Physics
Connecting Linear and Rotational Motion
Connecting linear and rotational motion involves understanding how linear motion (movement in a straight line) is related to rotational motion (movement around an axis). This connection is described by concepts such as torque, angular velocity, and moment of inertia. By understanding these relationships, physicists can analyze and predict the behavior of objects that undergo both linear and rotational motion.
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9 Key excerpts on "Connecting Linear and Rotational Motion"
- David V. Guerra(Author)
- 2023(Publication Date)
- CRC Press(Publisher)
13 Rotational Motion
DOI: 10.1201/9781003308065-1313.1 Introduction
Rotational motion is the motion of an object spinning around an axis that passes through the object itself. This is to be contrasted with translational motion, which is the motion of an object moving through space in a straight or curved path without rotation. As demonstrated in Figure 13.1 , a block sliding down an incline moves with only linear motion defined by a displacement, a velocity, and an acceleration. A disk rotating about a fixed axis moves with only rotational motion, but a ball rolling down an incline experiences both rotational and linear motion.FIGURE 13.1 Examples of different types of motion.As demonstrated in Chapter 12 , when an object is traveling in a circular path, the concepts of translation kinematics are commonly applied to the analysis, but sometimes the concepts of rotational motion, period, frequency, and angular frequency can be applied. So, circular motion provides a transition between the language of translational and rotational motion, which is formalized in this chapter. In addition, rotational dynamics will formalize the connection between the net rotational force, or net torque, on an object and the angular acceleration of the object.- Chapter question: There are bacteria that employ a rotating flagellum, tails that look a bit like a corkscrew, to propel themselves forward. In normal situations, these propulsion systems work well to move these bacteria forward through water. On the other hand, when a drop of water containing these bacteria is placed on a microscope slide, the bacteria begin to move in approximately circular paths at fairly constant speeds (Figure 13.2 ).
FIGURE 13.2
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
139 © 2010 Taylor & Francis Group, LLC Rotational Motion Translational motion and circular motion, uniform and nonuniform of an object, were discussed in Chapters 3 and 4. In this chapter, rotational motion of a point or extended object will be introduced. A force acting on an extended object creates a torque that rotates it about a fixed axis. A solid object of finite physical size is known as a rigid body. The axis of rotation could be about the center of mass of the rigid body or about other points where it is pivoted. In this chapter, Sections 8.2 through 8.7 discuss the kinematics of a rigid body, and Section 8.8 reviews the dynamics of the rigid body. In addition to drawing a parallel between Newton’s laws applied to point-like objects in linear motions and those applied to rotational motion of a rigid body, translational and rotational motions of a rigid body are discussed in detail. 8.1 ANGULAR KINEMATIC QUANTITIES Consider a point-like object moving in a circle of radius r (Figure 8.1). As the object moves from point P 1 at t 1 to point P 2 at t 2 , it sweeps through an arc Δ s that subtends an angle Δθ ( = θ 2 – θ 1 ) at the center, in a time interval Δ t = t 2 – t 1 . From geometry Δ s = r Δθ , (8.1) where Δθ is an angular displacement, measured in radians. To convert an angle expressed in degrees to radians, the following relation may be used: θ π θ rad deg deg 2 rad 360 = , (8.2a) and from radians to degrees, the conversion is θ π θ de g r ad 360 2 . = (8.2b) Dividing Equation 8.1 by Δ t gives the average linear velocity v ; that is, ∆ ∆ ∆ ∆ s t r t = θ or v r . = ω (8.3) For an object experiencing a constant linear acceleration, v v v 2 1 2 = + . 8 140 Essential Physics © 2010 Taylor & Francis Group, LLC Here, v 1 and v 2 are the instantaneous tangential, or linear, velocities of the object at points P 1 and P 2 , respectively.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
Many of the concepts describing translational motion have anal- ogous counterparts for circular motion. These will be developed in this chapter. The concepts in this chapter apply to both rota- tional and revolutionary motion. Rotation involves circular motion around an axis that passes through the object itself, for example, a rotating top. Revolution involves circular motion around an axis outside of the object. The Earth rotates on its axis and revolves around the Sun. In describing translational motion, a simple Cartesian coordinate system was 50 Rotational Motion used to reference position. This system can also be used for circular motion. When using the Cartesian coordinate system for circular motion, the position of an object can be referenced to the origin and the posi- tive x axis (Figure 4.1). Angles are measured counterclockwise starting from the positive x axis. The angular displacement is the angle swept out during a specific time period. For any two points along the circumference, the angular displacement is equal to A9. When starting from the positive x axis, where 0 is defined as zero, the angular displacement is just 0. The distance an object moves in one revolution around the circle is the circum- ference of the circle, 2irr. The time it takes for an object to make one revolution around the circle is called the period, symbolized by T. The tangential or linear speed of an object moving in a circular path of radius r around the origin is equal to the change in distance along the circumference divided by the change in time, As/At (if referenced from the positive x axis, the distance is just s). Since an object will travel a distance / equal to the circumference in one period, the tangential speed can be found by dividing the circumference by the period: 2rrr tangential speed = ~^r When the tangential speed is constant, the motion is described as uniform circular motion. The tangential velocity, v, is the vector analogy to tangential speed.
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
Our prior study of translational kinematics involved the investigation of the relationship between the displacement, velocity, and acceleration of a body. Completely analogous relationships exist for the rotational kinematics of a rigid body and are shown below. Since a rigid body can both translate and rotate, both sets of equations apply, given a certain set of circumstances. We will look at these sets of equations in detail in the upcoming sections. Specifically, we will investigate the relationship between a body’s angular displacement , its angular velocity , and its angular acceleration . Rotational Equations Translational Equations d dt ds v dt 2 2 d d dt dt 2 2 dv d s a dt dt d d a ds v dv What is the difference between the motion of a particle and the motion of a rigid body? Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 4 Conceptual Example 4.1-1 Draw possible paths for the motion of bar AB as it moves from point 1 to point 2 in pure translation. Draw the path of motion for bar AB as it rotates about point B in pure rotation. Conceptual Dynamics Kinematics: Chapter 4 – Kinematics of Rigid Bodies 4 - 5 4.2) PURE ROTATION 4.2.1) ROTATION ABOUT A FIXED AXIS We will first consider the case of a rigid body rotating about a fixed axis as shown in Figure 4.2 - 1. Pure rotation occurs when a body rotates about a fixed non-moving axis. Under the rigid-body assumption, the distance between any two points on the body remains constant. Therefore, each point on the body can be considered to be moving in concentric circles about the fixed axis. Fixed-point O, shown in Figure 4.2-1, is a point the coincides with the fixed axis, in this case, the z-axis. 4.2.2) ANGULAR KINEMATIC RELATIONSHIPS In previous chapters, we have studied linear position (s), linear velocity (v), and linear acceleration (a).
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Chapter 6 discusses their application to translational motion. These concepts are equally useful for rota- tional motion, provided they are expressed in terms of angular variables. TABLE 9.2 Analogies Between Rotational and Translational Concepts Physical Concept Rotational Translational Displacement s Velocity Acceleration a The cause of acceleration Torque Force F Inertia Moment of inertia I Mass m Newton’s second law Σ τ = Iα Σ F = ma Work τθ Fs Kinetic energy 1 2 Iω 2 1 2 mυ 2 Momentum L = Iω p = mυ F Rod (overhead view) F Axis (perpendicular to page) A F B C m m m m 2m 3m CYU FIGURE 9.6 242 CHAPTER 9 Rotational Dynamics The work W done by a constant force that points in the same direction as the displacement is W = Fs (Equation 6.1), where F and s are the magnitudes of the force and the displace- ment, respectively. To see how this expression can be rewritten using angular variables, consider Figure 9.20. Here a rope is wrapped around a wheel and is under a constant tension F. If the rope is pulled out a distance s, the wheel rotates through an angle = s/r (Equation 8.1), where r is the radius of the wheel and is in radians. Thus, s = r, and the work done by the tension force in turning the wheel is W = Fs = Fr. However, Fr is the torque applied to the wheel by the tension, so the rotational work can be written as follows: DEFINITION OF ROTATIONAL WORK The rotational work W R done by a constant torque in turning an object through an angle is W R = τθ (9.8) Requirement: must be expressed in radians. SI Unit of Rotational Work: joule (J) Section 6.2 discusses the work–energy theorem and kinetic energy. There we saw that the work done on an object by a net external force causes the translational kinetic energy ( 1 2 mυ 2 ) of the object to change. In an analogous manner, the rotational work done by a net external torque causes the rotational kinetic energy to change.
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 - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
However, our current formulation of Newton’s second law only applies to the center-of-mass (translational) motion of an object. In this unit we will define several new quantities and relationships to describe the rotational motion of rigid objects, objects that do not change their shape as they move. These quantities include rotational position, rotational velocity, rotational acceleration, rotational inertia, and torque (rotational force). We will then use these concepts to extend Newton’s second law to describe both translational and rota- tional motion. For now, we restrict our attention to situations where the axis of rotation remains fixed, which is the rotational equivalent of looking at one-dimensional translational motion. UNIT 12: ROTATIONAL MOTION 391 ROTATIONAL KINEMATICS 12.2 ROTATIONAL QUANTITIES FOR RIGID OBJECTS As an introduction to one of the central concepts in rotational motion, the fol- lowing activity considers the motion of two cars in a circular roundabout. 12.2.1. Activity: A Circular Roundabout a. Consider two cars driving in a circular roundabout with two lanes (see Fig. 12.1). If you’re not familiar with a roundabout, you can think of it as a circular racetrack. If two cars enter the roundabout side-by-side traveling at the same (constant) speed, will the cars remain next to each other as they travel around the circle? Explain. Andy F/Wikimedia Commons/CC BY 3.0 Fig. 12.1. b. Now suppose each car travels at a (possibly different) constant speed and traverses the roundabout in 40 seconds, with the two cars remaining next to each other the entire time. Determine the speed of each car if the circular path of the inner lane has a radius of 100 m and that of the outer lane has a radius of 105 m. c. Make a rough sketch of this situation showing the velocity vector of each car at two different locations on the circle. Do the cars have a con- stant velocity? If not, determine the directions and magnitudes of their (centripetal) accelerations.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
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. 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. Is it simply the net force? As it turns out, it is not the net external force, but rather the net external torque that causes the rotational velocity to change. Just as greater net forces cause greater linear accelerations, greater net torques cause greater rotational or angular accelerations. Interactive Figure 9.2 helps to explain the idea of torque. When you push on a door with a force → F , as in part a, the door opens more quickly when the force is larger. Other things being equal, a larger force generates a larger torque. However, the door does not open as quickly if you apply the same force at a point closer to the hinge, as in part b, because the force now produces less torque. Furthermore, if your push is directed nearly at the hinge, as in part c, you will have a hard time opening the door at all, because the torque is nearly zero. In summary, the torque depends on the magnitude of the force, on the point where the force is applied relative to the axis of rotation (the hinge in Interac- tive Figure 9.2), and on the direction of the force. For simplicity, we deal with situations in which the force lies in a plane that is per- pendicular to the axis of rotation.
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.
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