9/22/2011 - 1 - Uniform Circular Motion 1.0 Introduction Special cases often dominate our study of physics, and circular motion is certainly no exception. We see circular motion in many instances in the world; a bicycle rider on a circular track, a ball spun around by a string, and the rotation of a spinning wheel are just a few examples. Various planetary models described the motion of planets in circles before any understanding of gravitation. The motion of the moon around the earth is nearly circular. The motions of the planets around the sun are nearly circular. Our sun moves in nearly a circular orbit about the center of our galaxy, 50,000 light years from a massive black hole at the center of the galaxy. We shall describe the kinematics of circular motion, the position, velocity, and acceleration, as a special case of two-dimensional motion. We will see that unlike linear motion, where velocity and acceleration are directed along the line of motion, in circular motion the direction of velocity is always tangent to the circle. This means that as the object moves in a circle, the direction of the velocity is always changing. When we examine this motion, we shall see that the direction of change of the velocity is towards the center of the circle. This means that there is a non-zero component of the acceleration directed radially inward, which is called the centripetal acceleration. If our object is increasing its speed or slowing down, there is also a non-zero tangential acceleration in the direction of motion. But when the object is moving at a constant speed in a circle then only the centripetal acceleration is non-zero. In all of these instances, when an object is constrained to move in a circle, there must exist a force F! acting on the object directed towards the center. In 1666, twenty years before Newton published his Principia, he realized that the moon is always “falling” towards the center of the earth; otherwise, by the First Law, it would continue in some linear trajectory rather than follow a circular orbit. Therefore there must be a centripetal force, a radial force pointing inward, producing this centripetal acceleration. Since Newton’s Second Law m=F a!! is a vector equality, it can be applied to the radial direction to yield radial radialF m a=. (1.1) 2.0 Cylindrical Coordinate System We first choose an origin and an axis we call the z-axis with unit vector ˆz pointing in the increasing z-direction. The level surface of points such that z = zP define a plane. We shall choose coordinates for a point P in the plane z = zPas follows.9/22/2011 - 2 - One coordinate, r, measures the distance from the z-axis to the point P. The coordinate r ranges in value from 0 ! r ! ". In Figure 2.1 we draw a few surfaces that have constant values of r. These `level surfaces’ are circles. Figure 2.1 level surfaces for the coordinate r Our second coordinate measures an angular distance along the circle. We need to choose some reference point to define the angle coordinate. We choose a ‘reference ray’, a horizontal ray starting from the origin and extending to +! along the horizontal direction to the right. (In a typical Cartesian coordinate system, our ‘reference ray’ is the positive x-direction). We define the angle coordinate for the point P as follows. We draw a ray from the origin to the point P. We define the angle ! as the angle in the counterclockwise direction between our horizontal reference ray and the ray from the origin to the point P, (see Figure 2.2): Figure 2.2 the angle coordinate All the other points that lie on a ray from the origin to infinity passing through P have the same value as !. For any arbitrary point, our angle coordinate ! can take on values from 0 !"< 2#. In Figure 2.3 we depict other `level surfaces’ which are lines in the plane for the angle coordinate. The coordinates (r,!) in the plane z = zP are called polar coordinates.9/22/2011 - 3 - Figure 2.3 Level surfaces for the angle coordinate Unit Vectors: We choose two unit vectors in the plane at the point P as follows. We choose ˆr to point in the direction of increasing r, radially away from the z-axis. We choose ˆ! to point in the direction of increasing !. This unit vector points in the counterclockwise direction, tangent to the circle. Our complete coordinate system is shown in Figure 2.4. This coordinate system is called a ‘cylindrical coordinate system’. Essentially we have chosen two directions, radial and tangential in the plane and a perpendicular direction to the plane. Figure 2.4 Cylindrical coordinates If you are given polar coordinates (r,!) of a point in the plane, the Cartesian coordinates (x, y)can be determined from the coordinate transformations x = r cos! (2.1) y = r sin! (2.2)9/22/2011 - 4 - Conversely, if you are given the Cartesian coordinates (x, y), the polar coordinates (r,!) can be determined from the coordinate transformations r = +(x2+ y2)1 2 (2.3) 1tan ( / )y x!"= (2.4) Note that r ! 0so you always need to take the positive square root. Note also that tan!= tan(!+"). Suppose that 0 / 2! "# #, then x ! 0 and y ! 0. Then the point (! x,! y)will correspond to the angle !+". The unit vectors also are related by the coordinate transformations ˆ ˆˆcos sinr i j! != + (2.5) ˆ ˆ ˆsin cos! ! != " +i j (2.6) Similarly ˆˆˆcos sini r! ! != " (2.7) ˆˆˆsin cosj r! ! != + (2.8) One crucial difference between polar coordinates and Cartesian coordinates involves the choice of unit vectors. Suppose we consider a different point S in the plane. The unit vectors in Cartesian coordinates ˆ ˆ( , )S Si j at the point S have the same magnitude and point in the same direction as the unit vectors ˆ ˆ( , )P Pi j at P. Any two vectors that are equal in magnitude and point in the same direction are equal; therefore ˆ ˆ ˆ ˆ,S P S P= =i i j j (2.9) A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. In polar coordinates, the unit vectors at two
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