Physics
Rotational Dynamics
Rotational dynamics is the study of the motion of objects that rotate around an axis. It involves understanding the forces and torques that cause rotational motion, as well as the resulting angular acceleration and velocity. Key concepts in rotational dynamics include moment of inertia, angular momentum, and the relationship between torque and angular acceleration, as described by Newton's second law for rotation.
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5 Key excerpts on "Rotational Dynamics"
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
175 Rotational Dynamics I n Chapter 8 we considered rotational kinematics and pointed out that it contained no basic new features, the rotational parameters , , and being related to corresponding translational parameters x, v, and a for the particles that make up the rotating system. In this chapter, following the pattern of our study of translational motion, we consider the causes of rotation, a subject called Rotational Dynamics. Rotating systems are made up of particles, and we have already learned how to apply the laws of classical mechanics to the motion of particles. For this reason rotational dynam- ics, like kinematics, should contain no features that are fundamentally new. As in Chapter 8, however, it is very useful to recast the concepts of translational motion into a new form, especially chosen for its conve- nience in describing rotating systems. 9-1 TORQUE We began our study of dynamics in Chapter 3 by defining a force in terms of the acceleration it produced when acting upon a body of standard mass (Section 3-3). We were then able to obtain the mass of any other body in relation to the standard mass by measuring the acceleration produced when the same force acts on each body (Section 3-4). We incorporated our observations about force, mass, and accel- eration into Newton’s second law, according to which the net force acting on a body is equal to its mass times its ac- celeration. Our procedure for Rotational Dynamics is similar. We will begin by considering the angular acceleration produced when a force acts on a particular rigid body that is free to rotate about a fixed axis. In analogy with translational mo- tion, we will find that the angular acceleration is propor- tional to the magnitude of the applied force. However, a new feature emerges that was not present in translational motion: the angular acceleration also depends on where the force is applied to the body.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 9 Rotational Dynamics LEARNING OBJECTIVES After reading this module, you should be able to: 9.1 distinguish between torque and force 9.2 analyse rigid objects in equilibrium 9.3 determine the centre of gravity of rigid objects 9.4 analyse Rotational Dynamics using moments of inertia 9.5 apply the relation between rotational work and energy 9.6 solve problems using the conservation of angular momentum. INTRODUCTION The large counterweight on the right side (short end) of this tall tower crane ensures its boom remains balanced on its mast while lifting heavy loads. It is not equal weights on both sides of the tower that keep it in equilibrium, but equal torques. Torque is the rotational analog of force, and is an important topic of this chapter. Source: Mr. Green / Shutterstock 9.1 The action of forces and torques on rigid objects LEARNING OBJECTIVE 9.1 Distinguish between torque and force. The mass of most rigid objects, such as a propeller or a wheel, is spread out and not concentrated at a single point. These objects can move in a number of ways. Figure 9.1a illustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in figure 9.1b. FIGURE 9.1 Examples of (a) translational motion and (b) combined translational and rotational motions Translation ( ) a Combined translation and rotation ( ) b We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The large counterweight on the right side (short end) of this tall tower crane ensures its boom remains balanced on its mast while lifting heavy loads. It is not equal weights on both sides of the tower that keep it in equilibrium, but equal torques. Torque is the rotational analog of force, and is an important topic of this chapter. 9 | Rotational Dynamics Chapter | 9 LEARNING OBJECTIVES After reading this module, you should be able to... 9.1 | Distinguish between torque and force. 9.2 | Analyze rigid objects in equilibrium. 9.3 | Determine the center of gravity of rigid objects. 9.4 | Analyze Rotational Dynamics using moments of inertia. 9.5 | Apply the relation between rotational work and energy. 9.6 | Solve problems using the conservation of angular momentum. 9.1 | The Action of Forces and Torques on Rigid Objects The mass of most rigid objects, such as a propeller or a wheel, is spread out and not con- centrated at a single point. These objects can move in a number of ways. Figure 9.1a il- lustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9.1b. We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration. A net external force causes linear motion to change, but what causes rotational motion to change? For example, something causes the rotational velocity of a speedboat’s propeller to change when the boat accelerates.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
243 A hurricane, or typhoon, is composed of large rotating masses of air with a very low pressure eye at its center. When the rotating air near the outer edge of the storm is forced inward due to the lower pressure, its moment of inertia decreases. As a consequence of this, its angular speed increases to conserve angular momentum. This effect produces the very high and dangerous winds in the eye wall of the storm (see photograph). Moment of inertia and conservation of angular momentum are two important topics in Rotational Dynamics, which we will study in this chapter. Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects The mass of most rigid objects, such as a propeller or a wheel, is spread out and not concentrated at a single point. These objects can move in a num- ber of ways. Figure 9.1a illustrates one possibility called translational skeeze/Pixabay CHAPTER 9 LEARNING OBJECTIVES After reading this module, you should be able to... 9.1 Distinguish between torque and force. 9.2 Analyze rigid objects in equilibrium. 9.3 Determine the center of gravity of rigid objects. 9.4 Analyze Rotational Dynamics using moments of inertia. 9.5 Apply the relation between rotational work and energy. 9.6 Solve problems using the conservation of angular momentum. Translation ( ) a Combined translation and rotation ( ) b FIGURE 9.1 Examples of (a) translational motion and (b) combined translational and rotational motions. 244 CHAPTER 9 Rotational Dynamics motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because transla- tional motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9.1b.
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Dynamics of rigid bodies rotating about a fixed axis The human mind has first to construct forms, independently, before we can find them in things. Einstein, 1879–1955 20.1 A rigid body rotating about a fixed axis You might not be able to answer all these questions fully now, but the issues involved should become clearer as you work through this chapter. It is reasonable to treat a large object as a particle when every part of it is moving in the same direction with the same speed, but clearly this is not always the case. The particles in a rotating wheel have different velocities and accelerations and are subject to different forces. The laws of particle dynamics which you have used so far need to be developed so that they can be applied to the rotation of large objects. 20 Q UESTION 20.1 So far you have modelled moving objects as particles. In many circumstances this is reasonable, but how would you model the motion of the sails of a windmill or the other objects illustrated in the pictures above? Do the two children on the roundabout have the same kinetic energy? What is the kinetic energy of a rotating wheel? Definitions You are already familiar with many aspects of rotation such as the angular speed and acceleration of a particle and you have also taken moments to determine the turning effect of a force, but it is as well to be clear about what is meant by some of the terms involved before continuing with the discussion. Rigid bodies Wheels can be modelled as rigid bodies. A rigid body is such that each point within it is always the same distance from any other point. You are not a rigid body but a hard chair is one (molecular vibrations being ignored). The axis of rotation When you lean back on your chair, it might rotate about a point, say A, at the end of one leg. You will have more control, however, if it rotates about the axis formed by the line joining the ends, A and B, of two legs.
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