Department 0f Mathematical Sciences

MTS 411- Dynamics of a Rigid Body

COURSE PARTICULARS Course Code: MTS 411 Course Title: Dynamics of a Rigid Body No. of Units: 3 Course Duration: Two hours of lecture and one hour of tutorial per week for 15 weeks. Status: Compulsory Course Email Address: [email protected] Course Webpage: http://www.fwt.futa.edu.ng/courseschedule.php?coursecode=FWT%20204 Prerequisite: MTS 210

COURSE INSTRUCTORS Professor S. T. Oni Room 05, Alhaji (Dr.) Adamu Abdulahi Academic Staff Office Complex, Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria. Phone: +2348033502059 Email: [email protected]

COURSE DESCRIPTION This course begins with a clear statement of pertinent definitions, principles and theorems together with illustrative and other descriptive material in the subject of rigid body dynamics. This is followed by sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory and bring into sharp focus those fine points without which the student feels uncomfortable. The solved problems also serve to provide the repetition of basic principles that are so vital to effective learning. Numerous proofs of theorems and derivations of basic results are given under various topics. The course generally dwells on dynamics, kinematics and statics of a particle, systems of particles and rigid bodies.

COURSE OBJECTIVES 1

The objectives of this course are to:  Define and identify a particle and a rigid body and be able to state and derive the moment of inertia of various rigid bodies which arise in practice;  Specialize the motion of rigid bodies to one of translation of the center of mass plus rotation about an axis through the center of mass and perpendicular to a fixed plane  Treat the general motion of a rigid body in space which is composed of a translation of a fixed point of the body plus rotation about an axis through the fixed point which is not necessarily restricted in direction; and  Treat inertia and non-inertia frames of reference by considering the motion of particles relative to moving coordinate systems.

COURSE LEARNING OUTCOMES / COMPETENCIES Upon successful completion of this course, the student will be able to: (Knowledge based)  define the basic terms and concepts in dynamics of rigid bodies;  state and derive the moment of inertia of various rigid bodies which arise in practice ;  distinguish clearly between inertia and non-inertia frames of reference and solve some pertinent dynamical problems; and  solve various dynamical problems involving one of translation of the center of mass plus rotation about an axis through the center of mass and perpendicular to a fixed plane  solve various dynamical problems involving a translation of a fixed point of the body plus rotation about an axis through the fixed point which is not necessarily restricted in direction

GRADING SYSTEM FOR THE COURSE This course will be graded as follows: Class Attendance

A minimum of 65% attendance is required to sit for Examination

Assignments

10%

Test(s)

20%

Final Examination

70%

TOTAL

100%

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GENERAL INSTRUCTIONS Attendance: It is expected that every student will be in class for lectures and also participate in all practical exercises. Attendance records will be kept and used to determine each person’s qualification to sit for the final examination. In case of illness or other unavoidable cause of absence, the student must communicate as soon as possible with any of the instructors, indicating the reason for the absence. Academic Integrity: Violations of academic integrity, including dishonesty in assignments, examinations, or other academic performances are prohibited. You are not allowed to make copies of another person’s work and submit it as your own; that is plagiarism. All cases of academic dishonesty will be reported to the University Management for appropriate sanctions in accordance with the guidelines for handling students’ misconduct as spelt out in the Students’ Handbook. Assignments and Group Work: Students are expected to submit assignments as scheduled. Failure to submit an assignment as at when due will earn you zero for that assignment. Only under extenuating circumstances, for which a student has notified any of the instructors in advance, will late submission of assignments be permitted. Code of Conduct in Lecture Rooms and Laboratories: Students should turn off their cell phones during lectures. Students are prohibited from engaging in other activities (such as texting, watching videos, etc.) during lectures. Food and drinks are not permitted in the laboratories.

READING LIST Murray R. Spiegel(1967). Schaum’s Outline of Theorem and Problems of Theoretical Mechanics with an introduction to Lagrange’s Equations and Hamiltonian Theory. McGraw-Hill book Company. New York, St. Louis, San Francisco, Toronto, Sydney. 1

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Kibble, T. W. B (1973). Classical Mechanics. Second Edition. McGraw-Hill book Company(UK) Limited. Maidenhead. Berkshire.England.

Legend 1- Available in the University Library 2- Available in Departmental/School Libraries 3- Available on the Internet. 4- Available as Personal Collection 5- Available in local bookshops.

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COURSE OUTLINE Week

Topic

Remarks

1

Basic definitions and concepts. Particle, rigid body, inertia, moment of inertia and instantaneous axis. General motion of rigid body as a translation plus a rotation. Moment of inertia of discrete and continuous systems

Students will be given definitions of basic terms. Illustrations and examples will also be given Lectures will include definitions, illustrations. Various problems involving moment of inertia of continuous systems shall be treated These theorems shall be stated and proved. Illustrative examples shall be given with their solutions

2&3

4

5&6

7 &8

Parallel and Perpendicular axes theorems and their proofs. Illustrations and examples

Radius of gyrations. Angular momentum, kinetic energy of a rigid body and impulsive motions

The compound pendulum. Principles of conservation of energy. Stable and unstable equilibrium.

Students will be given definitions of basic terms. Illustrations and examples will also be given

Lectures involve the definitions and illustrations of pertinent terms. Some relevant theorems and proofs are given to the students. . MID-SEMESTER TEST

9 & 10

Space motion of rigid bodies. Moment of inertia, product of inertia and Kinetic energy. Derivations and examples.

4

Students will be taught to distinguish between moment of inertia and product of inertia. The formulae for calculating these shall be derived and illustrative examples shall be given. .

11 & 12

13 & 14

15

Principal axes of inertia and the directions of principal axis. Euler’s dynamical equations.

Frame of reference. Rotating and translating frames of reference. The symmetrical top and precession.

Lectures involve the definitions and illustrations of pertinent terms. Problems involving Principal axes of inertia and the directions of principal axis shall be solved for the students . Euler’s dynamical equations shall be derived. . The term ‘frame of reference’ shall be defined and illustrated to the students. Expressions for true, apparent, angular, coriolis and centripetal acceleration shall be derived. Illustrative examples shall be given.

REVISION This is the week preceding the final examination. At this time, evaluation will be done to assess how far the students’ expectations for the course have been met.

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