Chapter 17_1

Chapter Overview

• Course Title: Engineering Mechanics: Dynamics, Fifteenth Edition

• Chapter Focus: Planar Kinetics of a Rigid Body: Force and Acceleration

• Copyright: Copyright © 2022 Pearson Education, Inc. All Rights Reserved

Objective of the Lesson

• Today’s Objectives:

  • Determine the mass moment of inertia of rigid bodies.

  • In-Class Activities:

    • Check Homework

    • Reading Quiz

    • Applications

    • Mass Moment of Inertia

    • Parallel-Axis Theorem

    • Composite Bodies

    • Concept Quiz

    • Group Problem Solving

    • Attention Quiz

Reading Quiz

• Quiz Questions:

  1. Mass moment of inertia measures the resistance of a body to:A) Translational motionB) DeformationC) Angular accelerationD) Impulsive motion

  2. Mass moment of inertia is always:A) A negative quantityB) A positive quantityC) An integer valueD) Zero about an axis perpendicular to the plane of motion

Applications

• Flywheel Example:

  • A flywheel connected to a cutter provides uniform motion to the cutting blade.

  • Important Property: Mass moment of inertia, as mass is most impactful when concentrated near the circumference.

• Crank Example:

  • The crank on an oil-pump rig develops kinetic energy related to mass moment of inertia.

  • Inquiry: Is the mass moment of inertia about the rotation axis larger or smaller than about its center of mass?

Mass Moment of Inertia

• Definition: Mass moment of inertia (MMI) measures a body's resistance to angular acceleration.

• Equation: ( T = I \cdot \alpha )

  • where ( T ) is the torque, ( I ) is the mass moment of inertia about the z-axis, and ( \alpha ) is angular acceleration.

• General Formula:

  • ( I = \int r^2 dm )

  • where ( r ) is the moment arm (perpendicular distance from the axis).

• Units: kg⋅m² or slug⋅ft².

Procedure for Analysis

• Direct Integration:

  • Focus on symmetric bodies generated by revolving curves.

  • Shell Element:

    • Volume element: ( dV = 2 \pi y z dy )

  • Disk Element:

    • Volume element: ( dV = \pi y^2 dz )

Parallel-Axis Theorem

• Mass moment of inertia about a parallel axis: [ I = I_G + md^2 ]

  • where ( I_G ) is the MMI about the mass center, ( m ) is mass, and ( d ) is the distance between axes.

Radius of Gyration and Composite Bodies

• Radius of Gyration ( (k) ): Measures mass distribution about an axis.

  • [ I = mk^2 ]

• Composite Bodies: MMI can be found by summing individual moments for simple shapes.

Example Problems

• Example I: Determine MMI about the y-axis for a given volume.

• Example II: Calculate the radius of gyration for a pendulum consisting of a slender rod and a circular plate.

Concept Quiz

• Questions:

  1. The mass moment of inertia of a rod about a transverse axis at its end is: A) ( \frac{1}{12} m L^2 ) B) ( \frac{1}{6} m L^2 ) C) ( \frac{1}{3} m L^2 ) D) ( m L^2 )

  2. The mass moment of inertia of a thin ring about the z-axis is: A) ( \frac{1}{2} m R^2 ) B) ( m R^2 ) C) ( \frac{1}{4} m R^2 ) D) ( 2m R^2 )

Group Problem Solving

• Example of a Pendulum:

  • Given masses of parts, determine radius of gyration using MMI and mass values.

  • MMI calculations require the use of the parallel-axis theorem.

• Example II: Calculate MMI for a composite wheel comprising rings and rods using MMI data and ensure calculations reflect distance from either the center or parallel axes.

Attention Quiz

• Questions:

  1. Mass moment of inertia about the center of mass is always:A) MaximumB) MinimumC) ZeroD) None of the above

  2. For equal mass bodies A and B, if ( k_A ) and ( k_B ) are radii of gyration, then:A) ( I_A = I_B )B) ( I_A = \frac{1}{2} I_B )C) ( I_A = 4 I_B )D) ( I_A = \frac{1}{4} I_B )

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