MASS MOMENT OF INERTIA Today’s Objectives: Students will be able to : a) Explain the concept of the Mass Moment of Inertia (MMI) b) Determine the MMI of a composite body In-Class Activities: • • • • • • • • Statics, Fourteenth Edition R.C Hibbeler Check homework, if any Reading quiz Applications MMI: concept and definition Determining the MMI Concept quiz Group problem solving Attention quiz Copyright ©2016 by Pearson Education, Inc All rights reserved READING QUIZ The formula definition of the mass moment of inertia about an axis is _ A) ∫ r dm B) ∫ r dm C) ∫ m dr D) ∫ m dr The parallel-axis theorem can be applied to A) Only the MoI B) Only the MMI C) Both the MoI and MMI D) None of the above Note: MoI is the moment of inertia of an area and MMI is the mass moment inertia of a body Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS The large flywheel in the picture is connected to a large metal cutter The flywheel is used to provide a uniform motion to the cutting blade while it is cutting materials What property of the flywheel is most important for this use? How can we determine a value for this property? Why is most of the mass of the flywheel located near the flywheel’s circumference? Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) If a torque M is applied to a fan blade initially at rest, its angular speed (rotation) begins to increase Which property (which we will call P) of the fan blade you think effects the angular acceleration (α) the most? How can we determine a value for this property? What is the relationship between M, P, and α? Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved MASS MOMENT OF INERTIA Consider a rigid body with a center of mass at G It is free to rotate about the z axis, which passes through G Now, if we apply a torque T about the z axis to the body, the body begins to rotate with an angular acceleration α T and α are related by the equation T = I α In this equation, I is the mass moment of inertia (MMI) about the z axis The MMI of a body is a property that measures the resistance of the body to angular acceleration This is similar to the role of mass in the equation F = m a The MMI is often used when analyzing rotational motion (done in dynamics) Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved DEFINITION OF THE MMI Consider a rigid body and the arbitrary axis p shown in the figure The MMI about the p axis is defined as I = ∫m r dm, where r, the “moment arm,” is the perpendicular distance from the axis to the arbitrary element dm The MMI is always a positive quantity and has a unit of kg·m or slug·ft Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved RELATED CONCEPTS Parallel-Axis Theorem Just as with the MoI for an area, the parallel-axis theorem can be used to find the MMI about a parallel axis z that is a distance d from the z’ axis through the body’s center of mass G The formula is Iz = IG + (m) (d) (where m is the mass of the body) m The radius of gyration is similarly defined as k = √(I / m) Finally, the MMI can be obtained by integration or by the method for composite bodies The latter method is easier for many practical shapes Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE Given: The volume shown with ρ = slug/ft Find: The mass moment of inertia of this body about the y-axis Plan: Find the mass moment of inertia of a disk element about the y-axis, dIy, and integrate Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE (continued) Solution: The moment of inertia of a disk about an axis perpendicular to its plane is I = 0.5 m r Thus, for the disk element, we have dIy = 0.5 (dm) x where the differential mass dm = ρ dV = ρπx dy Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved CONCEPT QUIZ Consider a particle of mass kg z located at point P, whose coordinates •P (3,4,6) are given in meters Determine the MMI of that particle about the z axis A) kg·m C) 25 kg·m y B) 16 kg·m D) 36 kg·m x Consider a rectangular frame made of four slender bars with four axes (zP, zQ, zR and zS) perpendicular to the screen and passing through the points P Q P, Q, R, and S respectively About which of the four axes will the MMI of the • • frame be the largest? A) zP D) zS B) zQ S• C) zR E) Not possible to determine Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved •R GROUP PROBLEM SOLVING Given: The pendulum consists of a kg plate and a kg slender rod R Find: The radius of gyration of the pendulum about an axis P Plan: perpendicular to the screen and passing through point G Determine the MMI of the pendulum using the method for composite bodies Then determine the radius of gyration using the MMI and mass values Solution: Separate the pendulum into a square plate (P) and a slender rod (R) Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING (continued) The center of mass of the plate and rod are 2.25 m and m from point O, respectively R ∼ y = (Σ y m) / (Σ m ) = {(1) + (2.25) 5} / (3+5) = 1.781 m P The MMI data on plates and slender rods are given on the inside cover of the textbook Using those data and the parallel-axis theorem, 2 2 IP = (1/12) (0.5 + ) + (2.25 − 1.781) = 1.621 kg·m 2 IR = (1/12) (2) + (1.781 − 1) = 2.830 kg·m Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING (continued) IO = IP + IR = 1.621 + 2.830 = 4.45 kg·m R Total mass (m) equals kg Radius of gyration P k = √IO / m = √4.45 / = 0.746 m Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved ATTENTION QUIZ A particle of mass kg is located m down the y-axis What are the MMI of z the particle about the x, y, and z axes, respectively? A) (2, 0, 2) B) (0, 2, 2) C) (0, 2, 2) D) (2, 2, 0) 1m y • x Consider a rectangular frame made of four slender bars and four axes (zP, zQ, zR and zS) P Q • • perpendicular to the screen and passing through points P, Q, R, and S, respectively About which of the four axes will the S• MMI of the frame be the lowest? A) zP D) zS Statics, Fourteenth Edition R.C Hibbeler B) zQ C) zR E) Not possible to determine Copyright ©2016 by Pearson Education, Inc All rights reserved •R End of the Lecture Let Learning Continue Statics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved ... near the flywheel’s circumference? Statics, Fourteenth Edition R. C Hibbeler Copyright 201 6 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) If a torque M is applied to... MMI can be obtained by integration or by the method for composite bodies The latter method is easier for many practical shapes Statics, Fourteenth Edition R. C Hibbeler Copyright 201 6 by Pearson... 2.830 kg·m Statics, Fourteenth Edition R. C Hibbeler Copyright 201 6 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING (continued) IO = IP + IR = 1.621 + 2.830 = 4. 45 kg·m R Total