rotation around a fixed axis

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to: (C) analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples. (D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects. (4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to: (D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects. Students may get confused between deceleration and increasing acceleration in the negative direction. In the section on uniform circular motion, we discussed motion in a circle at constant speed and, therefore, constant angular velocity. However, there are times when angular velocity is not constant—rotational motion can speed up, slow down, or reverse directions. Angular velocity is not constant when a spinning skater pulls in her arms, when a child pushes a merry-go-round to make it rotate, or when a CD slows to a halt when switched off. In all these cases, angular acceleration occurs because the angular velocity 𝛚ωω changes. The faster the change occurs, the greater is the angular acceleration. Angular acceleration 𝛂αα is the rate of change of angular velocity. In equation form, average angular acceleration is These equations mean that the magnitudes of tangential acceleration and angular acceleration are directly proportional to each other. The greater the angular acceleration, the larger the change in tangential acceleration, and vice versa. For example, consider riders in their pods on a Ferris wheel at rest. A Ferris wheel with greater angular acceleration will give the riders greater tangential acceleration because, as the Ferris wheel increases its rate of spinning, it also increases its tangential velocity. Note that the radius of the spinning object also matters. For example, for a given angular acceleration 𝛂αα, a smaller Ferris wheel leads to a smaller tangential acceleration for the riders. So far, we have defined three rotational variables: 𝜃θθ, ωωω, and ααα. These are the angular versions of the linear variables x, v, and a. The following equations in the table represent the magnitude of the rotational variables and only when the radius is constant and perpendicular to the rotational variable. Table 6.2 shows how they are related. The kinematics of rotational motion describes the relationships between the angle of rotation, angular velocity, angular acceleration, and time. It only describes motion—it does not include any forces or masses that may affect rotation (these are part of dynamics). Recall the kinematics equation for linear motion: 𝐯 =𝐯0+𝐚⁢𝑡v =v0+a⁢tv =v0+at (constant a ). where 𝛚0ω0ω0 is the initial angular velocity. Notice that the equation is identical to the linear version, except with angular analogs of the linear variables. In fact, all of the linear kinematics equations have rotational analogs, which are given in Table 6.3 . These equations can be used to solve rotational or linear kinematics problem in which a and 𝛂αα are constant. Figure 6.10 Tornadoes descend from clouds in funnel-like shapes that spin violently. (Daphne Zaras, U.S. National Oceanic and Atmospheric Administration) Tornadoes are perfect examples of rotational motion in action in nature. They come out of severe thunderstorms called supercells, which have a column of air rotating around a horizontal axis, usually about four miles across. The difference in wind speeds between the strong cold winds higher up in the atmosphere in the jet stream and weaker winds traveling north from the Gulf of Mexico causes the axis of the column of rotating air to shift as the storm travels so that the axi