The moment of inertia, otherwise 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 tensor definition of moment of inertia. Inertia describes the relative amount of resistance to change that an object possesses. A verification procedure for mscnastran finite element. The concept of moment of inertia was introduced by leonhard euler. Then we have the following equation defining the rotation around. The rate of change of the inertia tensor is a linear function of rate of change of mass. They keep doing what they are doing until a force changes their speed or direction.
Inertia tensor article about inertia tensor by the free. Analogously, we can define the tensor of inertia about point o, by writing equation4 in matrix form. 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. Principal axes of inertia weve spent the last few lectures deriving the general expressions for l and trot in terms of the inertia tensor both expressions would be a great deal simpler if the inertia tensor was diagonal. I and j are used as symbols for denoting moment of inertia.
Inertia definition of inertia by the free dictionary. So far ive called the numbers i ij a tensor, but have provided no definition of what a tensor is. In the same way angular momentum can be exchanged between object within a closed system by means of equal and opposite torques. P442 analytical mechanics ii the tensor of inertia c alex r. An introduction for nonphysicists, by dan morris until recently, the depth of my understanding of the inertia tensor was that it tells you how the mass of an object is distributed. In this section students will learn about translational and rotational transformations of. The concept of inertia in physics is defined an d the measures of the mechanical inertia, the thermodynamic inertia and electrocinetic inertia are id entified. The tensor of inertia gives us an idea about how the mass is distributed in a rigid body.
By using the formulas for various components of inertia tensor, one can. Compute the moments of inertia of the bounding rectangle and halfcircle with respect to the x axis. The moment of inertia has two forms, a scalar form i used when the axis of rotation is known and a more general tensor form that does not require knowing the axis of rotation. Because the inertia tensor depends on the coordinates in a clear way, we can write down how it must behave under rotations. Area moments of inertia by integration second moments or moments of inertia of an area with respect to the x and y axes, x. Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the bodys actual distribution of mass, if the total mass of the body were concentrated. An introduction to tensors for students of physics and engineering joseph c. The quantity which allows us to calculate moment of inertia about arbitrary axes like z, etc. Inertia tensor linear momentum is defined as the product of the mass times the translational velocity mass is something that resists change in velocity angular momentum is defined as an angular mass times the angular velocity the angular mass is called the moment of inertia or inertia tensor of the rigid body.
Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. The products of inertia are zero when the body is symmetrical about the axes of. Inertia tensor in different coordinate systems so far ive called the numbers iij a tensor, but have provided no definition of what a tensor is to provide one, consider what happens to the iij if one rotates the coordinate system well start with the simpler case of observing how a vector. For your convenience, i present to you, in a single paragraph, the essence of tensor analysis. Define principal axes and principal moments of inertia. Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the bodys actual distribution of mass, if the total mass of the body were concentrated mathematically the radius of gyration is the root mean square distance of the objects parts from either its center of mass or a given axis. The analogous positive number quantities, moment of inertia of area and moment of inertia of mass, are involved in the analysis of problems of statics and dynamics respectively. Momentofinertiareg, pt, v pt v reg momentofinertiareg. Moment of inertia the moment of inertia of a more complicated object is found by adding up the moments of each individual piece. I xx is the moment of inertia of the body about the xaxis of the body frame and i. Inertia tensor and cross product in ndimensions space m. Mass is distributed uniformly on the ring, therefore, one can define a linear mass. Video created by georgia institute of technology for the course advanced engineering systems in motion. We have already seen that that linear momentum is conserved for a closed system but momentum can be exchanged between objects within the closed system by means of equal and opposite forces.
The inertia tensor is called a rank two tensor because it has two indices. If you think about mass as the difficulty it is to accelerate something linearly. Todays learning outcome is to derive the equation for the velocity of the same point relative to two different frames or bodies in planer motion. The moment of inertia of the shaded area is obtained by subtracting the moment of inertia of the halfcircle from the moment of inertia of the rectangle. The difficulty faced to change the angular motion of any object about an axis is shown or given or measured by calculating moment of inertia about that particular axis. The scalar moment of inertia i often called simply the moment of inertia allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of.
P442 analytical mechanics ii the tensor of inertia. Points, vectors, tensors, dyadics material points of the crystalline sample, of which x and y are examples, occupy a subset of the threedimensional euclidean point space. Vectors, matrices, rotations, axis transformations carnegie mellon mrsec most of the material in these slides originated in lecture notes by prof. Lecture 4 describing rigid bodies 1 the inertia tensor. Now if the two masses are each placed a distance rfrom the axis of rotation eq. In order to estimate the moment of inertia tensor, one needs to calculate the. The moment of inertia, otherwise known as the second moment of area, 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. Compute the product of inertia with respect to the xyaxes by dividing the section into three rectangles. Samuel dixon crossing the niagara river in 1890 the moment of inertia, otherwise known as the second moment of area, 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. Let us first define the position of the center of mass by generalizing the definition we had used for the. Inertia tensor linear momentum is defined as the product of the mass times the translational velocity mass is something that resists change in velocity angular momentum is defined as an angular mass times the angular velocity the angular mass is called the moment of.
The greater the mass the object possesses, the more inertia that it has, and the greater its tendency to not accelerate as much. But be careful you might fall in love with physics. But not all objects accelerate at the same rate when exposed to the same amount of unbalanced force. The inertia tensor is the rotational equivalent of mass in a way. Well the moment of inertia, or inertia tensor, is how difficult it is to rotate that object. Simply put, a tensor is a mathematical construction that eats a bunch of vectors, and spits out a scalar. An introduction to tensors for students of physics and. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. Physics the tendency of a body at rest to remain at rest or of a body in straight line motion to stay in motion in a straight line unless acted on by an outside force. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler, which characterizes simplified approaches to a problem as the poor mans way to do x. I could say that if someone asked me, but i didnt really know what it meant. The moment of inertia describes the angular acceleration produced by an applied torque. Moment of inertia is defined with respect to a specific rotation axis. It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change.
Inertia tensor fully describes the distribution of mass in a system diagonal elements are called moments of inertia offdiagonal elements are called products of inertia reference frame for a system of particles is almost complete. Since the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance from any rotation axis, the calculation of moments of inertia generally involves calculus, the discipline of mathematics which can handle such continuous variables. If any one of the three planes is a symmetric plane, then the product of inertia of the perpendicular planes are zero. Lets talk about where the moment of inertia tensor came from. If i x and i y be moment of inertia of the body about two. I depends on the geometry of the object relative to the axis. G is the tensor of inertia written in matrix form about the center of mass g and with respect to the xyz axes. A verification procedure for mscnastran finite element models.
We now define the components iij of the socalled inertia tensor i by. It depends on the shape and mass distribution of the body. First let me discuss intertia suppose you are riding a bike with high speed. The inertia tensor and after dinner tricks ubc math. A ball sitting still on a table wont start rolling around unless something pushes on it, be it your hand, a gust of. A disk and a point mass figure 1 shows a thin uniform disk of mass m and radius r. Moment of inertia tensor article about moment of inertia. Im going to write this equation in matrix form, and well play with it a little bit before we. The inertia tensor we found last time that the kinetic energy and angular momentumof a rotating object were. May 23, 2019 moment of inertia formula for perpendicular axes theoremthe sum of moment of inertia of a plane laminar body about two mutually perpendicular axes lying in its plane is equal to its moment of inertia about an axis passing through the point of intersection of these two axes and perpendicular to the plane of laminar type body. Determine the moment of inertia of the shaded area with respect to the x axis. The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The estimate inertia tensor block calculates the inertia tensor and the rate of change of the inertia tensor. The offdiagonal terms of is must be multiplied by 1.
Inertia is the resistance of any physical object to a change its state of motion, or the tendency of an object to resist any change in its motion. Let 1 where is the unit vector along the rotation axis of a rotating body, written componentwise as, then 2 where i is the moment of inertia tensor and its components. It illustrates the difference between a tensor and a matrix. Moment of inertia formula for perpendicular axes theoremthe sum of moment of inertia of a plane laminar body about two mutually perpendicular axes lying in its plane is equal to its moment of inertia about an axis passing through the point of intersection of these two axes and perpendicular to the plane of laminar type body. Moment of inertia tensor synonyms, moment of inertia tensor pronunciation, moment of inertia tensor translation, english dictionary definition of moment of inertia tensor. The resistance that is shown by the object to change its rotation is called moment of inertia. Tensor which depends on two directions one normal to the area and other that of the force on it. Moment of inertia definition of moment of inertia by the. Linear interpolation is used to estimate the inertia tensor as a function of mass. Looking up the positive yaxis, it is clear that the angular momentum vector l, defined from l rxp, points outwards. Moment of inertia definition is a measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the elements distance from the axis. Inertia ellipsoid from eric weissteins world of physics. Hagehassan universite libanaise, faculte des sciences section 1 hadathbeyrouth abstract we demonstrated using an elementary method that the inertia tensor of a material point. Were thinking here of an idealized solid, in which the distance between any two internal points stays the same as the body moves around.
The off diagonal elements of the inertia tensor are called the products of inertia. Principal axes of rotation university of texas at austin. I will insist to both science as well as nonscience background students to go through the answer. We have seen that the moment of inertia tensor, defined in section 8.
The weight should be the same in all three directions unless scalar masses are used. For example, the moment of inertia of the system shown in fig. One of the confusing aspects, i think, is that it seems like a completely separate quantity. Moment of inertia definition of moment of inertia by. In case of revolution bodies, the body will be symmetric about two axes, hence two planes will be symmetric. Moment of inertia wikimili, the best wikipedia reader. A quantity having magnitude only is called scalar and a quantity with magnitude and direction both, called vector. The easiest way to differentiate these quantities is through their units. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. What really happens is that when you compute the kinetic energy of a rigid body rotating th. Inertia is a quality of all objects made of matter that possess mass.
Oct 14, 20 the moment of inertia can also be called the mass moment of inertia especially by mechanical engineers to avoid confusion with the second moment of area, which is sometimes called the moment of inertia especially by structural engineers and denoted by the same symbol i. Learn how the distribution of mass can affect the difficulty of causing angular acceleration. Note that the inertia matrix, is, is not in tensor form. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. Also define 3 then 4 this can be transformed to the principal axes such that 5. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. The moments and product of inertia for an area are plotted as shown and used to construct mohrs circle, mohrs circle may be used to graphically or analytically determine the moments and product of inertia for any other rectangular axes including the principal axes and principal moments and products of inertia. Product of inertia of mass is the symmetric measure for a body. A relation between the area of a surface or the mass of a body to the position of a line.
Phys 211 lecture 23 inertia tensor 23 simon fraser university. Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. The secondorder cauchy stress tensor describes the stress forces experienced by a material at a given point. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. Note here that a local reference frame, the xyz system, is defined and fixed to the.