Figure 10.1 illustrates a view to spinning compact disc, or CD. Disk rotates on an axis fixed perpe

10.1 Position, Velocity, and Acceleration angle of 10.2 Analysis Model: Rigid Object In 10.3 Constant Angular Acceleration angle and magnitude Translational Rotational Kinetic Energy 10.4 10.5 10.6 Calculation of Moment of Inertia 10.7 Torque Model Analysis: Rigid Object Under Torque Total consideration 10.8 10.9 Energy in Motion Rotational Motion Rolling of Rigid Object
When an extended object air guns like a wheel spinning on its axis, the motion can not be analyzed by modeling the object as a particle because at any given time different parts of the object has a linear air guns velocity and linear acceleration are different. We can analyze the motion of an extended object modeling as a system of many particles, each having its own linear velocity and linear acceleration as discussed in Section 9.7. When dealing with a rotating object, analysis is greatly simplified by assuming a rigid object. A rigid object is one that is nondeformable (can not change shape), which is the relative location of the object that is composed of all particles remains constant. All real deformable objects to a certain extent, our rigid object model, however, is useful in many situations where deformation (shape change) are ignored. We have developed a model based on particle and system analysis. In this chapter, we introduce another class of models based on the analysis of the rigid object model. 10.1 Position, The rate and Acceleration angular (corner)
Figure 10.1 illustrates a view to spinning compact disc, or CD. Disk rotates on an axis fixed perpendicular to the image plane and passing through the center air guns of the disk in O. Small element of the disk is modeled as a particle at P is at a distance r from the origin and rotating remains surrounding the circle with radius r. (In fact, every element of the disk undergo rotary motion O.) It is convenient to represent the position of P with polar coordinates (r, q), where r is the distance from the origin to P and q are measured opposite from some reference line fixed in space as shown in Figure 10.1a. In this representation, the angle q changes in time while r remains constant. As the particle moves along the circle of the reference line, which is at an angle q = 0, moving through the arc length s as in Figure 10.1b. Arc length s associated with the angle q through air guns the relation: s = rq (10.1a) q = s / r (10.1b) Because q is the ratio of the arc length and radius of the circle, it is a pure number. Usually, we deliver q units of radians (rad), where one radian is the angle subtended by the arc length equal to the radius of the arc. Since the circumference of a circle is 2 pr, then from Equation air guns 10.1b that 360 0 equated with angle (2 pr / r) rad = 2 p rad. Therefore, 1 rad = 360 0/2 p @ 57.38. To convert the angle in degrees to angle in radians, we use p rad = 180 0, so
For example, 60 0 is equal to p / 3 rad and 45 0 equals p / 4 rad. Because the disk in Figure 10.1 is a rigid object, as the particle moves through an angle q from the reference line, every other particle on the object rotates through the same angle q. Therefore, we can associate the angle q with the entire rigid object as well as the individual particles, which allows us to determine the angular position of a rigid object in rotational motion. We choose the reference line on the object, such as the line connecting O and particles on the object selected. Angular position of a rigid object air guns q is the angle between the reference line on the object and the reference line fixed in space, which is often chosen as the x axis. Such identification is similar to the way we determine the position of an object in translational air guns motion as the distance x between the object and the reference position, air guns which is at the origin, x = 0. Therefore, the angle q plays the same role in rotational motion is the x position in translational motion. As the particles travel on a rigid object from position A to position B in the time interval Δt as shown in Figure 10.2, the reference line to a fixed object sweep angle Δ q = qf - qi. This quantity is defined as the angular displacement of the object air guns yes