Moment of inertia and equations of motion wiley online library. In this figure, 5 denotes the position vector of a small mass element dm from the center of mass. Rigidbody dynamics studies the movement of systems of interconnected bodies under the action of external forces. If c is an arbitrary vector, then i abc acb x i m ir 2 i c 2 r i c 03. Feu institute of technology civil engineering department characteristics of translation give the formulas, including the derivations. Mass moment of inertia of rigid bodies the following content is provided under a creative commons license.
Dynamics a constant horizontal force p is applied the light yoke attached to the center o of a uniform circular disk of mass m, which is initially at rest and rolls without slipping. To make a donation or to view additional materials from hundreds of mit courses, visit mit opencourseware at ocw. In a physics simulation of rigid bodies, if i have a cube with a known mass and moment of inertia tensor, and i attach it to another cube with a known mass and moment of inertia tensor such that its. Consider a rigid body as being made up of particles with masses m 1, m 2.
Your support will help mit opencourseware continue to offer high quality educational resources for free. Rigid bodies moment of inertia the inability of a body to change by itself its position of rest or uniform motion is called inertia. If dm represents the mass of any infinitesimal particle of the body and r is the perpendicular distance of this particle from the axis of. Notes on calculation of moment of inertia of rigid bodies. Its eigenvectors are special directions within the rigid body called the principal axes.
Rotational motion is more complicated than linear motion, and only the motion of rigid bodies will be considered here. The moment of inertia for continuous bodies this section is similar to the calculation of the center of mass for a continuous body, so i wont repeat all the boring steps. G is the mass moment of inertia for a body about an axis passing through the bodys mass center, g. Calculate the moment of inertia of a cylinder of mass m and radius r about an axis through the center of the cylinder. Continuous systems and rigid bodies the dynamical equations of motion are developed using classical eulerian and newtonian mechanics.
This rotational inertia of the body is called the m. The simulation of the dynamic interaction among rigid bodies takes into account various physical characteris tics such as elasticity, friction, mass, and moment of inertia to produce rolling and sliding contacts. Chapter 11 rotation of a rigid body about a fixed axis. If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis k perpendicular to.
Rigid body inertia about alternate points continuous. Objects deform elastically, but these deformation are negligible for a wide range of problems. Rigid body dynamics november 15, 2012 1 noninertial frames of reference so far we have formulated classical mechanics in inertial frames of reference, i. A rigid body is defined as an object that has fixed size and shape. Assume that the body is made of a large number of particles. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. Moment of inertia illinois institute of technology. Find the principal moment of inertia and the principal axes of inertia for the. Rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces. This section provides materials from a lecture session on angular momentum and motion of rotating rigid bodies.
Angular momentum and motion of rotating rigid bodies. Take measurements for at least 6 different r values spanning the length of the rod. Emphasis is placed on rigid body angular momentum and kinetic energy expression that are shown in a coordinate frame agnostic manner. The rigid body is rotating with an angular velocity. Chapter 11 dynamics of rigid bodies university of rochester.
Rigid body motion in this chapter we develop the dynamics of a rigid body, one in which all interparticle distances are xed by internal forces of constraint. The inertia tensor is really an invariant, thats it for this space craft has one inertia tensor. The motion of a translating body moving in a curve line. Rotation of a rigid body not all motion can be described as that of a particle. A method for the measurement of the inertia properties of rigid bodies is presented.
The angular mass is called the moment of inertia or inertia tensor of the rigid body. Rigid body inertia tensor continuous systems and rigid. The eigenvalues of the tensor, i 1, i 2 and i 3, are called. Mass moment of inertia of rigid bodies mit opencourseware. Mar 27, 2016 how to find moment of inertia of rigid bodies 1. A method for measuring the inertia properties of rigid bodies.
The radius of the inner cylinder is r and the outer is 2r. The tensor of inertia gives us an idea about how the mass is distributed in a rigid body. In this chapter we will consider the motion of solid objects under the application of forces and torques. Pdf inertial rotation of a rigid body researchgate. I g is the mass moment of inertia for a body about an axis passing through the bodys mass center, g. Chapter 11 rotation of a rigid body about a fixed axis we now broaden our interest to include the rotation of a rigid body about a fixed axis of rotation. If dm represents the mass of any infinitesimal particle of the body and r is the perpendicular distance of this particle from the axis of rotation, the moment of inertia of the object. The angular mass is called the moment of inertia or inertia tensor of the rigid body if you spin in your chair while extending your legs, and then suddenly pull your legs closer the chair spins faster reduces when you pull your legs closer. For a rigid body, the angular momentum l is the product of the moment of inertia and the angular velocity. Typical examples are the solar system, mechanisms in machines and living mechanisms such as the human body provided its individual members can be considered as rigid.
Moment of inertia of rigid body depends on the distribution of mass about the axis of rotation. Calculate the individual moment inertia of each shape around a the prescribed axis in the same coordinates system, and their individual origins. Thus the mass of the body is taken as a measure of its inertia for translatory. Eulers angles in many textbooks also this latter set of rotations is often referred to as eulers angles.
The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames. Objects deform elastically, but these deformation are negligible for a. Rigid bodies are such an important part of classical mechanics that we have developed special techniques to calculate their kinetic energies and angular momenta. Given a rigid body and its mass, the method allows to measure identify the centre of gravity location and the. Moment of inertia is known as the angular mass or rotational inertia,of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis. Moment of inertia is known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis. The simplest extendedbody model that can be treated is that of a rigid body, one in. Inertial rotation of a rigid body about one of the principal axes of inertia is very simple. Plane kinetics of rigid bodies relates external forces acting on a body with the translational and rotational motions of the body discussion restricted to motion in a single plane for this course body treated as a thin slab whose motion is confined to the plane of slab plane containing mass center is generally considered as plane of motion all forces that act on the body get projected. R is the angular velocity of the rod in reference frame e. The attempt at a solution b i thought of subtracting the big cylinder inertia from the small and adding the hanging body and. The greater the mass of the body, the greater its inertia as greater force is required to bring about a desired change in the body. To keep things simple, we wont consider a general rigid body right away. In the following analysis we will limit our study to planar kinetics to rigid bodies which, along with.
The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis. The definition of the moment of inertia is irdmz 2, where we think about the integral. The motion of a translating body moving in a straight line. Thus the mass of the body is taken as a measure of its inertia for translatory motion. Here the quantity i is called the objects moment of inertia. A rigid body is an object with a mass that holds a rigid shape, such as a phonograph turntable, in contrast to the sun, which is a ball of gas. Determine the velocity v of the center o in terms of t. Physics 191 calculating the moment of inertia rotational. Homework statement determine the moment of inertia of a rigid body on the picture. Physics 191 calculating the moment of inertia rotational inertia of rigid bodies. Let us consider a rigid body mounted on a rigid shaft as shown in figure a.
Chapter 11 dynamics of rigid bodies a rigid body is a collection of particles with fixed relative positions, independent of the motion carried out by the body. G is the tensor of inertia written in matrix form about the center of mass g and with respect to the xyz axes. Many of the equations for the mechanics of rotating objects are similar to the motion. Sunil kumar singh this work is produced by the connexions project and licensed under the creative commons attribution license abstract moment of inertia of rigid body depends on the distribution of mass about the axis of rotation. This is, of course, an idealization which ignores elastic and plastic deformations to which any real body is susceptible, but it is an excellent approximation for. In this section, we determine how to calculate the angular momentum and kinetic energy of a rigid body, and define two important quantities. Hence has to increase to keep angular momentum constant. Materials include a session overview, assignments, lecture videos, recitation videos and notes, and a problem set with solutions.
It depends on the bodys mass distribution and the axis chosen, with. Rotational motion chapters 9 and 10 are about rotation start with fixed axis motion rotational kinematics. As such, the inertia tensor is the angular analogue to mass. Plane kinetics of rigid bodies indian institute of. Many of the equations for the mechanics of rotating objects are similar to the motion equations for linear motion. In other words the ith particle will have a mass of m i and distance r i. Rotational motion of a rigid body notes rigid body dynamics. Advanced classical mechanicsrigid bodies wikiversity.
Lets calculate the kinetic energy of a mass in rigid body motion. Indeed, during this rotation directions of vectors l and. The moment of inertia mi of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis. Use parallel axis theorem to transform to inertia to unified object coordinates. In the module titled rotation of rigid body 1, we derived expressions of moments of inertia mi for di erent object forms as.
Why do nonrigid bodies try to increase their moment of inertia. Plane kinetics of rigid bodies mass moments of inertia radius of gyration k about an axis for which i is defined. L i for a point of mass, angular momentum can be expressed as the product of linear momentum and the radius r. An objects moment of inertia depends on the axis of rotation. So generally, to have a rigid body your inertia tensor, which is what im showing here, will be a three by three matrix when you numerically evaluate it. For the motion of the mass center g of the body with respect to the newtonian frame oxyz, f ma for the motion of the body with. The dynamics of a rigid body has been discussed in our introductory courses, and the techniques discussed in these courses allow us to solve many problems in which. The kinematical properties of a rigid body are fully determined by its mass, principal axes, and moments of inertia. Theres an infinity of answers, because i could pick a infinity of body frames to break out these vectors and do the tildes and compute this. If you spin in your chair while extending your legs, and then suddenly.
Forces and accelerations consider a rigid body acted upon by several external forces. Ir is the moment of inertia tensor of the rod relative to the center of mass and f. Realistic animation of rigid bodies computer graphics. When an object consists of a continuous distribution of mass, its moment of inertia can be calculated by integrating the moment of inertia of its small part. The moment of inertia about an axis going through point p a distance d away from the center of mass is equal to the moment of inertia about the center of mass plus md2. You can easily fill those in if you understand how to do multiple integration and you understand how to get the formula for the center of mass of a continuous body. When the rigid body rotates the speed v i of the ith particle is v i r i different particles have different speeds and.
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