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Rotational Motion 7 Rotational Motion 7 Once the translational motion of an object is accounted for, all the other motions of the object can best be described in the stationary reference frame of the center of mass. A reasonable image to keep in mind is to imagine following a seagull in a helicopter that tracks its translational motion. If you took a video of the seagull you would see quite dif- ferent motion than you would from the ground. The seagull would appear always ahead of you but would rotate and change its “shape” as it flapped its wings (e.g., see the film Winged Migration). You’ve probably seen such wildlife videos that can track animals and “subtract” their translational motion leaving only the other collective motions about their “centers”: lions seemingly running “in place” as the scenery flies by. In physics, we’ve already shown how to account for the translational motion of the center of mass. Aside from a possible constant velocity drift in the absence of any forces, motion of the center of mass is caused by external forces acting on the object. We now turn to the other motions about the center of mass as viewed from a reference frame fixed to the center of mass. These collective motions are of two types: coherent and incoherent. Coherent motions are those overall rotations or vibrations that occur within a solid in which the constituent particles making up the object interact with each other in a coordinated fashion. If the solid is rigid (with all the internal distances between constituent parts fixed) the only collective motion will be an overall rotation about the center of mass. For such a rigid body, a complete description of its motion includes the translational motion of the center of mass and the rotational motion about the center of mass. Because this nice separation of the problem can be made, we first present the descrip- tion, or kinematics, of pure rotational motion of a rigid body about a fixed axis, the axis of rotation. In this case all points of an object rotate in circles about some fixed point on the axis of rotation. This type of motion occurs, for example, when a door is opened, or for the wheels of a stationary bicycle, or when you lift an object by rotat- ing your forearm about a stationary elbow. Even if the solid is not rigid, its collective coherent motions can be described as a rigid body rotation (of the average-shaped body) as well as other coherent internal motions that can change the object’s shape. We next introduce the energy associated with rotational motion and the rotational analog of mass, known as the moment of inertia. We show that well-placed and directed forces can produce rotational motion and we introduce the notion of torque, the rotational analog of a force. For pure rotational motion there is an equation that is the rotational analog of Newton’s second law that can describe the dynamics of motion. Continuing with rotational analog quantities we introduce angular momentum, the rota- tional analog of (linear or translational) momentum and learn a new fundamental conservation law of angular momentum. Key in following the presentation of our under- standing of rotational motion is to keep in mind the strong analogy with what we have already learned. A preview glance at Table 7.2 below shows that the important new concepts in this chapter all have direct analogs with equations we have already studied. One of the new and revolutionary types of microscopy, atomic force microscopy, is discussed as an application of the material in this chapter. The technique allows J. Newman, Physics of the Life Sciences, DOI: 10.1007/978-0-387-77259-2_7, © Springer Science+Business Media, LLC 2008 ROTATIONAL M OTION 161 extremely high resolution maps of the microscopic surface topography, or structure, of materials and has been used extensively to study biological molecules and cells. After briefly considering the effects of diffusion on the rotational motion of macromolecules, the chapter concludes with a study of the special case of objects in static equilibrium. This is an important simplification of Newton’s laws and provides a powerful method of analyzing equilibrium situations. The other category of collective motions is known as incoherent. These are ran- dom motions of the atoms of the material, about the equilibrium positions in a solid or with no fixed average position in a fluid. Constituent fluid particles move about much more independently, in the ideal case not interacting with their neighbors at all. In Chapters 8 and 9 we discuss the flow of ideal fluids as well as some of the com- plications that occur in complex fluids in which there are strong interactions between constituents. Later in Chapters 12 and 13, we study the subject of thermodynamics concerned with describing the fundamental thermal properties of macroscopic sys- tems. We show that collective incoherent internal motions of an object give rise to an internal energy that is responsible for its temperature. 1. ROTATIONAL KINEMATICS A rigid body—one with a fixed shape—has motions that are limited to pure translation and pure rotation about its center of mass. Combinations of these can give rise to motions that appear more complex, such as rolling, but which can be simplified to pure rotations in a reference frame fixed to the center of mass. Since we’ve already learned how to handle translational motion in the previous chapter, here we first take up the problem of pure rotational motion about a fixed axis of rotation, leaving their synthe- sis leading to general rigid body motions for a discussion later in the chapter. Consider the motion of a point particle on the circumference of a circle, as shown in Figure 7.1. In order to describe its position and motion we could use its x- and y- coordinates or, better, its r and ␪ polar coordinates. These latter coordinates are pre- ferred because r is constant if the particle remains on the circle and so in polar coordinates there is really only one variable ␪, whereas both x and y change as the par- ticle moves on the circle. To describe the motion of the particle on the circle we could use its x- and y-components of velocity, both of which would continuously change, or, even better, we could use the ␪-component of velocity known as the angular velocity ␻, whose average value is defined as ¢u v ϭ . (7.1) ¢t In this expression, the particle has moved between two angular positions in a time ⌬t, where the angular displacement ⌬␪ must be measured in radian units and not in degrees. The fundamental unit of angular measure is the radian, because it is defined as the ratio of the arc length s to the radius r as ␪ ϭ s/r, and is a pure number with no units. FIGURE 7.1 A particle executing This definition of an angle in radians leads to the fact that there are 2␲ radians circular motion. in one revolution around a circle, given the circumference of s ϭ 2␲r. In one complete revolution there are also 360° and thus the radian is equal to 360°/2␲ or about 57.3°. The unit for ␻ is, according to Equation (7.1), the radian per second (rad/s); despite the fact that the radian unit is a pure number and we could write the units for angular velocity as 1/s, it is useful to retain the term yr “rad” in the numerator. (Note: Most pocket calculators can do calculations θ using either radians or degrees and because rad must be used here, some care x must be taken when first using a new calculator.) If our particle travels in a circle at a constant speed, executing uniform cir- cular motion, then the instantaneous value of ␻ is constant and equal to the aver- age value. Notice that because ␻ is an angular variable there are really only two possible directions of travel: clockwise or counterclockwise around the circle. Just as the sign (ϩ or Ϫ) for a linear quantity depends on the coordinate system, 162 ROTATIONAL M OTION we are free here to label the sign of the angular velocity in an arbitrary way, as long as we are self-consistent within the context of any particular discussion. In the more general case, when the particle does not travel at a constant speed, the angular velocity will vary and we need to introduce the concept of the instantaneous angular velocity, defined in a similar way to v, as ¢u v ϭ lim . (7.2) ¢t : 0 ¢t The distance traveled by the particle along the circumference ⌬s is proportional to the angular displacement ⌬␪ from the relation s ϭ r␪, therefore we have that ⌬s/⌬t ϭ r ⌬␪/⌬t so that v ϭ rv, (7.3) where r is the radius of the circle. We also need to introduce the concept of an angular acceleration ␣ in order to account for a changing ␻ in analogy with our introduction of a linear acceleration a to describe changes with time in the linear velocity v. We define the average and instantaneous angular acceleration in direct analogy with their linear counterparts as ¢v ¢v aϭ ; a ϭ lim . (7.4) ¢t ¢t : 0 ¢t Again, because Equation (7.3) implies the change in the magnitude of the linear velocity is proportional to the change in the angular velocity, we have a relationship between the linear and angular accelerations, ϭ a tang ra. (7.5) The linear acceleration in Equation (7.5) is called the tangential acceleration and is directed parallel (or antiparallel) to the tangential velocity.
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