Chapter 8. Rotational Motion

Size: px
Start display at page:

Download "Chapter 8. Rotational Motion"

Transcription

1 Chapter 8 Rotational Motion

2 Rotational Work and Energy W = Fs = s = rθ Frθ Consider the work done in rotating a wheel with a tangential force, F, by an angle θ. τ = Fr W =τθ

3 Rotational Work and Energy DEFINITION OF ROTATIONAL WORK The rotational work done by a constant torque in turning an object through an angle is W R =τθ Requirement: The angle must be expressed in radians. SI Unit of Rotational Work: joule (J)

4 Rotational Work and Energy According to the Work-Energy theorem: W = KE f - KE 0 So W R should be able to produce rotational kinetic energy. Calculate the kinetic energy of a mass m undergoing rotational motion at radius r and moving with tangential speed v T KE = T = 1 1 mv ω mr v T = rω For a system of rotating masses, the total kinetic energy is the sum over the kinetic energies of the individual masses, 1 KE = ( mr ω ) = 1 mr ( )ω = 1 Iω

5 Rotational Work and Energy DEFINITION OF ROTATIONAL KINETIC ENERGY The rotational kinetic energy of a rigid rotating object is KE R = 1 Iω Requirement: The angular speed must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J) Thus, the rotational version of the Work-Energy theorem is: { W R =τθ W R = KE Rf - KE R0 where KE R = 1 Iω

6 Example: A hanging mass rotating a solid disk. As seen in the figure, a.0 kg mass attached to a string is rotating a solid disk of mass 10.0 kg and radius 0.0 m pivoting around its center. If the system is initially at rest, what is the angular velocity of the disk after the mass falls 0.70 m? r M T -T m =.0 kg M =10.0 kg r = 0.0 m d = 0.70 m Find ω f Work done on the disk: W R = ΔKE R τθ = Trθ = 1 Iω f 1 Iω 0 Since: rθ = d, I disk = 1 Mr, ω 0 = 0 Td = 1 4 Mr ω f Work done on the hanging mass: { m d W NC = ΔE Td = 1 mv f + mgh f Since: v 0 = 0 v f = rω f h f h 0 = d Td = 1 mr ω f mgd ( ) ( 1 mv 0 + mgh ) 0 Add disk eq. + hanging mass eq. 0 = 1 4 Mr ω f + 1 mr ω f mgd ω f = r mgd M + m = 0.0 (.0) ( )( 0.70) ( ) = 9.9 rad/s

7 Total energy of a rotating and translating rigid body in a gravitational field M I ω X c.m. v CM M g total energy = E = E rotation about CM + E translation of CM = 1 Iω + 1 Mv CM + Mgh CM Since a gravitational field is a conservative force E f = E 0

8 Rotational Work and Energy Example: Rolling Cylinders A thin-walled hollow cylinder (mass = m, radius = r) and a solid cylinder (also, mass = m, radius = r) start from rest at the top of an incline. Determine which cylinder has the greatest translational speed upon reaching the bottom.

9 Rotational Work and Energy 1 1 E = mv + Iω + mgh ENERGY CONSERVATION f + Iω f + mghf = mvo i + Iωo i mv + mgh io 1 1 f + Iω f mv = mgh io ω f = v f r

10 Rotational Work and Energy 1 1 mv + I v r = f f mgh o i v f = mgh m + I r o The cylinder with the smaller moment of inertia will have a greater final translational speed. Since I solid = ½mr and I hollow = mr Then, I solid < I hollow è v f solid > v f hollow

11 Angular Momentum DEFINITION OF ANGULAR MOMENTUM The angular momentum L of a body rotating about a fixed axis is the product of the body s moment of inertia and its angular velocity with respect to that axis: L = Iω Requirement: The angular speed must be expressed in rad/s. SI Unit of Angular Momentum: kg m /s

12 Consider the rotational version of Newton s nd Law: ( ) τ EXT = Iα = I Δω Δt = Δ Iω Δt ( τ EXT )Δt = ΔL = ΔL Δt "angular impulse-angular momentum theorm" If ( τ ) EXT = 0 ΔL = 0 L f = L 0 Conservation of angular momentum

13 Angular Momentum PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero. L f = L 0

14 Angular Momentum Conceptual Example: A Spinning Skater An ice skater is spinning with both arms and a leg outstretched. She pulls her arms and leg inward and her spinning motion changes dramatically. Use the principle of conservation of angular momentum to explain how and why her spinning motion changes.

15 Angular Momentum Example: A Satellite in an Elliptical Orbit An artificial satellite is placed in an elliptical orbit about the earth. Its point of closest approach is 8.37 x 10 6 m from the center of the earth, and its point of greatest distance is 5.1 x 10 6 m from the center of the earth. The speed of the satellite at the perigee is 8450 m/s. Find the speed at the apogee.

16 Angular Momentum L = Iω Since no external torques are present in this case, we have angular momentum conservation I A ω = A I P ω P I = mr ω = v r mr A v r A A = mr P v r P P

17 Angular Momentum mr A v r A A = mr P v r P P r v = A A r P v P ( ) 8450m s v A = r Pv P = m r A m ( ) = 80m s

18 Example: A potter s wheel is rotating around a vertical axis through its center at a frequency of.00 rev/s. The wheel can be considered a uniform disk of mass 4.80 kg and diameter m. The potter then throws a 3.10 kg chunk of clay, approximately shaped as a flat disk of radius 11.0 cm, onto the center of the rotating wheel. (a) What is the frequency of the wheel after the clay sticks to it? (b) What fraction of the original mechanical energy of the wheel is lost to friction after the collision with the clay? ω 0 =.00 rev/s = 1.6 rad/s a) L f = L 0 I f ω f = I 0 ω 0 R = m r = m m = 3.10 kg M = 4.80 kg b) KE 0 KE f KE 0 ω f = I 0 I f ω 0 = = 1 MR 1 MR + 1 mr ω 0 ( 4.80) ( 0.180) ( ( 4.80) ( 0.180) + ( 3.10) ( 0.110) 1.6 ) =10. rad/s =1.6 rev/s =1 KE f KE 0 =1 1 I fω f 1 I 0 ω 0 =1 ω f ω 0 = = 0.190

19 The Vector Nature of Angular Variables Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity. we can express L as a vector in the direction of ω: L = I ω and write conservation of angular momentum in vector form: L f = L 0

20 Example: A person sitting on a chair that can rotate is initially at rest and holding a bicycle wheel which is spinning with its angular momentum vector in the vertically up direction and with magnitude 0 rad/s. The mass and radius of the bicycle wheel are 5.0 kg and 0.30 m, respectively, approximated as a solid disk. The mass and average radius of the person through a vertical axis are 90 kg and 0.35 m, respectively, approximated as a solid cylinder. If the person now flips the spinning wheel so that the angular momentum vector is vertically down, what is the angular velocity of the person? L f = L 0 L 1 + L = L 1 L = L 1, in upward direction I ω = I 1 ω 1 ω = I 1 I ω 1 ω = ( )( 0.30) ( 0) =1.6 rad/s ( )( 0.35) 1 90

Rotational Dynamics continued

Rotational Dynamics continued Chapter 9 Rotational Dynamics continued 9.4 Newton s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS I = ( mr 2

More information

Chapter 8 continued. Rotational Dynamics

Chapter 8 continued. Rotational Dynamics Chapter 8 continued Rotational Dynamics 8.6 The Action of Forces and Torques on Rigid Objects Chapter 8 developed the concepts of angular motion. θ : angles and radian measure for angular variables ω :

More information

Chapter 9. Rotational Dynamics

Chapter 9. Rotational Dynamics Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination

More information

= o + t = ot + ½ t 2 = o + 2

= o + t = ot + ½ t 2 = o + 2 Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

More information

Chapter 9. Rotational Dynamics

Chapter 9. Rotational Dynamics Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination

More information

Ch 8. Rotational Dynamics

Ch 8. Rotational Dynamics Ch 8. Rotational Dynamics Rotational W, P, K, & L (a) Translation (b) Combined translation and rotation ROTATIONAL ANALOG OF NEWTON S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS = τ Iα Requirement:

More information

Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations

Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations Chapter 9- Rotational Dynamics Torque Center of Gravity Newton s 2 nd Law- Angular Rotational Work & Energy Angular Momentum Angular

More information

Circular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics

Circular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics Circular Motion, Pt 2: Angular Dynamics Mr. Velazquez AP/Honors Physics Formulas: Angular Kinematics (θ must be in radians): s = rθ Arc Length 360 = 2π rads = 1 rev ω = θ t = v t r Angular Velocity α av

More information

Topic 1: Newtonian Mechanics Energy & Momentum

Topic 1: Newtonian Mechanics Energy & Momentum Work (W) the amount of energy transferred by a force acting through a distance. Scalar but can be positive or negative ΔE = W = F! d = Fdcosθ Units N m or Joules (J) Work, Energy & Power Power (P) the

More information

We define angular displacement, θ, and angular velocity, ω. What's a radian?

We define angular displacement, θ, and angular velocity, ω. What's a radian? We define angular displacement, θ, and angular velocity, ω Units: θ = rad ω = rad/s What's a radian? Radian is the ratio between the length of an arc and its radius note: counterclockwise is + clockwise

More information

Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum

Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion Torque and angular momentum In Figure, in order to turn a rod about a fixed hinge at one end, a force F is applied at a

More information

Rotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart

Rotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart Rotational Motion & Angular Momentum Rotational Motion Every quantity that we have studied with translational motion has a rotational counterpart TRANSLATIONAL ROTATIONAL Displacement x Angular Displacement

More information

Rotational Dynamics, Moment of Inertia and Angular Momentum

Rotational Dynamics, Moment of Inertia and Angular Momentum Rotational Dynamics, Moment of Inertia and Angular Momentum Now that we have examined rotational kinematics and torque we will look at applying the concepts of angular motion to Newton s first and second

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

PSI AP Physics I Rotational Motion

PSI AP Physics I Rotational Motion PSI AP Physics I Rotational Motion Multiple-Choice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from

More information

Chapter 8 Lecture Notes

Chapter 8 Lecture Notes Chapter 8 Lecture Notes Physics 2414 - Strauss Formulas: v = l / t = r θ / t = rω a T = v / t = r ω / t =rα a C = v 2 /r = ω 2 r ω = ω 0 + αt θ = ω 0 t +(1/2)αt 2 θ = (1/2)(ω 0 +ω)t ω 2 = ω 0 2 +2αθ τ

More information

AP Physics 1 Rotational Motion Practice Test

AP Physics 1 Rotational Motion Practice Test AP Physics 1 Rotational Motion Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A spinning ice skater on extremely smooth ice is able

More information

PHYSICS 220. Lecture 15. Textbook Sections Lecture 15 Purdue University, Physics 220 1

PHYSICS 220. Lecture 15. Textbook Sections Lecture 15 Purdue University, Physics 220 1 PHYSICS 220 Lecture 15 Angular Momentum Textbook Sections 9.3 9.6 Lecture 15 Purdue University, Physics 220 1 Last Lecture Overview Torque = Force that causes rotation τ = F r sin θ Work done by torque

More information

Slide 1 / 37. Rotational Motion

Slide 1 / 37. Rotational Motion Slide 1 / 37 Rotational Motion Slide 2 / 37 Angular Quantities An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360 in a circle or 2π radians.

More information

III. Work and Energy

III. Work and Energy Rotation I. Kinematics - Angular analogs II. III. IV. Dynamics - Torque and Rotational Inertia Work and Energy Angular Momentum - Bodies and particles V. Elliptical Orbits The student will be able to:

More information

Physics 111. Tuesday, November 2, Rotational Dynamics Torque Angular Momentum Rotational Kinetic Energy

Physics 111. Tuesday, November 2, Rotational Dynamics Torque Angular Momentum Rotational Kinetic Energy ics Tuesday, ember 2, 2002 Ch 11: Rotational Dynamics Torque Angular Momentum Rotational Kinetic Energy Announcements Wednesday, 8-9 pm in NSC 118/119 Sunday, 6:30-8 pm in CCLIR 468 Announcements This

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

Chapter 8. Rotational Equilibrium and Rotational Dynamics Chapter 8 Rotational Equilibrium and Rotational Dynamics Wrench Demo Torque Torque, τ, is the tendency of a force to rotate an object about some axis τ = Fd F is the force d is the lever arm (or moment

More information

Rolling without slipping Angular Momentum Conservation of Angular Momentum. Physics 201: Lecture 19, Pg 1

Rolling without slipping Angular Momentum Conservation of Angular Momentum. Physics 201: Lecture 19, Pg 1 Physics 131: Lecture Today s Agenda Rolling without slipping Angular Momentum Conservation o Angular Momentum Physics 01: Lecture 19, Pg 1 Rolling Without Slipping Rolling is a combination o rotation and

More information

PSI AP Physics I Rotational Motion

PSI AP Physics I Rotational Motion PSI AP Physics I Rotational Motion Multiple-Choice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from

More information

Rotational Motion and Torque

Rotational Motion and Torque Rotational Motion and Torque Introduction to Angular Quantities Sections 8- to 8-2 Introduction Rotational motion deals with spinning objects, or objects rotating around some point. Rotational motion is

More information

Chapter 9. Rotational Dynamics

Chapter 9. Rotational Dynamics Chapter 9 Rotational Dynamics In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation. 1) Torque Produces angular

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

Chapter 8. Rotational Equilibrium and Rotational Dynamics Chapter 8 Rotational Equilibrium and Rotational Dynamics 1 Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related 2 Torque The door is free to rotate

More information

31 ROTATIONAL KINEMATICS

31 ROTATIONAL KINEMATICS 31 ROTATIONAL KINEMATICS 1. Compare and contrast circular motion and rotation? Address the following Which involves an object and which involves a system? Does an object/system in circular motion have

More information

Chapter 8 - Rotational Dynamics and Equilibrium REVIEW

Chapter 8 - Rotational Dynamics and Equilibrium REVIEW Pagpalain ka! (Good luck, in Filipino) Date Chapter 8 - Rotational Dynamics and Equilibrium REVIEW TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. 1) When a rigid body

More information

Angular Momentum L = I ω

Angular Momentum L = I ω Angular Momentum L = Iω If no NET external Torques act on a system then Angular Momentum is Conserved. Linitial = I ω = L final = Iω Angular Momentum L = Iω Angular Momentum L = I ω A Skater spins with

More information

Chapter 8, Rotational Equilibrium and Rotational Dynamics. 3. If a net torque is applied to an object, that object will experience:

Chapter 8, Rotational Equilibrium and Rotational Dynamics. 3. If a net torque is applied to an object, that object will experience: CHAPTER 8 3. If a net torque is applied to an object, that object will experience: a. a constant angular speed b. an angular acceleration c. a constant moment of inertia d. an increasing moment of inertia

More information

AP Physics 1- Torque, Rotational Inertia, and Angular Momentum Practice Problems FACT: The center of mass of a system of objects obeys Newton s second law- F = Ma cm. Usually the location of the center

More information

CHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque

CHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque 7 1. Define Torque 2. State the conditions for equilibrium of rigid body (Hint: 2 conditions) 3. Define angular displacement 4. Define average angular velocity 5. Define instantaneous angular velocity

More information

Review questions. Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right.

Review questions. Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right. Review questions Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right. 30 kg 70 kg v (a) Is this collision elastic? (b) Find the

More information

PHYSICS 149: Lecture 21

PHYSICS 149: Lecture 21 PHYSICS 149: Lecture 21 Chapter 8: Torque and Angular Momentum 8.2 Torque 8.4 Equilibrium Revisited 8.8 Angular Momentum Lecture 21 Purdue University, Physics 149 1 Midterm Exam 2 Wednesday, April 6, 6:30

More information

Angular Momentum. Objectives CONSERVATION OF ANGULAR MOMENTUM

Angular Momentum. Objectives CONSERVATION OF ANGULAR MOMENTUM Angular Momentum CONSERVATION OF ANGULAR MOMENTUM Objectives Calculate the angular momentum vector for a moving particle Calculate the angular momentum vector for a rotating rigid object where angular

More information

Angular Momentum L = I ω

Angular Momentum L = I ω Angular Momentum L = Iω If no NET external Torques act on a system then Angular Momentum is Conserved. Linitial = I ω = L final = Iω Angular Momentum L = Iω Angular Momentum L = I ω A Skater spins with

More information

Rotation Quiz II, review part A

Rotation Quiz II, review part A Rotation Quiz II, review part A 1. A solid disk with a radius R rotates at a constant rate ω. Which of the following points has the greater angular velocity? A. A B. B C. C D. D E. All points have the

More information

General Definition of Torque, final. Lever Arm. General Definition of Torque 7/29/2010. Units of Chapter 10

General Definition of Torque, final. Lever Arm. General Definition of Torque 7/29/2010. Units of Chapter 10 Units of Chapter 10 Determining Moments of Inertia Rotational Kinetic Energy Rotational Plus Translational Motion; Rolling Why Does a Rolling Sphere Slow Down? General Definition of Torque, final Taking

More information

A) 1 gm 2 /s. B) 3 gm 2 /s. C) 6 gm 2 /s. D) 9 gm 2 /s. E) 10 gm 2 /s. A) 0.1 kg. B) 1 kg. C) 2 kg. D) 5 kg. E) 10 kg A) 2:5 B) 4:5 C) 1:1 D) 5:4

A) 1 gm 2 /s. B) 3 gm 2 /s. C) 6 gm 2 /s. D) 9 gm 2 /s. E) 10 gm 2 /s. A) 0.1 kg. B) 1 kg. C) 2 kg. D) 5 kg. E) 10 kg A) 2:5 B) 4:5 C) 1:1 D) 5:4 1. A 4 kg object moves in a circle of radius 8 m at a constant speed of 2 m/s. What is the angular momentum of the object with respect to an axis perpendicular to the circle and through its center? A)

More information

Recap: Solid Rotational Motion (Chapter 8) displacement velocity acceleration Newton s 2nd law τ = I.α N.s τ = F. l moment of inertia mass size

Recap: Solid Rotational Motion (Chapter 8) displacement velocity acceleration Newton s 2nd law τ = I.α N.s τ = F. l moment of inertia mass size Recap: Solid Rotational Motion (Chapter 8) We have developed equations to describe rotational displacement θ, rotational velocity ω and rotational acceleration α. We have used these new terms to modify

More information

Chapter 9. Rotational Dynamics

Chapter 9. Rotational Dynamics Chapter 9 Rotational Dynamics In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation. 1) Torque Produces angular

More information

Chapter 8 Rotational Motion and Equilibrium. 1. Give explanation of torque in own words after doing balance-the-torques lab as an inquiry introduction

Chapter 8 Rotational Motion and Equilibrium. 1. Give explanation of torque in own words after doing balance-the-torques lab as an inquiry introduction Chapter 8 Rotational Motion and Equilibrium Name 1. Give explanation of torque in own words after doing balance-the-torques lab as an inquiry introduction 1. The distance between a turning axis and the

More information

Rotational Dynamics. Slide 2 / 34. Slide 1 / 34. Slide 4 / 34. Slide 3 / 34. Slide 6 / 34. Slide 5 / 34. Moment of Inertia. Parallel Axis Theorem

Rotational Dynamics. Slide 2 / 34. Slide 1 / 34. Slide 4 / 34. Slide 3 / 34. Slide 6 / 34. Slide 5 / 34. Moment of Inertia. Parallel Axis Theorem Slide 1 / 34 Rotational ynamics l Slide 2 / 34 Moment of Inertia To determine the moment of inertia we divide the object into tiny masses of m i a distance r i from the center. is the sum of all the tiny

More information

Chapter 8 continued. Rotational Dynamics

Chapter 8 continued. Rotational Dynamics Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = s = rφ = Frφ Fr = τ (torque) = τφ r φ s F to s θ = 0 DEFINITION

More information

Chapter 10. Rotation

Chapter 10. Rotation Chapter 10 Rotation Rotation Rotational Kinematics: Angular velocity and Angular Acceleration Rotational Kinetic Energy Moment of Inertia Newton s nd Law for Rotation Applications MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised

More information

Rolling, Torque & Angular Momentum

Rolling, Torque & Angular Momentum PHYS 101 Previous Exam Problems CHAPTER 11 Rolling, Torque & Angular Momentum Rolling motion Torque Angular momentum Conservation of angular momentum 1. A uniform hoop (ring) is rolling smoothly from the

More information

Welcome back to Physics 211

Welcome back to Physics 211 Welcome back to Physics 211 Today s agenda: Moment of Inertia Angular momentum 13-2 1 Current assignments Prelecture due Tuesday after Thanksgiving HW#13 due next Wednesday, 11/24 Turn in written assignment

More information

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work Translational vs Rotational / / 1/ Δ m x v dx dt a dv dt F ma p mv KE mv Work Fd / / 1/ θ ω θ α ω τ α ω ω τθ Δ I d dt d dt I L I KE I Work / θ ω α τ Δ Δ c t s r v r a v r a r Fr L pr Connection Translational

More information

TORQUE. Chapter 10 pages College Physics OpenStax Rice University AP College board Approved.

TORQUE. Chapter 10 pages College Physics OpenStax Rice University AP College board Approved. TORQUE Chapter 10 pages 343-384 College Physics OpenStax Rice University AP College board Approved. 1 SECTION 10.1 PAGE 344; ANGULAR ACCELERATION ω = Δθ Δt Where ω is velocity relative to an angle, Δθ

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

p = mv L = Iω L =! r x p or, if we use translational parameters:

p = mv L = Iω L =! r x p or, if we use translational parameters: ANGULAR MOMENTUM Torque is the rotational counterpart to force. So whereas when a net force is applied, a body accelerates, when a net torque is applied, a body angularly accelerates. Angular momentum

More information

Basics of rotational motion

Basics of rotational motion Basics of rotational motion Motion of bodies rotating about a given axis, like wheels, blades of a fan and a chair cannot be analyzed by treating them as a point mass or particle. At a given instant of

More information

Rolling, Torque, Angular Momentum

Rolling, Torque, Angular Momentum Chapter 11 Rolling, Torque, Angular Momentum Copyright 11.2 Rolling as Translational and Rotation Combined Motion of Translation : i.e.motion along a straight line Motion of Rotation : rotation about a

More information

Unit 8 Notetaking Guide Torque and Rotational Motion

Unit 8 Notetaking Guide Torque and Rotational Motion Unit 8 Notetaking Guide Torque and Rotational Motion Rotational Motion Until now, we have been concerned mainly with translational motion. We discussed the kinematics and dynamics of translational motion

More information

Conservation of Angular Momentum

Conservation of Angular Momentum Lecture 23 Chapter 12 Physics I Conservation of Angular Momentum Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi IN THIS CHAPTER, you will continue discussing rotational dynamics

More information

Two-Dimensional Rotational Kinematics

Two-Dimensional Rotational Kinematics Two-Dimensional Rotational Kinematics Rigid Bodies A rigid body is an extended object in which the distance between any two points in the object is constant in time. Springs or human bodies are non-rigid

More information

Chapter 10.A. Rotation of Rigid Bodies

Chapter 10.A. Rotation of Rigid Bodies Chapter 10.A Rotation of Rigid Bodies P. Lam 7_23_2018 Learning Goals for Chapter 10.1 Understand the equations govern rotational kinematics, and know how to apply them. Understand the physical meanings

More information

Chapter 9-10 Test Review

Chapter 9-10 Test Review Chapter 9-10 Test Review Chapter Summary 9.2. The Second Condition for Equilibrium Explain torque and the factors on which it depends. Describe the role of torque in rotational mechanics. 10.1. Angular

More information

. d. v A v B. e. none of these.

. d. v A v B. e. none of these. General Physics I Exam 3 - Chs. 7,8,9 - Momentum, Rotation, Equilibrium Oct. 28, 2009 Name Rec. Instr. Rec. Time For full credit, make your work clear to the grader. Show the formulas you use, the essential

More information

Fall 2007 RED Barcode Here Physics 105, sections 1 and 2 Please write your CID Colton

Fall 2007 RED Barcode Here Physics 105, sections 1 and 2 Please write your CID Colton Fall 007 RED Barcode Here Physics 105, sections 1 and Exam 3 Please write your CID Colton -3669 3 hour time limit. One 3 5 handwritten note card permitted (both sides). Calculators permitted. No books.

More information

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning?

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning? 1. rpm is about rad/s. 7. ROTATIONAL MOTION 2. A wheel rotates with constant angular acceleration of π rad/s 2. During the time interval from t 1 to t 2, its angular displacement is π rad. At time t 2

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

Physics A - PHY 2048C

Physics A - PHY 2048C Physics A - PHY 2048C and 11/15/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapter 12 in the textbook on? 2 Must an object be rotating to have a moment

More information

Chapter 8 continued. Rotational Dynamics

Chapter 8 continued. Rotational Dynamics Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = rφ = Frφ Fr = τ (torque) = τφ r φ s F to x θ = 0 DEFINITION OF

More information

Physics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1

Physics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1 Physics 201 p. 1/1 Physics 201 Professor P. Q. Hung 311B, Physics Building Physics 201 p. 2/1 Rotational Kinematics and Energy Rotational Kinetic Energy, Moment of Inertia All elements inside the rigid

More information

Rolling, Torque, and Angular Momentum

Rolling, Torque, and Angular Momentum AP Physics C Rolling, Torque, and Angular Momentum Introduction: Rolling: In the last unit we studied the rotation of a rigid body about a fixed axis. We will now extend our study to include cases where

More information

Review for 3 rd Midterm

Review for 3 rd Midterm Review for 3 rd Midterm Midterm is on 4/19 at 7:30pm in the same rooms as before You are allowed one double sided sheet of paper with any handwritten notes you like. The moment-of-inertia about the center-of-mass

More information

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion Angular velocity and angular acceleration CHAPTER 9 ROTATION! r i ds i dθ θ i Angular velocity and angular acceleration! equations of rotational motion Torque and Moment of Inertia! Newton s nd Law for

More information

PHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011

PHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011 PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this

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

Physics 201 Midterm Exam 3

Physics 201 Midterm Exam 3 Physics 201 Midterm Exam 3 Information and Instructions Student ID Number: Section Number: TA Name: Please fill in all the information above. Please write and bubble your Name and Student Id number on

More information

Stability of rotating objects

Stability of rotating objects PHY131H1F - Hour 28 Today: We finish up Chapter 9! 9.5 Rotational Kinetic Energy (skip 9.6 on Tides and Earth s day) Stability of rotating objects If the rider's balance shifts a bit, the bike + rider

More information

3. A bicycle tire of radius 0.33 m and a mass 1.5 kg is rotating at 98.7 rad/s. What torque is necessary to stop the tire in 2.0 s?

3. A bicycle tire of radius 0.33 m and a mass 1.5 kg is rotating at 98.7 rad/s. What torque is necessary to stop the tire in 2.0 s? Practice 8A Torque 1. Find the torque produced by a 3.0 N force applied at an angle of 60.0 to a door 0.25 m from the hinge. What is the maximum torque this force could exert? 2. If the torque required

More information

Chapter 8 Lecture. Pearson Physics. Rotational Motion and Equilibrium. Prepared by Chris Chiaverina Pearson Education, Inc.

Chapter 8 Lecture. Pearson Physics. Rotational Motion and Equilibrium. Prepared by Chris Chiaverina Pearson Education, Inc. Chapter 8 Lecture Pearson Physics Rotational Motion and Equilibrium Prepared by Chris Chiaverina Chapter Contents Describing Angular Motion Rolling Motion and the Moment of Inertia Torque Static Equilibrium

More information

Test 7 wersja angielska

Test 7 wersja angielska Test 7 wersja angielska 7.1A One revolution is the same as: A) 1 rad B) 57 rad C) π/2 rad D) π rad E) 2π rad 7.2A. If a wheel turns with constant angular speed then: A) each point on its rim moves with

More information

Big Ideas 3 & 5: Circular Motion and Rotation 1 AP Physics 1

Big Ideas 3 & 5: Circular Motion and Rotation 1 AP Physics 1 Big Ideas 3 & 5: Circular Motion and Rotation 1 AP Physics 1 1. A 50-kg boy and a 40-kg girl sit on opposite ends of a 3-meter see-saw. How far from the girl should the fulcrum be placed in order for the

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

ΣF = ma Στ = Iα ½mv 2 ½Iω 2. mv Iω

ΣF = ma Στ = Iα ½mv 2 ½Iω 2. mv Iω Thur Oct 22 Assign 9 Friday Today: Torques Angular Momentum x θ v ω a α F τ m I Roll without slipping: x = r Δθ v LINEAR = r ω a LINEAR = r α ΣF = ma Στ = Iα ½mv 2 ½Iω 2 I POINT = MR 2 I HOOP = MR 2 I

More information

AP practice ch 7-8 Multiple Choice

AP practice ch 7-8 Multiple Choice AP practice ch 7-8 Multiple Choice 1. A spool of thread has an average radius of 1.00 cm. If the spool contains 62.8 m of thread, how many turns of thread are on the spool? "Average radius" allows us to

More information

Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular

Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.F.1: Only

More information

Angular Momentum Conservation of Angular Momentum

Angular Momentum Conservation of Angular Momentum Lecture 22 Chapter 12 Physics I Angular Momentum Conservation of Angular Momentum Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi IN THIS CHAPTER, you will continue discussing rotational

More information

Physics 111. Lecture 23 (Walker: 10.6, 11.1) Conservation of Energy in Rotation Torque March 30, Kinetic Energy of Rolling Object

Physics 111. Lecture 23 (Walker: 10.6, 11.1) Conservation of Energy in Rotation Torque March 30, Kinetic Energy of Rolling Object Physics 111 Lecture 3 (Walker: 10.6, 11.1) Conservation of Energy in Rotation Torque March 30, 009 Lecture 3 1/4 Kinetic Energy of Rolling Object Total kinetic energy of a rolling object is the sum of

More information

Practice Test 3. Multiple Choice Identify the choice that best completes the statement or answers the question.

Practice Test 3. Multiple Choice Identify the choice that best completes the statement or answers the question. Practice Test 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel rotates about a fixed axis with an initial angular velocity of 20 rad/s. During

More information

Webreview Torque and Rotation Practice Test

Webreview Torque and Rotation Practice Test Please do not write on test. ID A Webreview - 8.2 Torque and Rotation Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 0.30-m-radius automobile

More information

I 2 comω 2 + Rolling translational+rotational. a com. L sinθ = h. 1 tot. smooth rolling a com =αr & v com =ωr

I 2 comω 2 + Rolling translational+rotational. a com. L sinθ = h. 1 tot. smooth rolling a com =αr & v com =ωr Rolling translational+rotational smooth rolling a com =αr & v com =ωr Equations of motion from: - Force/torque -> a and α - Energy -> v and ω 1 I 2 comω 2 + 1 Mv 2 = KE 2 com tot a com KE tot = KE trans

More information

Chapter 10: Dynamics of Rotational Motion

Chapter 10: Dynamics of Rotational Motion Chapter 10: Dynamics of Rotational Motion What causes an angular acceleration? The effectiveness of a force at causing a rotation is called torque. QuickCheck 12.5 The four forces shown have the same strength.

More information

Angular Momentum and Its Conservation

Angular Momentum and Its Conservation Angular Momentum and Its Conservation Bởi: OpenStaxCollege Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by

More information

AP Physics QUIZ Chapters 10

AP Physics QUIZ Chapters 10 Name: 1. Torque is the rotational analogue of (A) Kinetic Energy (B) Linear Momentum (C) Acceleration (D) Force (E) Mass A 5-kilogram sphere is connected to a 10-kilogram sphere by a rigid rod of negligible

More information

CIRCULAR MOTION AND ROTATION

CIRCULAR MOTION AND ROTATION 1. UNIFORM CIRCULAR MOTION So far we have learned a great deal about linear motion. This section addresses rotational motion. The simplest kind of rotational motion is an object moving in a perfect circle

More information

Chapter 10. Rotation of a Rigid Object about a Fixed Axis

Chapter 10. Rotation of a Rigid Object about a Fixed Axis Chapter 10 Rotation of a Rigid Object about a Fixed Axis 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. A small

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

Chapter 8. Rotational Equilibrium and Rotational Dynamics Chapter 8 Rotational Equilibrium and Rotational Dynamics Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related Torque The door is free to rotate about

More information

Torque rotational force which causes a change in rotational motion. This force is defined by linear force multiplied by a radius.

Torque rotational force which causes a change in rotational motion. This force is defined by linear force multiplied by a radius. Warm up A remote-controlled car's wheel accelerates at 22.4 rad/s 2. If the wheel begins with an angular speed of 10.8 rad/s, what is the wheel's angular speed after exactly three full turns? AP Physics

More information

Write your name legibly on the top right hand corner of this paper

Write your name legibly on the top right hand corner of this paper NAME Phys 631 Summer 2007 Quiz 2 Tuesday July 24, 2007 Instructor R. A. Lindgren 9:00 am 12:00 am Write your name legibly on the top right hand corner of this paper No Books or Notes allowed Calculator

More information

Lecture 7 Chapter 10,11

Lecture 7 Chapter 10,11 Lecture 7 Chapter 10,11 Rotation, Inertia, Rolling, Torque, and Angular momentum Demo Demos Summary of Concepts to Cover from chapter 10 Rotation Rotating cylinder with string wrapped around it: example

More information

Physics 131: Lecture 21. Today s Agenda

Physics 131: Lecture 21. Today s Agenda Physics 131: Lecture 1 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 01: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia

More information

Physics 4A Solutions to Chapter 10 Homework

Physics 4A Solutions to Chapter 10 Homework Physics 4A Solutions to Chapter 0 Homework Chapter 0 Questions: 4, 6, 8 Exercises & Problems 6, 3, 6, 4, 45, 5, 5, 7, 8 Answers to Questions: Q 0-4 (a) positive (b) zero (c) negative (d) negative Q 0-6

More information

Uniform Circular Motion

Uniform Circular Motion Uniform Circular Motion Motion in a circle at constant angular speed. ω: angular velocity (rad/s) Rotation Angle The rotation angle is the ratio of arc length to radius of curvature. For a given angle,

More information

Forces of Rolling. 1) Ifobjectisrollingwith a com =0 (i.e.no netforces), then v com =ωr = constant (smooth roll)

Forces of Rolling. 1) Ifobjectisrollingwith a com =0 (i.e.no netforces), then v com =ωr = constant (smooth roll) Physics 2101 Section 3 March 12 rd : Ch. 10 Announcements: Mid-grades posted in PAW Quiz today I will be at the March APS meeting the week of 15-19 th. Prof. Rich Kurtz will help me. Class Website: http://www.phys.lsu.edu/classes/spring2010/phys2101-3/

More information

Energy and Angular Momentum

Energy and Angular Momentum Notes 13 Rotation Page 1 Energy and Angular Momentum The kinetic energy associate with a rotating object is simply the sum of the regular kinetic energies. Our goal is to state the rotational kinetic energy

More information