Introduction Figure 14.2 Rotation of a compact disc about a fixed axis.Figure 14.3 Fixed axis rotation and center of mass translation for a bicycle wheel.Figure 14.4 Coordinate system for fixed-axis rotation.Figure 14.5 Sign conventions for rotational motion. Figure 14.7 Tangential force acting on a volume element.IntroductionChapter 14 Rotational Dynamics Introduction Let’s consider a rigid rod thrown in the air (Figure 14.1) so that the rod is spinning as its center of mass moves with velocity cmvG. (Recall in the Introduction to Chapter 4 that 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-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. Figure 14.1 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. 14.1 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. 7/15/2008 1A simple example 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 creates a torque on the disc that produces angular acceleration, causing the disc to spin. As the disc is set in motion, a frictional torque is produced that opposes the applied torque. When these two torques are equal in magnitude, 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 14.2). This type of motion is called fixed-axis rotation. Figure 14.2 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 14.3). 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 14.3 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 7/15/2008 2we 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 7, we introduced an angle coordinate θ, and then defined the angular velocity (Equation 7.2.7) as ddtθω≡ , (14.1.1) and angular acceleration (Equation 7.3.4) as 22ddtθα≡ . (14.1.2) For a rigid body undergoing fixed-axis rotation, we can divide the body up into small volume elements with mass imΔ. Each of these volume elements is moving in a circle of radius about the axis of rotation (Figure 14.4). ,ir⊥ Figure 14.4 Coordinate system for fixed-axis rotation. We will adopt the notation implied in Figure 14.4, and denote the vector from the axis to the point where the mass element is located as ,i⊥rG, with ,,iir⊥⊥= rG. Because the body is rigid, all the volume elements will have the same angular velocity ω and hence the same angular acceleration α. If the bodies did not have the same angular velocity, the volume elements would “catch up to” or “pass” each other, precluded by the rigid-body assumption. Sign Convention: Angular Velocity and Angular Acceleration 7/15/2008 3Suppose we choose θ to be increasing in the counterclockwise direction as shown in Figure 14.5. Figure 14.5 Sign conventions for rotational motion. If the rigid body rotates in the counterclockwise direction, then the angular velocity is positive, 0ddtωθ≡>. If the rigid body rotates in the clockwise direction, then the angular velocity is negative, 0ddtωθ≡<. • If the rigid body increases its rate of rotation in the counterclockwise (positive) direction then the angular acceleration is positive, 22/0ddtddtαθ ω≡=>. • If the rigid body decreases its rate of rotation in the counterclockwise (positive) direction then the angular acceleration is negative, 22/0ddtddtαθ ω≡=<. • If the rigid body increases its rate of rotation in the clockwise (negative) direction then the angular acceleration is negative, 22/0ddtddtαθ ω≡=<. • If the rigid body decreases its rate of rotation in the clockwise (negative) direction then the angular acceleration is positive, 22/0ddtddtαθ ω≡=>. To phrase this more generally, if α and ω have the same sign, the body is speeding up; if opposite signs, the body is slowing down. This general result is independent of the choice of positive direction of rotation. Note that in Figure 14.2, the CD has a negative
View Full Document
Unlocking...