Rotational Inertia

12 Key excerpts on "Rotational Inertia"

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  Advanced Mechanics (Important Branch of Physics)

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  The configuration space of a nonfixed (with non-zero translational motion) rigid body is E + (3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations and rotations). ________________________ WORLD TECHNOLOGIES ________________________ Moment of inertia In classical mechanics, moment of inertia, also called mass moment of inertia , rota-tional inertia , polar moment of inertia of mass , or the angular mass , (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia. While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion. The concept was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765. In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia. Overview The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. Therefore, it encompasses not just how much mass the object has overall, but how far each bit of mass is from the axis. The farther out the object's mass is, the more Rotational Inertia the object has, and the more force is required to change its rotation rate. For example, consider two hoops, A and B, made of the same material and of equal mass.

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  Reading Physics

  A Guide to Understanding Basic Classical Mechanics without Mathematical Expressions

  • Jae Jun Kim, Ph.D.(Authors)
  • 2023(Publication Date)
  • Universal Publishers

    (Publisher)

  Let us look at the colors. You might already have noticed that the items in light gray are ones that change as a function of time and the ones in gray are not. For instance, in a linear motion, the positions in the horizontal and vertical directions change, so they are colored gray. In a rotational motion, the size does not change, so it is colored gray. Given all that, here comes an important question. If angle is something that is going to change, but the distance and the mass are not, why not combine the ones that do not change together? Answer: That is exactly where we introduce the moment of inertia. In a rotational motion, the moment of inertia is not going to change as a function of time. In short, we can treat the moment of inertia just like we do mass in a linear motion.

  Remember that we do not always need to introduce the moment of inertia to describe rotational motion. It is just that introducing the quantity makes studying and understanding the rotational motion easier, so do so. We take the quantities that change as a function of time from the ones that do not change. When combining all the quantities that do not change, as colored in gray in Figure 33.1 , we end up with the moment of inertia in a rotational motion.

  How do we define the moment of inertia mathematically? It is mass multiplied by distance squared. Is that the end of the story? Probably not. For a test particle, it is as simple as that. If there is a point and if we are to describe its moment of inertia, that is all we need to understand. However, when calculating the size of the moment of inertia, we calculate the momentum of inertia for an individual piece of mass within the object and add them up together in the end, which is where we need to introduce calculus.

  For instance, the moment of inertia for a ring with respect to a line passing the center is just mass multiplied by the radius of the ring. However, when you do the same but for a solid sphere, an individual chunk of mass within the sphere has a different distance with respect to the center of the sphere, so we need to introduce calculus to calculate the total moment of inertia associated with the shape of the object; you do the same for any other object with a different shape.

  Figure 33.1: In a two-dimensional linear motion, we need two quantities in length and one in mass to describe its motion. In a rotational motion, what do we need instead?

  Figure 33.2: Imagine that we have a solid sphere that rotates with respect to an axis that passes through the center. The two small circles in the figure represent the two infinitesimally small pieces within the sphere. In terms of the displacement associated with the two small pieces per rotation of the sphere, the size of the displacement is going to be larger for the one that is farther from the axis, which is the one with the larger size arrow in light gray. Question: how do you accommodate for the difference in the distance to the rotational axis when calculating the size of the moment of inertia? We cannot simply assume that the moment of inertia is going to be simply the total mass times the radius squared. Why? All the small pieces shown in the figure have different sizes for distance.

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  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The forces in such a pair have the same line of action, so they have identical lever arms and produce torques of equal magnitudes. One torque is counterclockwise, while the other is clockwise, the net torque from the pair being zero. It can be seen from equation 9.6 that the moment of inertia depends on both the mass of each particle and its distance from the axis of rotation. The farther a particle is from the axis, the greater is its contribution to the moment of inertia. Therefore, although a rigid object possesses a unique total mass, it does not have a unique moment of inertia, as indicated by example 9. This example shows how the moment of inertia can change when the axis of rotation changes. The procedure illustrated in example 9 can be extended using integral calculus to evaluate the moment of inertia of a rigid object with a continuous mass distribution, and table 9.1 gives some typical results. These results depend on the total mass of the object, its shape, and the location and orientation of the axis. When forces act on a rigid object, they can affect its motion in two ways. They can produce a translational acceleration a (components a x and a y ). The forces can also produce torques, which can cause the object to have an angular acceleration . In general, we can deal with the resulting combined motion by using Newton’s second law. For the translational motion, we use the law in the form ∑F = ma. For the rotational motion of a rigid object about a fixed axis, we use the law in the form ∑ = I. When a (both components) and  are zero, there is no acceleration of any kind, and the object is in equilibrium. This is the situation CHAPTER 9 Rotational dynamics 229 already discussed in section 9.2. If any component of a is nonzero or if  is nonzero, we have accelerated motion, and the object is not in equilibrium. Examples 10 and 11 deal with this type of situation.

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  Scientific Foundations of Engineering

  • 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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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The latter quantity is the moment of inertia I of the body: Moment of inertia of a body I 5 m 1 r 1 2 1 m 2 r 2 2 1 . . . 1 m N r N 2 5 Smr 2 (9.6) In this equation, r is the perpendicular radial distance of each particle from the axis of rotation. Combining Equation 9.6 with Equation 9.5 gives the following result: (b) Axis f 43 f 34 f 34 = – f 43 Internal forces m 4 m 3 Axis m 4 m 3 m 1 m 2 r 2 r 1 (a) Figure 9.16 (a) A rigid body consists of a large number of particles, four of which are shown. (b) The internal forces that particles 3 and 4 exert on each other obey Newton’s law of action and reaction. Rotational Analog of Newton’s Second Law for a Rigid Body Rotating About a Fixed Axis Net external torque 5 St 5 Ia (9.7) Requirement: a must be expressed in rad/s 2 . Moment of inertia Angular acceleration 3 a a b b The version of Newton’s second law given in Equation 9.7 applies only for rigid bodies. The word “rigid” means that the distances r 1 , r 2 , r 3 , etc. that locate each particle m 1 , m 2 , m 3 , etc. (see Figure 9.16a) do not change during the rotational motion. In other words, a rigid body is one that does not change its shape while undergoing an angular acceleration in response to an applied net external torque. The form of the second law for rotational motion, St 5 Ia, is similar to the equation for translational (linear) motion, SF 5 ma, and is valid only in an inertial frame. The moment of inertia I plays the same role for rotational motion that the mass m does for 232 Chapter 9 | Rotational Dynamics translational motion. Thus, I is a measure of the Rotational Inertia of a body. When using Equation 9.7, a must be expressed in rad/s 2 , because the relation a T 5 ra (which requires radian measure) was used in the derivation. When calculating the sum of torques in Equation 9.7, it is necessary to include only the external torques, those applied by agents outside the body.

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  Introduction to Physics

  • 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.

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  An Introduction to System Modeling and Control

  • John Chiasson(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  5 Rigid Body Rotational Dynamics 5.1 Moment of Inertia The equation s of motion of a rigid body that is constrained to rotate about a fixed axis are reviewed here briefly. Consider the cylinder shown in Figure 5.1. Figure 5.1 Cylinder constrained to rotate about a fixed axis. The approach here is to obtain the equations of motion of the cylinder by first obtaining an expression for its kinetic energy. To do so, denote the angular speed of the cylinder by and the mass density of the material making up the cylinder by. Then consider the cylinder to be made up of a large number of small pieces of material where the th piece has mass This is illustrated in Figure 5.2. Each piece of mass is rotating at the same angular speed so that the linear speed of is where is the distance of from the axis of rotation. The kinetic energy of is given by Figure 5.2 Cylinder is considered to be made up of small masses The total kinetic energy is then Dividing the cylinder into f iner and finer pieces so that and, the sum becomes the integral The quantity is called the moment of inertia. Using the kinetic energy of the cylinder may now be written as Taking the axle radius to be zero, the moment of inertia of the cylinder (assuming the mass density is constant) is computed to be where is the total mass of the cylinder. 5.2 Newton's Law of Rotational Motion The kinetic energy is now used to derive a relationship between torque and angular acceleration. Recall from elementary mechanics that the work done on a mass by an external force equals the change in its kinetic energy

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The SI unit for moment of inertia is kg · m 2 . If all objects were single particles, it would be just as convenient to use the second law in the form F T = ma T as in the form  = I. The advantage in using  = I is that it can be applied to any rigid body rotating about a fixed axis, and not just to a particle. To illustrate how this advantage arises, Figure 9.16a shows a flat sheet of material that rotates about an axis perpendicular to the sheet. The sheet is composed of a number of mass particles, m 1 , m 2 , . . . , m N , where N is very large. Only four particles are shown for the sake of clarity. Each particle behaves in the same way as the model airplane in Figure 9.15 and obeys the relation  = (mr 2 ): τ 1 = ( m 1 r 1 2 ) α τ 2 = ( m 2 r 2 2 ) α ⋅ ⋅ ⋅ τ N = ( m N r N 2 ) α In these equations each particle has the same angular acceleration , since the rotating object is assumed to be rigid. Adding together the N equations and factoring out the common value of , we find that Σ τ = (Σ mr 2 ) α (9.5) { Moment of inertia I { Moment of inertia Net external torque { FIGURE 9.15 A model airplane on a guide- line has a mass m and is flying on a circle of radius r (top view). A net tangential force F T → acts on the plane. F T r 90° (b) Axis f 43 f 34 f 34 = – f 43 Internal forces m 4 m 3 Axis m 4 m 3 m 1 m 2 r 2 r 1 (a) FIGURE 9.16 (a) A rigid body consists of a large number of particles, four of which are shown. (b) The internal forces that particles 3 and 4 exert on each other obey Newton’s law of action and reaction. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis 237 where the expression Σ τ = τ 1 + τ 2 + ⋅ ⋅ ⋅ + τ N is the sum of the external torques, and Σ mr 2 = m 1 r 1 2 + m 2 r 2 2 + ⋅ ⋅ ⋅ + m N r N 2 represents the sum of the individual moments of inertia. The latter quantity is the moment of inertia I of the body: Moment of inertia of a body I = m 1 r 1 2 + m 2 r 2 2 + .

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  Available until 16 Feb |Learn more

  Classical Mechanics

  A Computational Approach with Examples Using Mathematica and Python

  • Christopher W. Kulp, Vasilis Pagonis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 11

  Rigid Body Motion

  We begin this chapter by reviewing the rotational motion of a single particle around an arbitrary axis, and the concepts of the moment of inertia and the center of mass. In particular we focus on how the center of mass simplifies the description of the translational and rotational motion of a system of particles. After the review, we explore generalized definitions of the moment of inertia, including products of inertia and the inertia tensor, and we demonstrate how to calculate these quantities for a variety of solids. We will see that the moment of inertia tensor of a solid depends on the choice of the coordinate system. Furthermore, we will discuss the parallel axis theorem for rigid bodies and show how it can be used to calculate the moment of inertia tensor. This is followed by a discussion of eigenvalues and eigenvectors of matrices, and how they can be used to describe the principal axes of a rigid body. The chapter will conclude with a discussion of the Euler equations and how they can be used to describe the precessional motion of spinning tops and gyroscopes.

  11.1Rotational motion of particles around a fixed axis

  In this section, we will review rotational motion concepts that you likely learned in your introductory physics course. Here we will focus on the rotational motion of a particle that is confined to move in a circle of radius r in the xy-plane with an angular velocity

  ω =

  ω z

  k ^

  , as illustrated in Figure 11.1

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  Physics

  • 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.

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  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  10.6 Calculation of Moments of Inertia The moment of inertia of a system of discrete particles can be calculated in a straightforward way with Equation 10.19. On the other hand, suppose we consider a continuous rigid object. We can evaluate its moment of inertia by imagining the Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 264 Chapter 10 Rotation of a Rigid Object About a Fixed Axis object to be divided into many small elements, each of which has mass Dm i . We use the definition I 5 o i r i 2 Dm i and take the limit of this sum as Dm i S 0. In this limit, the sum becomes an integral over the volume of the object: I 5 lim Dm i S 0 o i r i 2 Dm i 5 # r 2 dm (10.20) It is usually easier to calculate moments of inertia in terms of the volume of the elements rather than their mass, and we can easily make that change by using Equation 1.1, r ; m/V, where r is the density of the object and V is its volume. From this equation, the mass of a small element is dm 5 r dV. Substituting this result into Equation 10.20 gives I 5 # rr 2 dV (10.21) If the object is homogeneous, r is constant and the integral can be evaluated for a known geometry. If r is not constant, its variation with position must be known to complete the integration. The density given by r 5 m/V sometimes is referred to as volumetric mass density because it represents mass per unit volume. Often we use other ways of express- ing density. For instance, when dealing with a sheet of uniform thickness t , we can define a surface mass density s 5 m/ A 5 rt , which represents mass per unit area.

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  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  For example, if the rotation of a body in a problem takes place about a fixed axis, the problem involves pure rotation; we may then ignore any translational aspects of the rotating body. The kinetic energy of the body is then given by K = 1/2 I ω 2. If the system under consideration is a rigid body rotating about a fixed axis, any particle located at a distance r from the axis of rotation has translational displacement, tangential speed, and tangential acceleration related to its angular variables by s = r θ, v t = r ω, and a t = r α. If a rotating body in a problem is not rigid, that is, if the distances between particles that comprise the body change, the problem is probably one that utilizes the principle of conservation of angular momentum. Before we conclude that the angular momentum of a system is conserved, we must be certain that external forces produce no net torque on the system (the necessary condition for conservation of angular momentum). When angular momentum of a system is conserved, the equation I i ω i = I f ω f relates the angular velocities and moments of inertia of the system before (initially) and after (finally) the configuration has been changed to give the system a different moment of inertia. Rolling (or unwinding) bodies translate and rotate simultaneously. If they roll without slipping, the equations s = r θ, v t = r ω, and a t = r α relate the translational and rotational variables of points on or within the rolling body

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