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© 2004 Penn State University Physics 211R: Lab – The Kinematics and Dynamics of Circular and Rotational Motion Physics 211R: Lab The Kinematics and Dynamics of Circular and Rotational Motion Reading Assignment: Chapter 6 – Sections 2, 3, and 5 Chapter 10 – Sections 2-5 Introduction: When discussing motion, it is important to be aware that there are several different types and that each type is dependent upon a particular frame of reference. Linear motion is the most basic form, involving the translation of a particle from one point to another in one dimension. Projectile motion, although two-dimensional, is easily analyzed by separating it into its linear components. Circular motion is a special case of two-dimensional motion in which an object translates in a circular path. Rotational motion is observed when an object itself rotates about some internal axis. Consider the following examples: • A car traveling along a straight road exhibits linear motion from the perspective of a person standing on the side of the road. The wheels of the car exhibit rotational motion about the axle. A pebble embedded in the tread of the car exhibits circular motion about the axle. • A baseball spins towards a batter at home plate. From the perspective of a spectator, the ball is undergoing translational motion across the infield. However, the ball itself is rotating about its own axis. The stitches of thread on the outside of the ball are moving in a circle relative to the spinning axis of the ball. (These circles differ depending upon the location of the particular stitch.) • The earth rotates on its own axis as it also moves in a circle from the perspective of a stationary sun. In addition, people located at the surface of the earth undergo a different circular motion each day around the axis of rotation of the earth. (These circles differ depending upon the latitude of each person.) It is important to notice that circular motion connects the concepts of linear and rotational motion. For any object that is rotating, a particular point on that object is moving in a circle. One of the goals of this lab activity is to explore and understand this connection. The translational motion of any point particle can be described in terms of standard Cartesian coordinates. In other words, Cartesian coordinates can describe both linear and circular motion. However, in the case of circular motion, the particle moves an arc length, s, around a circle with a (constant) radius, r. Therefore, in this case it is simpler to use Polar coordinates where the position of the particle can be specified by r and the angular position, θ, rather than x and y.© 2004 Penn State University Physics 211R: Lab – The Kinematics and Dynamics of Circular and Rotational Motion Notice that the right triangle defines the relationship between the location of the particle in Cartesian and Polar coordinates. The following equations describe how to transform between Polar and Cartesian coordinates. θcosrx= θsinry= 22yxr += rsyx==−1tanθ The radius of a particle undergoing circular motion is always a constant. The angular position, however, will change with time depending on the motion of the particle. Since only the angular position changes with time its behavior is exactly analogous to the behavior of the position in one-dimensional motion that was studied previously. Thus, the angular equivalent of the kinematic quantities for one-dimensional motion can be defined as follows: Arc distance traveled rss =⇒θ Angular position Linear (tangential) velocity rvdtddtdsv ==⇒=θω Angular velocity Linear (tangential) acceleration radtddtdvatt==⇒=ωα Angular acceleration The relationships between the angular position, velocity, and acceleration are exactly the same as the relationships previously determined for one-dimensional motion. For example, for motion with constant angular velocity, ω: otθωθ+= For motion with constant angular acceleration, α : otωαω+= oottθωαθ++=221 The Net Force that causes an object of mass, m, to move in a circular path is called the centripetal force, Fc. At a particular constant linear speed, v, the following equation (Newton’s Second Law) describes the dynamics of the object’s circular motion: ccamFrr= S (x, y) = (r cosθ , r sinθ) X Y r θ© 2004 Penn State University Physics 211R: Lab – The Kinematics and Dynamics of Circular and Rotational Motion where the magnitude of the centripetal acceleration is given by: rvac2= For circular motion that is not uniform, the total acceleration (magnitude and direction) of the object, at every moment, is determined by adding the centripetal (radial component) and linear (tangential component) accelerations together. Note: ac and at are vectors that form a right triangle when added together because they are perpendicular to each other. tctotalaaarrr+= Remember that tangential acceleration is a consequence of any change in the linear (and therefore, angular) speed of the object. Centripetal acceleration is a consequence of the rate at which the direction of the object changes at every moment. The centripetal force, the component of the net force directed towards the center of the circle, is caused by different types of forces. Here are some examples for the case of a single force causing circular motion: • Tension in a string (such as when a ball is whirled in a horizontal circle at the end of a rope) • The normal force (such as on the Rotor ride found at many amusement parks) • Gravity (such as the orbits of planets around the sun) • Static friction (such as a car traveling around a level curve) Here are some examples for the case of two forces in combination causing circular motion: • Tension and gravity (such as when a ball is whirled in a vertical circle at the end of a rope) • The normal force and gravity (such as when a person rides a vertical loop on a

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