring Semester 2012 Lecture 23 Circular Motion, Angular Velocity (with a little help from Jack Sparrow) February 28, 2012 Physics for Scientists&Engineers 1 2 Circular Motion Circular Motion Circular motion is motion along the perimeter of a circle Circular motion is surprisingly common • CD, DVD, Bluray, Indycar racing, carousel, ferris wheel, etc. February 28, 2012 Physics for Scientists&Engineers 1 3 Circular Motion Circular Motion February 28, 2012 Physics for Scientists&Engineers 1 4 Polar Coordinates Polar Coordinates During an object’s circular motion, its x and y coordinates change continuously, but the distance from the object to the center of the circular path stays the same We can take advantage of this fact by using polar coordinates The position vector of an object in circular motion changes as a function of time but its tip always moves on the circumference of a circle February 28, 2012 Physics for Scientists&Engineers 1 5 Polar Coordinates Polar Coordinates We can specify the position vector by giving its x and ycomponents We can also specify the same vector by giving two other numbers, r and θ The relationship between Cartesian coordinates and polar coordinates is r = x 2 + y 2 θ = tan − 1 y x x = r cos θ y = r sin θ February 28, 2012 Physics for Scientists&Engineers 1 6 Polar Coordinates Polar Coordinates Using polar coordinates to describe circular motion reduces twodimension motion on the circumference of a circle to one dimension motion involving θ In the figure, two unit vectors are shown The radial unit vector can be written as radial unit vector ˆ r tangential unit vector ˆ t ˆ r = x r ˆ x + y r ˆ y = cos θ ( ) ˆ x + sin θ ( ) ˆ y = cos θ ,sin θ ( ) 2 February 28, 2012 Physics for Scientists&Engineers 1 7 Polar Coordinates Polar Coordinates The tangential unit vector is The radial and tangential unit vectors are perpendicular to each other These two unit vectors have a length of 1 ˆ t = − y r ˆ x + x r ˆ y = − sin θ ( ) ˆ x + cos θ ( ) ˆ y = − sin θ ,cos θ ( ) ˆ r • ˆ t = ˆ r • ˆ r = cos θ ,sin θ ( ) • cos θ ,sin θ ( ) = cos 2 θ + sin 2 θ = 1 ˆ t • ˆ t = − sin θ ,cos θ ( ) • − sin θ ,cos θ ( ) = sin 2 θ + cos 2 θ = 1 Angular Coordinates and Angular Displacement Angular Coordinates and Angular Displacement Polar coordinates allow us to describe and analyze circular motion, where the distance to the origin of the object in motion stays constant and the angle θ varies as a function of time The angle θ is measured with respect to the positive xaxis A move in the counterclockwise direction away from the positive...
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This note was uploaded on 04/02/2012 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.
 Spring '08
 Wolf
 Physics, Circular Motion

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