Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa

Size: px
Start display at page:

Download "Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa"

Transcription

1 Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa

2 Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t. Pcture a grndstone on a smooth axle. BUT the axle must be exactly at the center of gravty otherwse gravty wll provde a torque, and the rotaton wll not be at constant velocty! A

3 How s Angular Acceleraton Related to Torque? Thnk about a tangental force F appled to a mass m attached to a lght dsk whch can rotate about a fxed axs. (A radally drected force has zero torque, does nothng.) The relevant equatons are: F = ma, a = rα, τ = rf. Therefore F = ma becomes τ = mr 2 α Vhas zero Lght dsk axle r m F

4 Newton s Second Law for Rotatons For the specal case of a mass m constraned by a lght dsk to crcle around an axle, the angular acceleraton α s proportonal to the torque τ exactly as n the lnear case the acceleraton a s proportonal to the force F: τ = mr 2 α F = ma The angular equvalent of nertal mass m s the moment of nerta mr 2.

5 More Complcated Rotatng Bodes Suppose now a lght dsk has A several dfferent masses attached at dfferent places, and varous forces act on them. As before, radal components cause no rotaton, we have a sum of torques. BUT the rgdty of the dsk ensures that a force appled to one mass wll cause a torque on the others! How do we handle that? m 2 F 2 r 1 m 1 F 1

6 Newton s Thrd Law for a Rgd Rotatng Body If a rgd body s made up of many masses m connected by rgd rods, the force exerted along the rod of m on m j s equal n magntude, opposte n drecton and along the same lne as that of m j on m, therefore the nternal torques come n equal and opposte pars, and cannot contrbute to the body s angular acceleraton. It follows that the angular acceleraton s generated by the sum of the external torques.

7 Moment of Inerta of a Sold Body Consder a flat square plate rotatng about a perpendcular axs wth angular acceleraton α. One small part of t, Δm, dstance r from the axle, has equaton of moton τ = τ + τ = mr α ext nt 2 Addng contrbutons from all parts of the wheel ext 2 τ = τ = mr α = Iα I s the Moment of Inerta. Z Δm

8 Calculatng Moments of Inerta A thn hoop of radus R (thnk a bcycle wheel) has all the mass dstance R from a perpendcular axle through ts center, so ts moment of nerta s I = m r = MR 2 2 A unform rod of mass M, length L, has moment of nerta about one end L I = x ( M / L) dx = ML v x dx L R Mass of length dx of rod s (M/L)dx

9 Dsks and Cylnders A dsk: mass M, radus R, s a sum of nested rngs. The red rng, radus r and thckness dr, has area 2πrdr, hence mass dm = M(2πrdr/ πr 2 ). Addng up rngs to make a dsk, R R 2 2( 2) / I r dm r M R rdr MR = = = A cylnder s just a stack of dsks, so t s also ½MR 2 about the axle. c

10 Parallel Axs Theorem If we already know I CM about some lne through the CM (we take t as the z- axs), then I about a parallel lne at a dstance h s I = I CM + Mh 2 A r h y dm r CM at O x Here s the proof: Moment 2 I = mr = m r + h ( ) 2 mr = + 2h mr + Mh 2 2 = + (Snce = 0.) 2 ICM Mh mr of nerta I about perpendcular axs through A We prove t for a 2D object the proof n 3D s exactly the same, takng the lne through the CM as the z-axs.

11 Clcker Queston We found the moment of nerta of a rod about a 1 perpendcular lne through one end was ML 2 3. Use the parallel axs theorem to fgure out what t s about a perpendcular lne through the center of the rod. A B C D E 1 ML ML ML ML 14 ML 2 12

12 Perpendcular Axs Theorem For a 2D object (a thn plate) the moment of nerta I z about a perpendcular axs equals the sum of the moments of nerta about any two axes at rght angles through the same pont n the plane, a x z y I z = I x + I y Proof: ( ) = = + = + I mr m x y I I z x y

13 Clcker Queston Gven that the moment of nerta of a dsk about ts 1 axle s MR 2, use the perpendcular axs theorem to fnd 2 the moment of nerta of a dsk about a lne through ts center and n ts plane. A B C 1 MR MR 2 MR

14 Rotatonal Knetc Energy Imagne a rotatng body as composed of many small masses m at dstances r from the axs of rotaton. The mass m has speed v = ωr, so KE = ½m r 2 ω 2. The total KE of the rotatng body (assumng the axs s at rest) s ( ) K = 2mr ω = 2 Iω

Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation!

Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation! Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post

More information

Chapter 11 Angular Momentum

Chapter 11 Angular Momentum Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle

More information

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta

More information

Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Section 1-5 Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

More information

So far: simple (planar) geometries

So far: simple (planar) geometries Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector

More information

Spring 2002 Lecture #13

Spring 2002 Lecture #13 44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-rotation coupling of the angularly accelerated rigid body Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

More information

10/9/2003 PHY Lecture 11 1

10/9/2003 PHY Lecture 11 1 Announcements 1. Physc Colloquum today --The Physcs and Analyss of Non-nvasve Optcal Imagng. Today s lecture Bref revew of momentum & collsons Example HW problems Introducton to rotatons Defnton of angular

More information

Physics 111: Mechanics Lecture 11

Physics 111: Mechanics Lecture 11 Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

Chapter 11: Angular Momentum

Chapter 11: Angular Momentum Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding. Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays

More information

MEASUREMENT OF MOMENT OF INERTIA

MEASUREMENT OF MOMENT OF INERTIA 1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us

More information

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004 Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a

More information

Chapter 11 Torque and Angular Momentum

Chapter 11 Torque and Angular Momentum Chapter Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector

More information

τ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1

τ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1 A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor

More information

Dynamics of Rotational Motion

Dynamics of Rotational Motion Dynamcs of Rotatonal Moton Torque: the rotatonal analogue of force Torque = force x moment arm = Fl moment arm = perpendcular dstance through whch the force acts a.k.a. leer arm l F l F l F l F = Fl =

More information

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons

More information

PHYS 1443 Section 003 Lecture #17

PHYS 1443 Section 003 Lecture #17 PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!

More information

The classical spin-rotation coupling

The classical spin-rotation coupling LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd

More information

Study Guide For Exam Two

Study Guide For Exam Two Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force

More information

Physics 207: Lecture 27. Announcements

Physics 207: Lecture 27. Announcements Physcs 07: ecture 7 Announcements ake-up labs are ths week Fnal hwk assgned ths week, fnal quz next week Revew sesson on Thursday ay 9, :30 4:00pm, Here Today s Agenda Statcs recap Beam & Strngs» What

More information

Rotational and Translational Comparison. Conservation of Angular Momentum. Angular Momentum for a System of Particles

Rotational and Translational Comparison. Conservation of Angular Momentum. Angular Momentum for a System of Particles Conservaton o Angular Momentum 8.0 WD Rotatonal and Translatonal Comparson Quantty Momentum Ang Momentum Force Torque Knetc Energy Work Power Rotaton L cm = I cm ω = dl / cm cm K = (/ ) rot P rot θ W =

More information

Conservation of Angular Momentum = "Spin"

Conservation of Angular Momentum = Spin Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts

More information

Modeling of Dynamic Systems

Modeling of Dynamic Systems Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

Physics 207: Lecture 20. Today s Agenda Homework for Monday Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems

More information

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m) 7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to

More information

Linear Momentum. Center of Mass.

Linear Momentum. Center of Mass. Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html

More information

Rigid body simulation

Rigid body simulation Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Chapter 10 Rotational motion

Chapter 10 Rotational motion Prof. Dr. I. Nasser Chapter0_I November 6, 07 Important Terms Chapter 0 Rotatonal moton Angular Dsplacement s, r n radans where s s the length of arc and r s the radus. Angular Velocty The rate at whch

More information

Physics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4

Physics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4 Physcs 6 ecture 6 Conservaton o Angular Momentum SJ 7 th Ed.: Chap.4 Comparson: dentons o sngle partcle torque and angular momentum Angular momentum o a system o partcles o a rgd body rotatng about a xed

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 705: Classical Mechanics. Newtonian Mechanics 1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]

More information

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system). EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

Angular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 )

Angular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 ) Angular momentum Instructor: Dr. Ho Lam TAM ( 譚海嵐 ) Physcs Enhancement Programme or Gted Students The Hong Kong Academy or Gted Educaton and Department o Physcs, HKBU Department o Physcs Hong Kong Baptst

More information

10/23/2003 PHY Lecture 14R 1

10/23/2003 PHY Lecture 14R 1 Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth

More information

1 Hz = one cycle per second

1 Hz = one cycle per second Rotatonal Moton Mchael Fowler, UVa Physcs, 14E Sprng 009 Mar 5 Prelmnares: Unts for Angular Velocty The tachometer on your car dashboard tells you your car engne s angular speed n rpm, revolutons per mnute,

More information

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16 0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03 -- Lecture 7 0/4/03

More information

Mechanics Cycle 3 Chapter 9++ Chapter 9++

Mechanics Cycle 3 Chapter 9++ Chapter 9++ Chapter 9++ More on Knetc Energy and Potental Energy BACK TO THE FUTURE I++ More Predctons wth Energy Conservaton Revst: Knetc energy for rotaton Potental energy M total g y CM for a body n constant gravty

More information

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons

More information

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2 Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of

More information

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular

More information

Physics 181. Particle Systems

Physics 181. Particle Systems Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system

More information

PHYSICS 231 Review problems for midterm 2

PHYSICS 231 Review problems for midterm 2 PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October

More information

Linear Momentum. Center of Mass.

Linear Momentum. Center of Mass. Lecture 16 Chapter 9 Physcs I 11.06.2013 Lnear oentu. Center of ass. Course webste: http://faculty.ul.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.ul.edu/danylov2013/physcs1fall.htl

More information

EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics

EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.

More information

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15 NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound

More information

ROTATIONAL MOTION. dv d F m m V v dt dt. i i i cm i

ROTATIONAL MOTION. dv d F m m V v dt dt. i i i cm i ROTATIONAL MOTION Consder a collecton of partcles, m, located at R relatve to an nertal coordnate system. As before wrte: where R cm locates the center of mass. R Rcm r Wrte Newton s second law for the

More information

Chapter 9. The Dot Product (Scalar Product) The Dot Product use (Scalar Product) The Dot Product (Scalar Product) The Cross Product.

Chapter 9. The Dot Product (Scalar Product) The Dot Product use (Scalar Product) The Dot Product (Scalar Product) The Cross Product. The Dot Product (Scalar Product) Chapter 9 Statcs and Torque The dot product of two vectors can be constructed by takng the component of one vector n the drecton of the other and multplyng t tmes the magntude

More information

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force. Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

More information

Classical Mechanics ( Particles and Biparticles )

Classical Mechanics ( Particles and Biparticles ) Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles

More information

3 d Rotations Rotating dumbbells in lab frame Moment of Inertial Tensor

3 d Rotations Rotating dumbbells in lab frame Moment of Inertial Tensor d Rotatons Rotatng dumells n la frame Moment of Inertal Tensor Revew of BCS and FCS sstems Component notaton for I β Moments and Products of Inerta I x, I, I z P x, P xz, P z Moment of Inerta for a cue

More information

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions. Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu

More information

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The

More information

Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation

Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 20.1 Introducton... 1 20.2 Constraned Moton: Translaton and Rotaton... 1 20.2.1 Rollng wthout slppng... 5 Example 20.1

More information

Physics 207 Lecture 6

Physics 207 Lecture 6 Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free

More information

Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7

Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7 Electrcty and Magnetsm - Physcs 11 Lecture 10 - Sources of Magnetc Felds (Currents) Y&F Chapter 8, Sec. 1-7 Magnetc felds are due to currents The Bot-Savart Law Calculatng feld at the centers of current

More information

Physics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1

Physics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. Mult-Partcle

More information

SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

SYSTEMS OF PARTICLES AND ROTATIONAL MOTION CHAPTER SEVEN SYSTES OF PARTICLES AND ROTATIONAL OTION 7.1 Introducton 7.2 Centre of mass 7.3 oton of centre of mass 7.4 Lnear momentum of a system of partcles 7.5 Vector product of two vectors 7.6 Angular

More information

SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

SYSTEMS OF PARTICLES AND ROTATIONAL MOTION CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 Introducton 7. Centre of mass 7.3 Moton of centre of mass 7.4 Lnear momentum of a system of partcles 7.5 Vector product of two vectors 7.6 Angular

More information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before .1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments

More information

PES 1120 Spring 2014, Spendier Lecture 6/Page 1

PES 1120 Spring 2014, Spendier Lecture 6/Page 1 PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons -> charged rod -> charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura

More information

Physics 114 Exam 3 Spring Name:

Physics 114 Exam 3 Spring Name: Physcs 114 Exam 3 Sprng 015 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem 4. Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse

More information

LAB # 4 - Torque. d (1)

LAB # 4 - Torque. d (1) LAB # 4 - Torque. Introducton Through the use of Newton's three laws of moton, t s possble (n prncple, f not n fact) to predct the moton of any set of partcles. That s, n order to descrbe the moton of

More information

a) No books or notes are permitted. b) You may use a calculator.

a) No books or notes are permitted. b) You may use a calculator. PHYS 050 Sprng 06 Name: Test 3 Aprl 7, 06 INSTRUCTIONS: a) No books or notes are permtted. b) You may use a calculator. c) You must solve all problems begnnng wth the equatons on the Inormaton Sheet provded

More information

SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ

SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSED-BOOK

More information

Physics for Scientists & Engineers 2

Physics for Scientists & Engineers 2 Equpotental Surfaces and Lnes Physcs for Scentsts & Engneers 2 Sprng Semester 2005 Lecture 9 January 25, 2005 Physcs for Scentsts&Engneers 2 1 When an electrc feld s present, the electrc potental has a

More information

Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.

Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f. Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal

More information

PHYS 1441 Section 001 Lecture #15 Wednesday, July 8, 2015

PHYS 1441 Section 001 Lecture #15 Wednesday, July 8, 2015 PHYS 1441 Secton 001 Lecture #15 Wednesday, July 8, 2015 Concept of the Center of Mass Center of Mass & Center of Gravty Fundamentals of the Rotatonal Moton Rotatonal Knematcs Equatons of Rotatonal Knematcs

More information

For a 1-weight experiment do Part 1. For a 2-weight experiment do Part 1 and Part 2

For a 1-weight experiment do Part 1. For a 2-weight experiment do Part 1 and Part 2 Page of 6 THE GYROSCOPE The setup s not connected to a computer. You cannot get measured values drectly from the computer or enter them nto the lab PC. Make notes durng the sesson to use them later for

More information

Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3. r r. Position, Velocity, and Acceleration Revisited Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

More information

Homework 2: Kinematics and Dynamics of Particles Due Friday Feb 7, 2014 Max Score 45 Points + 8 Extra Credit

Homework 2: Kinematics and Dynamics of Particles Due Friday Feb 7, 2014 Max Score 45 Points + 8 Extra Credit EN40: Dynamcs and Vbratons School of Engneerng Brown Unversty Homework : Knematcs and Dynamcs of Partcles Due Frday Feb 7, 014 Max Score 45 Ponts + 8 Extra Credt 1. An expermental mcro-robot (see a descrpton

More information

5 The Physics of Rotating Bodies

5 The Physics of Rotating Bodies Prnceton Unversty 1996 Ph101 Laboratory 5 1 5 The Physcs of Rotatng Bodes Introducton Newton s law F = m a descrbes the moton of the center of mass of an object. But n general the moton of a rgd body conssts

More information

SUMMARY Phys 2113 (General Physics I) Compiled by Prof. Erickson. v = r t. v = lim t 0. p = mv. a = v. a = lim

SUMMARY Phys 2113 (General Physics I) Compiled by Prof. Erickson. v = r t. v = lim t 0. p = mv. a = v. a = lim SUMMARY Phys 2113 (General Physcs I) Compled by Prof. Erckson Poston Vector (m): r = xˆx + yŷ + zẑ Average Velocty (m/s): v = r Instantaneous Velocty (m/s): v = lm 0 r = ṙ Lnear Momentum (kg m/s): p =

More information

Iterative General Dynamic Model for Serial-Link Manipulators

Iterative General Dynamic Model for Serial-Link Manipulators EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general

More information

PY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg

PY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays

More information

PHY2049 Exam 2 solutions Fall 2016 Solution:

PHY2049 Exam 2 solutions Fall 2016 Solution: PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now

More information

THERMAL DISTRIBUTION IN THE HCL SPECTRUM OBJECTIVE

THERMAL DISTRIBUTION IN THE HCL SPECTRUM OBJECTIVE ame: THERMAL DISTRIBUTIO I THE HCL SPECTRUM OBJECTIVE To nvestgate a system s thermal dstrbuton n dscrete states; specfcally, determne HCl gas temperature from the relatve occupatons of ts rotatonal states.

More information

Lecture 23: Newton-Euler Formulation. Vaibhav Srivastava

Lecture 23: Newton-Euler Formulation. Vaibhav Srivastava Lecture 23: Newton-Euler Formulaton Based on Chapter 7, Spong, Hutchnson, and Vdyasagar Vabhav Srvastava Department of Electrcal & Computer Engneerng Mchgan State Unversty Aprl 10, 2017 ECE 818: Robotcs

More information

Chapter 12 Equilibrium & Elasticity

Chapter 12 Equilibrium & Elasticity Chapter 12 Equlbrum & Elastcty If there s a net force, an object wll experence a lnear acceleraton. (perod, end of story!) If there s a net torque, an object wll experence an angular acceleraton. (perod,

More information

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

11. Dynamics in Rotating Frames of Reference

11. Dynamics in Rotating Frames of Reference Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy

More information

TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 2211, Exam 2 Section 1 Version 1 October 18, 2013 Total Weight: 100 points

TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 2211, Exam 2 Section 1 Version 1 October 18, 2013 Total Weight: 100 points TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS, Exam Secton Verson October 8, 03 Total Weght: 00 ponts. Check your examnaton or completeness pror to startng. There are a total o nne

More information

Lesson 5: Kinematics and Dynamics of Particles

Lesson 5: Kinematics and Dynamics of Particles Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and

More information

Classical Mechanics Virtual Work & d Alembert s Principle

Classical Mechanics Virtual Work & d Alembert s Principle Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often

More information

coordinates. Then, the position vectors are described by

coordinates. Then, the position vectors are described by Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,

More information

Physics 114 Exam 2 Fall 2014 Solutions. Name:

Physics 114 Exam 2 Fall 2014 Solutions. Name: Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,

More information

PHYS 1443 Section 002

PHYS 1443 Section 002 PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS

More information

Module 11 Design of Joints for Special Loading. Version 2 ME, IIT Kharagpur

Module 11 Design of Joints for Special Loading. Version 2 ME, IIT Kharagpur Module 11 Desgn o Jonts or Specal Loadng Verson ME, IIT Kharagpur Lesson 1 Desgn o Eccentrcally Loaded Bolted/Rveted Jonts Verson ME, IIT Kharagpur Instructonal Objectves: At the end o ths lesson, the

More information

Physics 111 Final Exam, Fall 2013, Version A

Physics 111 Final Exam, Fall 2013, Version A Physcs 111 Fnal Exam, Fall 013, Verson A Name (Prnt): 4 Dgt ID: Secton: Honors Code Pledge: For ethcal and farness reasons all students are pledged to comply wth the provsons of the NJIT Academc Honor

More information

Page 1. SPH4U: Lecture 7. New Topic: Friction. Today s Agenda. Surface Friction... Surface Friction...

Page 1. SPH4U: Lecture 7. New Topic: Friction. Today s Agenda. Surface Friction... Surface Friction... SPH4U: Lecture 7 Today s Agenda rcton What s t? Systeatc catagores of forces How do we characterze t? Model of frcton Statc & Knetc frcton (knetc = dynac n soe languages) Soe probles nvolvng frcton ew

More information

= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]

= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W] Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:

More information

Slide. King Saud University College of Science Physics & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1-D (PART 2) LECTURE NO.

Slide. King Saud University College of Science Physics & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1-D (PART 2) LECTURE NO. Slde Kng Saud Unersty College of Scence Physcs & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1-D (PART ) LECTURE NO. 6 THIS PRESENTATION HAS BEEN PREPARED BY: DR. NASSR S. ALZAYED Lecture

More information

CONDUCTORS AND INSULATORS

CONDUCTORS AND INSULATORS CONDUCTORS AND INSULATORS We defne a conductor as a materal n whch charges are free to move over macroscopc dstances.e., they can leave ther nucle and move around the materal. An nsulator s anythng else.

More information

Physics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V.

Physics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V. Physcs 4 Solutons to Chapter 3 HW Chapter 3: Questons:, 4, 1 Problems:, 15, 19, 7, 33, 41, 45, 54, 65 Queston 3-1 and 3 te (clockwse), then and 5 te (zero), then 4 and 6 te (counterclockwse) Queston 3-4

More information