The polar moment of inertia j of an element about an axis perpendicular to its plane is the product of the area of the element and the square. Assume we have an extended body rotating about some. It will not attempt to teach you the calculus involved since you are presumed to have learned it in another course. I is relative to the centroid of the crosssection area more generally moment of inertia is defined as i zz. Physics 111 lecture 09 rotation, moment of inertia sj 8th ed. What is the moment of inertia of the threeobject system. Lecture 11 deflection in beams by moment area method lecture 12 numerical examples on deflection of beams by moment area method module 4 lecture 1 closed coiled helical springs. The moment of inertia, also known as the second moment of area, is basically a geometric property related to some object resistance to rotational movements. The smallest moment of inertia about any axis passes through the centroid. The greater the distance the mass is from the axis of rotation, the larger i g will be. Here, the moment of inertia iplays the same role as the objects mass min f ma. The more mass something has, the more it resists acceleration. Change in rotation from torque moment of inertia i mr2 for a single mass the total moment of inertia is due to the sum of masses at a distance from the axis of rotation.
The moment of inertia, i, of a beam depends on the geometrical properties of the crosssection area a of a beam. Lecture 12 moment of inertia lecture 12 moment of inertia. For example, flywheels have a heavy outer flange that locates as much mass as possible at a greater distance from the hub. Moment of inertia 5 an example of this is the concrete tbeam shown. The centroid, or center of gravity, of any object is the point within that object from which the force of gravity. The inertia tensor we found last time that the kinetic energy and angular momentumof a rotating object were. I m i r i 2 when the distribution of mass is continuous, the discrete sum becomes. Enr202 mechanics of materials lecture 5a slides and notes. Axial load, axial torque, strain energy in spring, numerical examples. N i mi ri 2 i 1 two spheres a spun baton has a moment of inertia due to each separate mass. The moment of inertia of the area about the yaxis and the radius of gyration ky. If this first moment of area is again multiplied by the perpendicular distance between them, the product is known as. Of a body about an axis is the sum of the products of its mass elements and the squares of their distances to the axis.
The letter i is a standard notation for moment of inertia. Physics 111 lecture 09 rotation, moment of inertia sj 8th ed chap 10. Pdf lecture notes on engineering mechanics properties of area. Nov 15, 20 the moment of inertia is a geometrical property of a beam and depends on a reference axis. Lecture notes physics i physics mit opencourseware. The moment of inertia measures the resistance to a change in rotation. If we have a distributed mass we compute the moment of inertia by summing the contribu tions of each of its parts. The moment of inertia about any point or axis is the product of the area and the perpendicular distance between the point or axis to the centre of gravity of the area. Viken kiledjian gives the definition and meaning of moment of inertia and how to find the moment of inertia of 1 or more objects around any axis. Integral represent the moment of inertia of area about the y axis. For a point mass, m, the moment of inertia about the line is. Moments of inertia of geometric areas frame 281 introduction this unit will deal with the computation of second moments, or moments of inertia, of areas.
The polar moment of inertia is the sum of any two moments of inertia about axes at right angles to each other, e. Mass moment of inertia, i g contd i g for a body depends on the bodys mass and the location of the mass. However, if we found the moment of inertia of each section about some. However, the cylinder with larger diameter will have greater moment of inertia larger lever 825 arm than the one with smaller. 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. Here the quantity i is called the objects moment of inertia. The moment of inertia of a composite area a about a given axis is obtained by. Finding moments of inertia, rolling cylinder with hole example finding moments of inertia figure 1. Determine the moment of inertia of the shaded area with respect to the x axis. Introductory moment of inertia example problem example. Calculate first and second moment of area of simple geometry. This is known as the parallel axis theorem 24 examples of moment of inertia.
Lecture notes on engineering mechanics properties of area, first and second moments of area or inertia, transformation, mohrs circle diagram. Mass moment of inertia and area moment of inertia duration. In civil engineering, it is often combined this geometric property with material properties to determine physical quantities such as stiffness. Lecture 2 spring under impact loading and numerical examples. Although it is a simple matter to determine the moment of inertia of each rectangular section that makes up the beam, they will not reference the same axis, thus cannot be added. Notes on torque weve seen that if we define torque as rfsin. Moment of inertia ph1011 physics week 6 lecture 12. This means that moment of inertia or rotational mass is a. Centre of gravity it is defined as the point through which whole weight of body supposed to be act.
It tells us how difficult is to set an object in rotational motion. It is defined as the point through which whole area is assumed to be concentrated. System of particle and rotational motion is an important topic from jee main jee advanced exam point of view. Pdf ge8292 engineering mechanics lecture notes, books. Every year there are questions asked from this topic. Polar moment of inertia moment of inertia about the z axis the polar area moment of inertia of a beams crosssectional area measures the beams ability to resist torsion. Thisclassisusuallyreferredtoasstatics,butwellbecoveringsomeextra. Pdf moment of inertia and friction find, read and cite all the research. The transverse loads cause internal shear forces and bending moments in the beams as shown in figure 1 below. Welcome to lecture summary 5a, which will concentrate on centroid and moments of inertia of cross sectional shape of members. 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. The particular skills you will need are in establishing.
View notes lecture 12 moment of inertia from ph 1011 at nanyang technological university. Lecture 1 introduction, units, linear algebra 0introduction welcometoengineeringmechanicsi. Lecture notes, lecture 11 center of gravity, centroid and moment of inertia. The moment of inertia about an axis of a body is calculated by the summation of mr 2 for every particle in the body, where m is the mass of the particle and r is the perpendicular distance from the axis. Lecture series on strength of materials5 moment of inertia. Compute the moments of inertia of the bounding rectangle and halfcircle with respect to the x axis. Mass is an intrinsic property of an object, or system of objects. This section provides the lecture notes for the course, information on lecture topics, and question posed to the class to stimulate discussion. In the figure below, the two cylinders have the same masses.
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