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MIT 8 01T - Rotational Dynamics

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Introduction Figure 13.2 Rotation of a compact disc about a fixed axis. Figure 13.3 Fixed axis rotation and center of mass translation for a bicycle wheel. Figure 13.4 Coordinate system for fixed-axis rotation. Figure 13.5 Sign conventions for rotational motion. Figure 13.7 Vector direction for the torque Figure 13.9 The moment arm about the point associated with a force acting at the point is the perpendicular distance from to the line of action of the force passing through the point Figure 13.10a Positive torque Figure 13.10b Negative torque Figure 13.12 Tangential force acting on a volume element. IntroductionChapter 13 Rotational Dynamics He sighed with the difficulty of talking mechanics to an unmechanical person. "There's a torque," he said. "It ain't balanced ---" Any mechanic would have understood his drift at once. If a three-bladed propeller loses a blade, there are two blades left on one-third of its circumference, and nothing on the other two-thirds. All the resistance to its rotation under water is consequently concentrated upon one small section of the shaft, and a smooth revolution would be rendered impossible ...1 C.S. Forester The African Queen Introduction The physical objects that we encounter in the world consist of collections of atoms that are bound together to form systems of particles. When forces are applied, the shape of the body may be stretched or compressed like a spring, or sheared like jello. In some systems the constituent particles are very loosely bound to each other as in fluids and gasses, and the distances between the constituent particles will vary. We shall begin our study of extended objects by restricting ourselves to an ideal category of objects, rigid bodies, which do not stretch, compress, or shear. A body is called a rigid body if the distance between any two points in the body does not change in time. Rigid bodies, unlike point masses, can have forces applied at different points in the body. For most objects, treating as a rigid body is an idealization, but a very good one. In addition to forces applied at points, forces may be distributed over the entire body. Distributed forces are difficult to analyze; however, for example, we regularly experience the effect of the gravitational force on bodies. Based on our experience observing the effect of the gravitational force on rigid bodies, we note that the gravitational force can be concentrated at a point in the rigid body called the center of gravity, which for small bodies (so that gG may be taken as constant within the body) is identical to the center of mass of the body (we shall prove this fact in Appendix 13.A). Let’s consider a rigid rod thrown in the air (Figure 13.1) so that the rod is spinning as its center of mass moves with velocity cmvG. Rigid bodies, unlike point-like objects, can have forces applied at different points in the body. We have explored the physics of translational motion; now, we wish to investigate the properties of rotation exhibited in the rod’s motion, beginning with the notion that every particle is rotating about the center of mass with the same angular (rotational) velocity. 1 The authors of these notes suspect either a math error on Mr. Forester’s part or an oversight by his editors. 8/25/2008 1Figure 13.1 The center of mass of a thrown rigid rod follows a parabolic trajectory while the rod rotates about the center of mass. We can use Newton’s Second Law to predict how the center of mass will move. Since the only external force on the rod is the gravitational force (neglecting the action of air resistance), the center of mass of the body will move in a parabolic trajectory. How was the rod induced to rotate? In order to spin the rod, we applied a torque with our fingers and wrist to one end of the rod as the rod was released. The applied torque is proportional to the angular acceleration. The constant of proportionality is called the moment of inertia. When external forces and torques are present, the motion of a rigid body can be extremely complicated while it is translating and rotating in space. We shall begin our study of rotating objects by considering the simplest example of rigid body motion, rotation about a fixed axis. 13.1 Fixed Axis Rotation: Rotational Kinematics Fixed Axis Rotation When we studied static equilibrium, we demonstrated the need for two conditions: The total force acting on an object is zero, as is the total torque acting on the object. If the total torque is non-zero, then the object will start to rotate. A simple example of rotation about a fixed axis is the motion of a compact disc in a CD player, which is driven by a motor inside the player. In a simplified model of this motion, the motor produces angular acceleration, causing the disc to spin. As the disc is set in motion, resistive forces oppose the motion until the disc no longer has any angular acceleration, and the disc now spins at a constant angular velocity. Throughout this process, the CD rotates about an axis passing through the center of the disc, and is perpendicular to the plane of the disc (see Figure 13.2). This type of motion is called fixed-axis rotation. 8/25/2008 2Figure 13.2 Rotation of a compact disc about a fixed axis. When we ride a bicycle forward, the wheels rotate about an axis passing through the center of each wheel and perpendicular to the plane of the wheel (Figure 13.3). As long as the bicycle does not turn, this axis keeps pointing in the same direction. This motion is more complicated than our spinning CD because the wheel is both moving (translating) with some center of mass velocity, cmvG, and rotating. Figure 13.3 Fixed axis rotation and center of mass translation for a bicycle wheel. When we turn the bicycle’s handlebars, we change the bike’s trajectory and the axis of rotation of each wheel changes direction. Other examples of non-fixed axis rotation are the motion of a spinning top, or a gyroscope, or even the change in the direction of the earth’s rotation axis. This type of motion is much harder to analyze, so we will restrict ourselves in this chapter to considering fixed axis rotation, with or without translation. Angular Velocity and Angular Acceleration When we considered the rotational motion of a point-like object in Chapter 6, we introduced an angle coordinate θ, and then defined the angular velocity (Equation 6.2.7) as


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MIT 8 01T - Rotational Dynamics

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