Physics
Rotational Motion Equations
Rotational motion equations describe the motion of objects that rotate around an axis. They include equations for angular velocity, angular acceleration, and rotational kinetic energy. These equations are analogous to the linear motion equations and are used to analyze and predict the behavior of rotating objects.
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11 Key excerpts on "Rotational Motion Equations"
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- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
214 CHAPTER 8 Rotational Kinematics are the same and that they are related to the angular acceleration of the wheel relative to the axle: a = a T = rα ( α in rad /s 2 ) (8.13) Equations 8.12 and 8.13 may be applied to any rolling motion, because the object does not slip against the surface on which it is rolling. Check Your Understanding (The answers are given at the end of the book.) 16. The speedometer of a truck is set to read the linear speed of the truck, but uses a device that actually measures the angular speed of the rolling tires that came with the truck. However, the owner replaces the tires with larger-diameter versions. Does the reading on the speedometer after the replacement give a speed that is (a) less than, (b) equal to, or (c) greater than the true linear speed of the truck? 17. Rolling motion is an example that involves rotation about an axis that is not fixed. Give three other examples of rotational motion about an axis that is not fixed. 8.7 *The Vector Nature of Angular Variables We have presented angular velocity and angular acceleration by taking advantage of the analogy between angular variables and linear variables. Like the linear velocity and the linear accelera- tion, the angular quantities are also vectors and have a direction as well as a magnitude. As yet, however, we have not discussed the directions of these vectors. When a rigid object rotates about a fixed axis, it is the axis that identifies the motion, and the angular velocity vector points along this axis. Interactive Figure 8.16 shows how to determine the direction using a right-hand rule: Right-Hand Rule Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity vector. No part of the rotating object moves in the direction of the angular velocity vector.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
Table 4.2 lists the main Rotational Motion 65 Table 4.2 Kinematic Equations for Translational and Rotational Motion Final velocity Displacement Displacement Change in velocity Acceleration Momentum Newton's second law Translational Motion v f = v. + at Vf + Vi Ax = 1 2 1 2 Ax = v t + - a t 2 2aAx = v f 2 — Vi 2 _ A v a ~ A t mv F =ma Rotational Motion co f = to. + af A6- t*' + ftll) 2 AO = w t + - a t 2 2 2aA0 = ct)r 2 — w / Aw a = A t Leo T = La equations for translational and rotational motion. The first four equations are the kinematic equations presented in Table 3.1. Each of the equations for translational motion has a corresponding equation for rotational motion. The four equations for rotational motion apply when the angular acceleration is constant, for example, uni- form circular motion. The time interval, t, for the first four equations (both transla- tional and rotational) is the time over which the change in motion takes place. This page intentionally left blank
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, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 8 Rotational kinematics LEARNING OBJECTIVES After reading this module, you should be able to: 8.1 define angular displacement 8.2 define angular velocity and angular acceleration 8.3 solve rotational kinematics problems 8.4 relate angular and tangential variables 8.5 distinguish between centripetal and tangential accelerations 8.6 analyse rolling motion 8.7 use the right‐hand rule to determine the direction of angular vectors. INTRODUCTION The figure shows the front view of a turbine jet engine on a commercial aircraft. The rotating fan blades collect air into the engine before it is compressed, mixed with fuel, and ignited to produce thrust. The rotational motion of the blades can be described using the concepts of angular displacement, angular velocity, and angular acceleration within the framework of rotational kinematics. Source: Mikhail Starodubov / Shutterstock 8.1 Rotational motion and angular displacement LEARNING OBJECTIVE 8.1 Define angular displacement. FIGURE 8.1 When a rigid object rotates, points on the object, such as A, B, or C, move on circular paths. The centres of the circles form a line that is the axis of rotation. A B C Axis of rotation In the simplest kind of rotation, points on a rigid object move on circular paths. In figure 8.1, for example, we see the circular paths for points A, B, and C on a spinning skater. The centres of all such circular paths define a line, called the axis of rotation. The angle through which a rigid object rotates about a fixed axis is called the angular displacement. Figure 8.2 shows how the angular displacement is measured for a rotating compact disc (CD). Here, the axis of rotation passes through the centre of the disc and is perpendicular to its surface. On the surface of the CD we draw a radial line, which is a line that intersects the axis of rotation perpendicularly. As the CD turns, we observe the angle through which this line moves relative to a convenient reference line that does not rotate.
eBook - PDF- Jerry Ginsberg(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
(6.1.6) 6.1 Fundamental Equations 299 One can use the repetitive pattern of Euler’s equations to remember the individual com-ponents by a mnemonic algorithm based on permutations of the alphabetical order. Eu-ler’s equations are particularly useful when it is only necessary to address the moment exerted about one axis. A basic aspect of the force and moment components is their dual interpretation. One way of evaluating them is to form the resultant as the vector sum of the contribution of each force acting on the body. Alternatively, we may sum the contribution of each force to a specific force or moment component. The latter is very useful for the moment components, which are the moments each force exerts about each of the xyz coordinate axes. The previous section made it evident that a spatial rotation will require that ¯ H A change, even if the all rates of rotation are constant, as a consequence of changing the orientation of ¯ H A . The moment equation merely requires that the force system apply a moment that balances the rate at which the angular momentum changes. The moment required to balance the portion of d ¯ H A / dt that features products of rotation rates, and therefore is present even if the rotation rates are constant, is often referred to as the gyroscopic moment . Various questions may be investigated with the equations of motion. In the simplest case, the motion of a rigid body is fully specified. This permits complete evaluation of the right side of the translational and rotational equations. The forcing effects, which appear on the left side of the equations, originate from known excitations, as well as reactions. The latter are particularly important to characterize. A free-body diagram, in which the body is isolated from its surroundings, is essential to the correct description of the reactions. As an aid in drawing a free-body diagram, recall that reactions are the physical manifestations of kinematical constraints.
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
3 Kinematics of Rotational Motion But though the professed aim of all scientific work is to unravel the secrets of nature, it has another effect, not less valuable, on the mind of the worker. It leaves him in possession of methods which nothing but scientific work could have led him to invent. James Clerk Maxwell, The Theory of Molecules The dynamics of rigid bodies is a chapter of classical mechanics that deserves to be highlighted not only owing to its intrinsic physical interest but also because it involves important mathematical techniques. Before, however, embarking on the study of dynamics, it is necessary to formulate efficacious methods to describe the motion of rigid bodies. A considerable space will be dedicated to the study of rotational kinematics in the perspective that several of the mathematical tools to be developed are of great generality, finding wide application in other domains of theoretical physics. 3.1 Orthogonal Transformations A rigid body has, in general, six degrees of freedom. Obviously, three of them correspond to translations of the body as a whole, whereas the other three degrees of freedom describe the orientations of the body relative to a system of axes fixed in space. A simple way to specify the orientation of the rigid body consists in setting up a Cartesian system of axes fixed in the body, which move along with it, and consider the angles that these axes make with axes parallel to those that remain fixed in space, represented by dashed lines in Fig. 3.1. Direction Cosines Let be a Cartesian coordinate system (x 1 , x 2 , x 3 ) with corresponding unit vectors ˆ e 1 , ˆ e 2 , ˆ e 3 representing axes fixed in space, and let be a Cartesian coordinate system (x 1 , x 2 , x 3 ) with unit vectors ˆ e 1 , ˆ e 2 , ˆ e 3 whose axes remain attached to the rigid body, as in Fig.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The kinetic energy of the entire rotating body, then, is the sum of the kinetic energies of the particles: Rotational KE = Σ ( 1 _ 2 mr 2 ω 2 ) = 1 _ 2 (Σmr 2 ) ω 2 In this result, the angular speed ω is the same for all particles in a rigid body and, there- fore, has been factored outside the summation. According to Equation 9.6, the term in parentheses is the moment of inertia, I = Σ mr 2 , so the rotational kinetic energy takes the following form: DEFINITION OF ROTATIONAL KINETIC ENERGY The rotational kinetic energy KE R of a rigid object rotating with an angular speed ω about a fixed axis and having a moment of inertia I is KE R = 1 _ 2 Iω 2 (9.9) Requirement: ω must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J) Kinetic energy is one part of an object’s total mechanical energy. The total mechanical energy is the sum of the kinetic and potential energies and obeys the principle of conserva- tion of mechanical energy (see Section 6.5). Specifically, we need to remember that transla- tional and rotational motion can occur simultaneously. When a bicycle coasts down a hill, for instance, its tires are both translating and rotating. An object such as a rolling bicycle tire has both translational and rotational kinetic energies, so that the total mechanical energy is E = 1 _ 2 mυ 2 + 1 _ 2 Iω 2 + mgh � Moment of inertia, I � Total mechanical energy � Translational kinetic energy � Rotational kinetic energy � Gravitational potential energy F Axis of rotation r Rope θ s FIGURE 9.21 The force → F does work in rotating the wheel through the angle θ. m r 1 r 2 FIGURE 9.22 The rotating wheel is composed of many particles, two of which are shown. 264 CHAPTER 9 Rotational Dynamics EX AMPLE 13 Rolling Cylinders A thin-walled hollow cylinder (mass = m h , radius = r h ) and a solid cylinder (mass = m s , radius = r s ) start from rest at the top of an incline (Figure 9.23).
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The kinetic energy of the entire rotating body, then, is the sum of the kinetic energies of the particles: Rotational KE 5 S ( 1 2 mr 2 v 2 ) 5 1 2 ( Smr 2 )v 2 u Moment of inertia, I In this result, the angular speed v is the same for all particles in a rigid body and, therefore, has been factored outside the summation. According to Equation 9.6, the term in parenthe- ses is the moment of inertia, I 5 Smr 2 , so the rotational kinetic energy takes the following form: Definition of Rotational Kinetic Energy The rotational kinetic energy KE R of a rigid object rotating with an angular speed v about a fixed axis and having a moment of inertia I is KE R 5 1 2 Iv 2 (9.9) Requirement: v must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J) Kinetic energy is one part of an object’s total mechanical energy. The total mechan- ical energy is the sum of the kinetic and potential energies and obeys the principle of conservation of mechanical energy (see Section 6.5). Specifically, we need to remember that translational and rotational motion can occur simultaneously. When a bicycle coasts down a hill, for instance, its tires are both translating and rotating. An object such as a rolling bicycle tire has both translational and rotational kinetic energies, so that the total mechanical energy is E 5 1 2 mv 2 1 1 2 Iv 2 1 mgh u Total mechanical energy u Translational kinetic energy u Rotational kinetic energy u Gravitational potential energy Figure 9.20 The force F B does work in rotating the wheel through the angle u. F Axis of rotation r Rope θ s B m r 1 r 2 Figure 9.21 The rotating wheel is composed of many particles, two of which are shown. 238 Chapter 9 | Rotational Dynamics Here m is the mass of the object, v is the translational speed of its center of mass, I is its moment of inertia about an axis through the center of mass, v is its angular speed, and h is the height of the object’s center of mass relative to an arbitrary zero level.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The kinetic energy of the entire rotating body, then, is the sum of the kinetic energies of the particles: Rotational KE 5 S ( 1 2 mr 2 v 2 ) 5 1 2 ( Smr 2 )v 2 u Moment of inertia, I In this result, the angular speed v is the same for all particles in a rigid body and, therefore, has been factored outside the summation. According to Equation 9.6, the term in parenthe- ses is the moment of inertia, I 5 Smr 2 , so the rotational kinetic energy takes the following form: Definition of Rotational Kinetic Energy The rotational kinetic energy KE R of a rigid object rotating with an angular speed v about a fixed axis and having a moment of inertia I is KE R 5 1 2 Iv 2 (9.9) Requirement: v must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J) Kinetic energy is one part of an object’s total mechanical energy. The total mechan- ical energy is the sum of the kinetic and potential energies and obeys the principle of conservation of mechanical energy (see Section 6.5). Specifically, we need to remember that translational and rotational motion can occur simultaneously. When a bicycle coasts down a hill, for instance, its tires are both translating and rotating. An object such as a rolling bicycle tire has both translational and rotational kinetic energies, so that the total mechanical energy is E 5 1 2 mv 2 1 1 2 Iv 2 1 mgh u Total mechanical energy u Translational kinetic energy u Rotational kinetic energy u Gravitational potential energy Figure 9.19 The force F B does work in rotating the wheel through the angle u. F Axis of rotation r Rope θ s B m r 1 r 2 Figure 9.20 The rotating wheel is composed of many particles, two of which are shown. 212 Chapter 9 | Rotational Dynamics Here m is the mass of the object, v is the translational speed of its center of mass, I is its moment of inertia about an axis through the center of mass, v is its angular speed, and h is the height of the object’s center of mass relative to an arbitrary zero level.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
3 Rotational motion While we have been discussing Newton’s Laws so far, mass – resistance to acceleration – has been considered as a scalar quantity. No matter which direction you apply a force to a point object, its mass is the same. This simple formalism breaks down when we consider rotational motion. If we have a long cylinder, it matters very much whether we rotate it about an axis down the center of the cylinder or an axis which is perpendicular to the cylinder! This will require us to extend the concept of resistance to acceleration to depend on the direction of the rotation. 3.1 Rotational motion and moment of inertia Consider a mass attached to a string rotating in a circle one complete rotation each second, as shown in Figure 3.1. The speed, the magnitude of the velocity, is constant. Geometry gives us that the arc length s is related to the radius and the angle that the arc transects by s ¼ θr where θ is the angle in radians as shown in Figure 3.2. (Note if θ is 2π radians – 360 degrees – this gives us arc length ¼ circumference ¼ 2πr, and we recover our familiar relation that circumference ¼ π diameter.) The speed is the time derivative of the distance traveled (the arc length): |v| ¼ ds/dt ¼ r dθ/dt ¼ rω where ω ¼ dθ/dt is the angular velocity in radians per second. At each moment, the ball has a velocity which is tangential to the circle. Since the direction of the velocity is constantly changing – even if the speed is constant – the ball is undergoing constant acceleration. This centripetal acceleration is caused by the force exerted on the string to pull the ball into a circular path. As illustrated in Figure 3.3, when the angle changes by a small dθ, the change in velocity d v ! is directed toward the center of the circle.
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