Rigid bodies - general theory

Size: px
Start display at page:

Download "Rigid bodies - general theory"

Transcription

1 Rigid bodies - general theory Kinetic Energy: based on FW-26 Consider a system on N particles with all their relative separations fixed: it has 3 translational and 3 rotational degrees of freedom. Motion with One Arbitrary Fixed Point: continuum limit: it is convenient to define the inertia tensor: = 0 in a it is constant and depends only on the mass distribution the kinetic energy has a simple form: both I and ω depend on the choice of the coordinate system, but T does not! Angular momentum: General motion with No Fixed Point: with the origin at the center of mass a parallel copy of the with the same origin as the body fixed frame - the center-of-mass frame (in general it is not an ) continuum limit: using the inertia tensor it can be written as: and it is related to the kinetic energy: I, ω and L depend on the choice of the coordinate system! 0 for rigid bodies

2 Kinetic Energy and Angular Momentum: with the origin at the center of mass a parallel copy of the with the same origin as the body fixed frame - the center-of-mass frame (in general it is not an ) Review: Inertia tensor based on FW-26 We have to learn how to evaluate the inertia tensor in a body-fixed frame. First, let s define the inertia tensor in a with the origin at the center of mass: internal motion about the CM Then the inertia tensor in a with the origin displaced by a is given by: center-of-mass relations Formulas for T and L are identical to the case of a motion with one fixed point! parallel-axis theorem! Example (a uniform disk of radius a): Principal axes (a coordinate system in which the inertia tensor is diagonal): Diagonalizing a real symmetric matrix: all the formulas simplify in such a coordinate system moment of inertia about a perpendicular axis through the edge of the disk (would be tedious to calculate directly) moment of inertia about a perpendicular axis through the center of the disk (easy to calculate in cylindrical polar coordinates) non-trivial solution only if: similar to solving for normal modes, v I and m 1 we get 3eigenvalues, 3 eigenvectors, and we can form the modal matrix: T new orthonormal basis - principal axes

3 Modal matrix diagonalizes the inertia tensor: Angular Momentum in the new coordinate system: we can define a new set of angular velocities: T projections of ω along the principal axes Principal moments of inertia: Kinetic Energy becomes: projections of r along principal axes Euler s Equations Applications - Compound Pendulum Describing rotational motion: based on FW-27 valid in any but also in the cm frame: in terms of body-fixed observables: Euler s equations: a rigid body constrained to rotate about a fixed axis: stationary both in the inertial frame and in we can choose the coordinate system so that: one degree of freedom ignoring friction, the only torque comes from gravity: projecting onto the principal body axes simple formulas, but limited use, since both angular velocities and torques are evaluated in a time dependent (principal axes); used mostly for torque-free motion or partially constrained motion (examples follow) in both inertial and body frame perp. distance from the 3rd axis

4 equation of motion of an arbitrary compound pendulum: oscillation frequency about the axis through Q: for small oscillations, the motion is simple harmonic: length of a simple pendulum with the same frequency center of percussion Let s rewrite it in terms of the moment of inertia about the axis going through the center of mass: radius of gyration equivalent radial distance of a point mass M leading to the same moment of inertia radius of gyration about the axis through Q invert the pendulum and suspend it from the parallel axis going through P The oscillations frequencies and periods about the axis through Q and P are identical: is used to measure g! Response of a baseball bat to a transverse force: motion of the center of mass (N-2nd): torque equation: reaction force at the point of support the only direction R can move due to constraints applied force (we neglect gravity) Rolling and Sliding Billiard Ball a ball is struck at the center (h=0) with a horizontal force. When does it start rolling? initial conditions:.. x = 0, x = v, ϕ = 0, ϕ = 0 force equation: solution: 0 torque equation: Pure rolling without sliding occurs only for: if the force is applied at the center of percussion the reaction force at the point of support vanishes! if applied beyond that point, the reaction force has the same direction as applied force a = 5 2 µgt at this time:

5 Rolling and Sliding Billiard Ball a ball is struck at h above the center with a horizontal force. When does it start rolling? the same equations of motion: initial conditions: initial angular velocity Pure rolling without sliding occurs only for: rolls immediately, no sliding (friction) slides, then rolls rolls too fast?! independent of the impulse! 147

STATICS Chapter 1 Introductory Concepts

STATICS Chapter 1 Introductory Concepts Contents Preface to Adapted Edition... (v) Preface to Third Edition... (vii) List of Symbols and Abbreviations... (xi) PART - I STATICS Chapter 1 Introductory Concepts 1-1 Scope of Mechanics... 1 1-2 Preview

More information

Constrained motion and generalized coordinates

Constrained motion and generalized coordinates Constrained motion and generalized coordinates based on FW-13 Often, the motion of particles is restricted by constraints, and we want to: work only with independent degrees of freedom (coordinates) k

More information

PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)

PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when

More information

6-1. Conservation law of mechanical energy

6-1. Conservation law of mechanical energy 6-1. Conservation law of mechanical energy 1. Purpose Investigate the mechanical energy conservation law and energy loss, by studying the kinetic and rotational energy of a marble wheel that is moving

More information

Problem 1. Mathematics of rotations

Problem 1. Mathematics of rotations Problem 1. Mathematics of rotations (a) Show by algebraic means (i.e. no pictures) that the relationship between ω and is: φ, ψ, θ Feel free to use computer algebra. ω X = φ sin θ sin ψ + θ cos ψ (1) ω

More information

θ + mgl θ = 0 or θ + ω 2 θ = 0 (2) ω 2 = I θ = mgl sinθ (1) + Ml 2 I = I CM mgl Kater s Pendulum The Compound Pendulum

θ + mgl θ = 0 or θ + ω 2 θ = 0 (2) ω 2 = I θ = mgl sinθ (1) + Ml 2 I = I CM mgl Kater s Pendulum The Compound Pendulum Kater s Pendulum The Compound Pendulum A compound pendulum is the term that generally refers to an arbitrary lamina that is allowed to oscillate about a point located some distance from the lamina s center

More information

Classical Mechanics III (8.09) Fall 2014 Assignment 3

Classical Mechanics III (8.09) Fall 2014 Assignment 3 Classical Mechanics III (8.09) Fall 2014 Assignment 3 Massachusetts Institute of Technology Physics Department Due September 29, 2014 September 22, 2014 6:00pm Announcements This week we continue our discussion

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 IT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical echanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ASSACHUSETTS INSTITUTE

More information

AP PHYSICS 1 Learning Objectives Arranged Topically

AP PHYSICS 1 Learning Objectives Arranged Topically AP PHYSICS 1 Learning Objectives Arranged Topically with o Big Ideas o Enduring Understandings o Essential Knowledges o Learning Objectives o Science Practices o Correlation to Knight Textbook Chapters

More information

12. Rigid Body Dynamics I

12. Rigid Body Dynamics I University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 1. Rigid Body Dynamics I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License

More information

13. Rigid Body Dynamics II

13. Rigid Body Dynamics II University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 13. Rigid Body Dynamics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License

More information

FINAL EXAM CLOSED BOOK

FINAL EXAM CLOSED BOOK Physics 7A- Section 2, Fall 2008. Instructor Lanzara FINAL EXAM CLOSED BOOK GOOD LUCK! Print Name Discussion Section# or Time Signature Discussion Section GSI Student ID# Problem Points Score 1 20 2 20

More information

Lecture D20-2D Rigid Body Dynamics: Impulse and Momentum

Lecture D20-2D Rigid Body Dynamics: Impulse and Momentum J Peraire 1607 Dynamics Fall 004 Version 11 Lecture D0 - D Rigid Body Dynamics: Impulse and Momentum In lecture D9, we saw the principle of impulse and momentum applied to particle motion This principle

More information

2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity

2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity 2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics

More information

Rotation and Translation Challenge Problems Problem 1:

Rotation and Translation Challenge Problems Problem 1: Rotation and Translation Challenge Problems Problem 1: A drum A of mass m and radius R is suspended from a drum B also of mass m and radius R, which is free to rotate about its axis. The suspension is

More information

Video 2.1a Vijay Kumar and Ani Hsieh

Video 2.1a Vijay Kumar and Ani Hsieh Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Analytical Mechanics Aristotle Galileo Bernoulli

More information

Physics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top

Physics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem

More information

PROBLEM 2 10 points. [ ] increases [ ] decreases [ ] stays the same. Briefly justify your answer:

PROBLEM 2 10 points. [ ] increases [ ] decreases [ ] stays the same. Briefly justify your answer: PROBLEM 2 10 points A disk of mass m is tied to a block of mass 2m via a string that passes through a hole at the center of a rotating turntable. The disk rotates with the turntable at a distance R from

More information

16. Rotational Dynamics

16. Rotational Dynamics 6. Rotational Dynamics A Overview In this unit we will address examples that combine both translational and rotational motion. We will find that we will need both Newton s second law and the rotational

More information

Rotational motion problems

Rotational motion problems Rotational motion problems. (Massive pulley) Masses m and m 2 are connected by a string that runs over a pulley of radius R and moment of inertia I. Find the acceleration of the two masses, as well as

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

More information

AP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems

AP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems AP Physics C: Rotation II (Torque and Rotational Dynamics, Rolling Motion) Problems 1980M3. A billiard ball has mass M, radius R, and moment of inertia about the center of mass I c = 2 MR²/5 The ball is

More information

For a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is

For a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

More information

Lecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws

Lecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,

More information

UPDATED: OCTOBER Problem #1

UPDATED: OCTOBER Problem #1 UVA PHYSICS DEPARTMENT PHD QUALIFYING EXAM PROBLEM FILE CLASSICAL MECHANICS UPDATED: OCTOBER 008 1. Consider the system shown below of two masses, a fixed wall, and three massless, ideal springs. Motion

More information

Homework 1. Due Tuesday, January 29.

Homework 1. Due Tuesday, January 29. Homework 1. Due Tuesday, January 29. Problem 1. An ideal rope (no friction) lying on a table slides from its edge down to a scales lying on the floor. The table s height is h. Find a stationary velocity

More information

Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration

Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,

More information

Torque and Rotation Lecture 7

Torque and Rotation Lecture 7 Torque and Rotation Lecture 7 ˆ In this lecture we finally move beyond a simple particle in our mechanical analysis of motion. ˆ Now we consider the so-called rigid body. Essentially, a particle with extension

More information

Physics for Scientists and Engineers 4th Edition, 2017

Physics for Scientists and Engineers 4th Edition, 2017 A Correlation of Physics for Scientists and Engineers 4th Edition, 2017 To the AP Physics C: Mechanics Course Descriptions AP is a trademark registered and/or owned by the College Board, which was not

More information

N mg N Mg N Figure : Forces acting on particle m and inclined plane M. (b) The equations of motion are obtained by applying the momentum principles to

N mg N Mg N Figure : Forces acting on particle m and inclined plane M. (b) The equations of motion are obtained by applying the momentum principles to .004 MDEING DNMIS ND NTR I I Spring 00 Solutions for Problem Set 5 Problem. Particle slides down movable inclined plane. The inclined plane of mass M is constrained to move parallel to the -axis, and the

More information

Lecture 6 Physics 106 Spring 2006

Lecture 6 Physics 106 Spring 2006 Lecture 6 Physics 106 Spring 2006 Angular Momentum Rolling Angular Momentum: Definition: Angular Momentum for rotation System of particles: Torque: l = r m v sinφ l = I ω [kg m 2 /s] http://web.njit.edu/~sirenko/

More information

9 Kinetics of 3D rigid bodies - rotating frames

9 Kinetics of 3D rigid bodies - rotating frames 9 Kinetics of 3D rigid bodies - rotating frames 9. Consider the two gears depicted in the figure. The gear B of radius R B is fixed to the ground, while the gear A of mass m A and radius R A turns freely

More information

CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017

CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017 CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and

More information

Homework 1. Due whatever day you decide to have the homework session.

Homework 1. Due whatever day you decide to have the homework session. Homework 1. Due whatever day you decide to have the homework session. Problem 1. Rising Snake A snake of length L and linear mass density ρ rises from the table. It s head is moving straight up with the

More information

UPDATED: NOVEMBER A particle of mass m and energy E moving in one dimension comes in from - and encounters the repulsive potential:

UPDATED: NOVEMBER A particle of mass m and energy E moving in one dimension comes in from - and encounters the repulsive potential: UVA PHYSICS DEPARTMENT PHD QUALIFYING EXAM PROBLEM FILE CLASSICAL MECHANICS UPDATED: NOVEMBER 007 1. A particle of mass m and energy E moving in one dimension comes in from - and encounters the repulsive

More information

Physics 106b/196b Problem Set 9 Due Jan 19, 2007

Physics 106b/196b Problem Set 9 Due Jan 19, 2007 Physics 06b/96b Problem Set 9 Due Jan 9, 2007 Version 3: January 8, 2007 This problem set focuses on dynamics in rotating coordinate systems (Section 5.2), with some additional early material on dynamics

More information

16.07 Dynamics Final Exam

16.07 Dynamics Final Exam Name:... Massachusetts Institute of Technology 16.07 Dynamics Final Exam Tuesday, December 20, 2005 Problem 1 (8) Problem 2 (8) Problem 3 (10) Problem 4 (10) Problem 5 (10) Problem 6 (10) Problem 7 (10)

More information

Chapters 10 & 11: Rotational Dynamics Thursday March 8 th

Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Review of rotational kinematics equations Review and more on rotational inertia Rolling motion as rotation and translation Rotational kinetic energy

More information

PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION

PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION I. Moment of Inertia: Since a body has a definite size and shape, an applied nonconcurrent force system may cause the body to both translate and rotate.

More information

Chapter 9 Notes. x cm =

Chapter 9 Notes. x cm = Chapter 9 Notes Chapter 8 begins the discussion of rigid bodies, a system of particles with fixed relative positions. Previously we have dealt with translation of a particle: if a rigid body does not rotate

More information

Physics 351, Spring 2018, Homework #9. Due at start of class, Friday, March 30, 2018

Physics 351, Spring 2018, Homework #9. Due at start of class, Friday, March 30, 2018 Physics 351, Spring 218, Homework #9. Due at start of class, Friday, March 3, 218 Please write your name on the LAST PAGE of your homework submission, so that we don t notice whose paper we re grading

More information

The... of a particle is defined as its change in position in some time interval.

The... of a particle is defined as its change in position in some time interval. Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity

Chapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity Chapter 8 Rotational Equilibrium and Rotational Dynamics 1. Torque 2. Torque and Equilibrium 3. Center of Mass and Center of Gravity 4. Torque and angular acceleration 5. Rotational Kinetic energy 6. Angular

More information

STATICS & DYNAMICS. Engineering Mechanics. Gary L. Gray. Francesco Costanzo. Michael E. Plesha. University of Wisconsin-Madison

STATICS & DYNAMICS. Engineering Mechanics. Gary L. Gray. Francesco Costanzo. Michael E. Plesha. University of Wisconsin-Madison Engineering Mechanics STATICS & DYNAMICS SECOND EDITION Francesco Costanzo Department of Engineering Science and Mechanics Penn State University Michael E. Plesha Department of Engineering Physics University

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 AERONAUTICAL ENGINEERING DEFINITIONS AND TERMINOLOGY Course Name : ENGINEERING MECHANICS Course Code : AAEB01 Program :

More information

28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)

28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) 28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.

More information

Lab #4 - Gyroscopic Motion of a Rigid Body

Lab #4 - Gyroscopic Motion of a Rigid Body Lab #4 - Gyroscopic Motion of a Rigid Body Last Updated: April 6, 2007 INTRODUCTION Gyroscope is a word used to describe a rigid body, usually with symmetry about an axis, that has a comparatively large

More information

Phys 7221 Homework # 8

Phys 7221 Homework # 8 Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with

More information

is acting on a body of mass m = 3.0 kg and changes its velocity from an initial

is acting on a body of mass m = 3.0 kg and changes its velocity from an initial PHYS 101 second major Exam Term 102 (Zero Version) Q1. A 15.0-kg block is pulled over a rough, horizontal surface by a constant force of 70.0 N acting at an angle of 20.0 above the horizontal. The block

More information

Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)

Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas) Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in

More information

Phys 270 Final Exam. Figure 1: Question 1

Phys 270 Final Exam. Figure 1: Question 1 Phys 270 Final Exam Time limit: 120 minutes Each question worths 10 points. Constants: g = 9.8m/s 2, G = 6.67 10 11 Nm 2 kg 2. 1. (a) Figure 1 shows an object with moment of inertia I and mass m oscillating

More information

Introduction. Physics E-1a Expt 5: The Sweet Spot Fall 2006

Introduction. Physics E-1a Expt 5: The Sweet Spot Fall 2006 Physics E-1a Expt 5: The Sweet Spot all 2006 The center of percussion () is the place on a bat where it may be struck without causing a reaction at the point of support. When a ball is hit at the the contact

More information

Physical Dynamics (PHY-304)

Physical Dynamics (PHY-304) Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.

More information

PHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements.

PHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements. PHYSICS Course Structure Unit Topics Marks I Physical World and Measurement 1 Physical World 2 Units and Measurements II Kinematics 3 Motion in a Straight Line 23 4 Motion in a Plane III Laws of Motion

More information

Revolve, Rotate & Roll:

Revolve, Rotate & Roll: I. Warm-UP. Revolve, Rotate & Roll: Physics 203, Yaverbaum John Jay College of Criminal Justice, the CUNY Given g, the rate of free-fall acceleration near Earth s surface, and r, the radius of a VERTICAL

More information

1 MR SAMPLE EXAM 3 FALL 2013

1 MR SAMPLE EXAM 3 FALL 2013 SAMPLE EXAM 3 FALL 013 1. A merry-go-round rotates from rest with an angular acceleration of 1.56 rad/s. How long does it take to rotate through the first rev? A) s B) 4 s C) 6 s D) 8 s E) 10 s. A wheel,

More information

General Physics I. Lecture 10: Rolling Motion and Angular Momentum.

General Physics I. Lecture 10: Rolling Motion and Angular Momentum. General Physics I Lecture 10: Rolling Motion and Angular Momentum Prof. WAN, Xin (万歆) 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Rolling motion of a rigid object: center-of-mass motion

More information

CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5

CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5 1 / 36 CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Date: 2 / 36 EQUATIONS OF MOTION: ROTATION

More information

J07M.1 - Ball on a Turntable

J07M.1 - Ball on a Turntable Part I - Mechanics J07M.1 - Ball on a Turntable J07M.1 - Ball on a Turntable ẑ Ω A spherically symmetric ball of mass m, moment of inertia I about any axis through its center, and radius a, rolls without

More information

Table of Contents. Pg. # Momentum & Impulse (Bozemanscience Videos) Lab 2 Determination of Rotational Inertia 1 1/11/16

Table of Contents. Pg. # Momentum & Impulse (Bozemanscience Videos) Lab 2 Determination of Rotational Inertia 1 1/11/16 Table of Contents g. # 1 1/11/16 Momentum & Impulse (Bozemanscience Videos) 2 1/13/16 Conservation of Momentum 3 1/19/16 Elastic and Inelastic Collisions 4 1/19/16 Lab 1 Momentum 5 1/26/16 Rotational tatics

More information

7. The gyroscope. 7.1 Introduction. 7.2 Theory. a) The gyroscope

7. The gyroscope. 7.1 Introduction. 7.2 Theory. a) The gyroscope K 7. The gyroscope 7.1 Introduction This experiment concerns a special type of motion of a gyroscope, called precession. From the angular frequency of the precession, the moment of inertia of the spinning

More information

Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics. Lectures 3+4 Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

More information

Physical Dynamics (SPA5304) Lecture Plan 2018

Physical Dynamics (SPA5304) Lecture Plan 2018 Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle

More information

Scalar product Work Kinetic energy Work energy theorem Potential energy Conservation of energy Power Collisions

Scalar product Work Kinetic energy Work energy theorem Potential energy Conservation of energy Power Collisions BLOOM PUBLIC SCHOOL Vasant Kunj, New Delhi Lesson Plan Class: XI Subject: Physics Month: August No of Periods: 11 Chapter No. 6: Work, energy and power TTT: 5 WT: 6 Chapter : Work, energy and power Scalar

More information

kx m x B N 1 C L, M Mg θ

kx m x B N 1 C L, M Mg θ .004 MODELING DYNAMICS AND CONTROL II Spring 00 Solutions to Problem Set No. 7 Problem 1. Pendulum mounted on elastic support. This problem is an execise in the application of momentum principles. Two

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

More information

Fundamentals Physics. Chapter 10 Rotation

Fundamentals Physics. Chapter 10 Rotation Fundamentals Physics Tenth Edition Halliday Chapter 10 Rotation 10-1 Rotational Variables (1 of 15) Learning Objectives 10.01 Identify that if all parts of a body rotate around a fixed axis locked together,

More information

Homework 1. Due Thursday, January 21

Homework 1. Due Thursday, January 21 Homework 1. Due Thursday, January 21 Problem 1. Rising Snake A snake of length L and linear mass density ρ rises from the table. It s head is moving straight up with the constant velocity v. What force

More information

DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS

DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS LSN 8-7: ROTATIONAL KINETIC ENERGY Questions From Reading Activity? Big Idea(s): The interactions of an object with other objects can be described by

More information

Dynamics. Dynamics of mechanical particle and particle systems (many body systems)

Dynamics. Dynamics of mechanical particle and particle systems (many body systems) Dynamics Dynamics of mechanical particle and particle systems (many body systems) Newton`s first law: If no net force acts on a body, it will move on a straight line at constant velocity or will stay at

More information

Chapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:

Chapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum: linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)

More information

Physics 312, Winter 2007, Practice Final

Physics 312, Winter 2007, Practice Final Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight.

More information

AP Physics C Mechanics Objectives

AP Physics C Mechanics Objectives AP Physics C Mechanics Objectives I. KINEMATICS A. Motion in One Dimension 1. The relationships among position, velocity and acceleration a. Given a graph of position vs. time, identify or sketch a graph

More information

PY205N Spring The vectors a, b, and c. are related by c = a b. The diagram below that best illustrates this relationship is (a) I

PY205N Spring The vectors a, b, and c. are related by c = a b. The diagram below that best illustrates this relationship is (a) I PY205N Spring 2013 Final exam, practice version MODIFIED This practice exam is to help students prepare for the final exam to be given at the end of the semester. Please note that while problems on this

More information

Lecture 41: Highlights

Lecture 41: Highlights Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final

More information

Angular Momentum System of Particles Concept Questions

Angular Momentum System of Particles Concept Questions Question 1: Angular Momentum Angular Momentum System of Particles Concept Questions A non-symmetric body rotates with an angular speed ω about the z axis. Relative to the origin 1. L 0 is constant. 2.

More information

PHYSICS Final Exam

PHYSICS Final Exam Name: Answer Key PHYSICS 1210-04 Final Exam University of Wyoming 2 May 2012 1. (10 points) A cannonball is fired with a speed of 75 m/s at an angle of 30 above horizontal. It lands at its starting height

More information

PHYSICS 221 SPRING 2014

PHYSICS 221 SPRING 2014 PHYSICS 221 SPRING 2014 EXAM 2: April 3, 2014 8:15-10:15pm Name (printed): Recitation Instructor: Section # INSTRUCTIONS: This exam contains 25 multiple-choice questions plus 2 extra credit questions,

More information

Chapter 8- Rotational Motion

Chapter 8- Rotational Motion Chapter 8- Rotational Motion Assignment 8 Textbook (Giancoli, 6 th edition), Chapter 7-8: Due on Thursday, November 13, 2008 - Problem 28 - page 189 of the textbook - Problem 40 - page 190 of the textbook

More information

16.07 Dynamics. Problem Set 10

16.07 Dynamics. Problem Set 10 NAME :..................... Massachusetts Institute of Technology 16.07 Dynamics Problem Set 10 Out date: Nov. 7, 2007 Due date: Nov. 14, 2007 Problem 1 Problem 2 Problem 3 Problem 4 Study Time Time Spent

More information

Contents. Objectives Torque on an Object Rotational Kinetic Energy Yo yo Rolling on an Incline Physical Pendulum Angular Momentum and Torque Recap

Contents. Objectives Torque on an Object Rotational Kinetic Energy Yo yo Rolling on an Incline Physical Pendulum Angular Momentum and Torque Recap Physics 121 for Majors Class 21 Rotating Objects Last Class We learned to find angular momentum and torques of point masses and objects. We learned how to use torques and forces to solve problems with

More information

6. Find the net torque on the wheel in Figure about the axle through O if a = 10.0 cm and b = 25.0 cm.

6. Find the net torque on the wheel in Figure about the axle through O if a = 10.0 cm and b = 25.0 cm. 1. During a certain period of time, the angular position of a swinging door is described by θ = 5.00 + 10.0t + 2.00t 2, where θ is in radians and t is in seconds. Determine the angular position, angular

More information

Rotational Kinematics and Dynamics. UCVTS AIT Physics

Rotational Kinematics and Dynamics. UCVTS AIT Physics Rotational Kinematics and Dynamics UCVTS AIT Physics Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position,

More information

CEE 271: Applied Mechanics II, Dynamics Lecture 24: Ch.17, Sec.1-3

CEE 271: Applied Mechanics II, Dynamics Lecture 24: Ch.17, Sec.1-3 1 / 38 CEE 271: Applied Mechanics II, Dynamics Lecture 24: Ch.17, Sec.1-3 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Tuesday, Nov. 13, 2012 2 / 38 MOMENT OF

More information

Concept Question: Normal Force

Concept Question: Normal Force Concept Question: Normal Force Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is 1. larger than 2. identical

More information

1. Which of the following is the unit for angular displacement? A. Meters B. Seconds C. Radians D. Radian per second E. Inches

1. Which of the following is the unit for angular displacement? A. Meters B. Seconds C. Radians D. Radian per second E. Inches AP Physics B Practice Questions: Rotational Motion Multiple-Choice Questions 1. Which of the following is the unit for angular displacement? A. Meters B. Seconds C. Radians D. Radian per second E. Inches

More information

Q2. A machine carries a 4.0 kg package from an initial position of d ˆ. = (2.0 m)j at t = 0 to a final position of d ˆ ˆ

Q2. A machine carries a 4.0 kg package from an initial position of d ˆ. = (2.0 m)j at t = 0 to a final position of d ˆ ˆ Coordinator: Dr. S. Kunwar Monday, March 25, 2019 Page: 1 Q1. An object moves in a horizontal circle at constant speed. The work done by the centripetal force is zero because: A) the centripetal force

More information

PHY 5246: Theoretical Dynamics, Fall Assignment # 10, Solutions. (1.a) N = a. we see that a m ar a = 0 and so N = 0. ω 3 ω 2 = 0 ω 2 + I 1 I 3

PHY 5246: Theoretical Dynamics, Fall Assignment # 10, Solutions. (1.a) N = a. we see that a m ar a = 0 and so N = 0. ω 3 ω 2 = 0 ω 2 + I 1 I 3 PHY 54: Theoretical Dynamics, Fall 015 Assignment # 10, Solutions 1 Graded Problems Problem 1 x 3 a ω First we calculate the moments of inertia: ( ) a I 1 = I = m 4 + b, 1 (1.a) I 3 = ma. b/ α The torque

More information

Course Name: AP Physics. Team Names: Jon Collins. Velocity Acceleration Displacement

Course Name: AP Physics. Team Names: Jon Collins. Velocity Acceleration Displacement Course Name: AP Physics Team Names: Jon Collins 1 st 9 weeks Objectives Vocabulary 1. NEWTONIAN MECHANICS and lab skills: Kinematics (including vectors, vector algebra, components of vectors, coordinate

More information

Module 17: Systems, Conservation of Momentum and Center of Mass

Module 17: Systems, Conservation of Momentum and Center of Mass Module 17: Systems, Conservation of Momentum and Center of Mass 17.1 External and Internal Forces and the Change in Momentum of a System So far we have restricted ourselves to considering how the momentum

More information

Tutorial 4. Figure 1: Rod and spindle. Dog. Figure 2: I have no idea what I m doing. Dog

Tutorial 4. Figure 1: Rod and spindle. Dog. Figure 2: I have no idea what I m doing. Dog Tutorial 4 Question 1 Figure 1: Rod and spindle A uniform disk rotates at 3.60 rev/s around a frictionless spindle. A non-rotating rod, of the same mass as the disk and equal in length to the disk s diameter,

More information

4 A mass-spring oscillating system undergoes SHM with a period T. What is the period of the system if the amplitude is doubled?

4 A mass-spring oscillating system undergoes SHM with a period T. What is the period of the system if the amplitude is doubled? Slide 1 / 52 1 A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its velocity is a maximum in magnitude? A 0 B + or - A C

More information

Rotations and vibrations of polyatomic molecules

Rotations and vibrations of polyatomic molecules Rotations and vibrations of polyatomic molecules When the potential energy surface V( R 1, R 2,..., R N ) is known we can compute the energy levels of the molecule. These levels can be an effect of: Rotation

More information

Use the following to answer question 1:

Use the following to answer question 1: Use the following to answer question 1: On an amusement park ride, passengers are seated in a horizontal circle of radius 7.5 m. The seats begin from rest and are uniformly accelerated for 21 seconds to

More information

PHYSICS I RESOURCE SHEET

PHYSICS I RESOURCE SHEET PHYSICS I RESOURCE SHEET Cautions and Notes Kinematic Equations These are to be used in regions with constant acceleration only You must keep regions with different accelerations separate (for example,

More information

Lecture 9 Kinetics of rigid bodies: Impulse and Momentum

Lecture 9 Kinetics of rigid bodies: Impulse and Momentum Lecture 9 Kinetics of rigid bodies: Impulse and Momentum Momentum of 2-D Rigid Bodies Recall that in lecture 5, we discussed the use of momentum of particles. Given that a particle has a, and is travelling

More information

Problem Set x Classical Mechanics, Fall 2016 Massachusetts Institute of Technology. 1. Moment of Inertia: Disc and Washer

Problem Set x Classical Mechanics, Fall 2016 Massachusetts Institute of Technology. 1. Moment of Inertia: Disc and Washer 8.01x Classical Mechanics, Fall 2016 Massachusetts Institute of Technology Problem Set 10 1. Moment of Inertia: Disc and Washer (a) A thin uniform disc of mass M and radius R is mounted on an axis passing

More information

SOLUTIONS, PROBLEM SET 11

SOLUTIONS, PROBLEM SET 11 SOLUTIONS, PROBLEM SET 11 1 In this problem we investigate the Lagrangian formulation of dynamics in a rotating frame. Consider a frame of reference which we will consider to be inertial. Suppose that

More information

PHYSICS GRADUATE SCHOOL QUALIFYING EXAMINATION

PHYSICS GRADUATE SCHOOL QUALIFYING EXAMINATION PHYSICS GRADUATE SCHOOL QUALIFYING EXAMINATION Partl INSTRUCTIONS: Work all problems. This is a closed book examination. Start each problem on a new pack of yellow paper and use only one side of each sheet.

More information