Rotation. Moment of inertia of a rotating body: w I = r 2 dm

Save this PDF as:

Size: px
Start display at page:

Download "Rotation. Moment of inertia of a rotating body: w I = r 2 dm"

Transcription

1 Rotation Moment of inertia of a rotating body: w I = r 2 dm Usually reasonably easy to calculate when Body has symmetries Rotation axis goes through Center of mass Exams: All moment of inertia will be given! No need to copy the table from the book.

2 Rotation Parallel axis theorem: Assume the body rotates around an axis through P. Let the COM be the center of our coordinate system. b COM. P. dm P has the coordinates (a,b) a w w I = r 2 dm = (x-a) 2 +(y-b) 2 dm w w w w = (x 2 +y 2 )dm 2a xdm 2b ydm + (a 2 +b 2 ) dm = I COM h 2 M I = I COM + h 2 M This is something you might need

3 Rotation Parallel axis theorem: Assume the body rotates around an axis through P. Let the COM be the center of our coordinate system. b COM. P. dm P has the coordinates (a,b) a I = I COM +Mh 2 The moment of inertia of a body rotating around an arbitrary axis is equal to the moment of inertia of a body rotating around a parallel axis through the center of mass plus the mass times the perpendicular distance between the axes h squared.

4 Example: Moment of inertia Solid sphere of radius R rotating around symmetry axis: I = 2MR 2 /5 I = I COM +Mh 2 If R<<h I COM <<< Mh 2 I COM,A > I COM,B > I COM,C h=1m 36kg h=2m h=3m 9kg All have essentially the same moment of inertia: I ~ 36 kg m 2 4kg

5 Example: Moment of inertia.p w I = r 2 dm Lets calculate the moment of inertia for an annular homogeneous cylinder rotating around the central axis: Parameters: Mass M, Length L Outer and Inner Radii R 1, R 2 Step1: Replace dm with an integration over a volume element dv. Step 2: Express the volume element in useful coordinates and find the boundaries for the integration. Step 3: Integrate

6 Example: Moment of inertia.p w I = r 2 dm V cyl = L (R 1 2 -R 2 2 ) w I = r 2 dv Lets calculate the moment of inertia for an annular homogeneous cylinder rotating around the central axis: Parameters: Mass M, Length L Outer and Inner Radii R 1, R 2 Step1: Replace dm with an integration over a volume element dv. Homogeneous: Density = M/V dm = dv and is independent of the coordinates (As long as we only integrate over the body itself and not over empty space)

7 Example: Moment of inertia.p w I = r 2 dv Step 2: Express the volume element in useful coordinates and find the boundaries for the integration. dv = dxdydz in cartesian coordinates Better coordinates: cylindrical coordinates Lets calculate the moment of inertia for an annular homogeneous cylinder rotating around the central axis: Parameters: Mass M, Length L Outer and Inner Radii R 1, R 2 y ds dr x

8 Volume element in cylindrical coordinates w I = r 2 dv dv = dxdydz in cartesian coordinates Better coordinates for this problem: Cylindrical Coordinates dz is the same in both only change how we describe things in the y x-y plane Area element in x-y plane ds dr dsdr = rd dr x dv = dxdydz = rd drdz

9 Example: Moment of inertia w.p I = r 2 dv w I = r 2 rd drdz Step 2: Express the volume element in useful coordinates and find the boundaries for the integration. Integration boundaries: z: -L/2 to L/2, : 0 to 2 r: R 2 to R 1 I = L 2 (R 1 4 -R 2 4 )/4 Use (R 1 4 -R 2 4 ) = (R 1 2 +R 2 2 ) (R 1 2 -R 2 2 ) z-integration -integration r-integration Recall: V cyl = L (R 12 -R 22 ) I = M (R 1 2 +R 2 2 )/2 See Table in book, also Sample Problem 10-7

10 Kinetic Energy Kinetic energy of a rotating body: Lets split up the body into a collection of particles K = 0.5 ( m i r i2 ) 2 ṛ i Rotational Inertia: I = ( m i r i 2 ) m v K = 0.5 I 2 Compare with translation: K = 0.5 m v 2 Correspondences: v <-> I <-> m

11 Example: Kinetic Energy Energy storage in an electric flywheel: K rot = 0.5 I 2 To reduce mass w/o reducing I it might be useful to use an annular cylinder. But material stress favors solid wheels for high end applications.

12 Example: Kinetic Energy Energy storage in an electric flywheel: K rot = 0.5 I 2 Example (solid wheel): M = 600kg, R=0.5m I = 75kg m 2 Rotates 30000rpm: f = 30000rpm/60s/min = 500Hz = 2 f = * 1000 rad/s K rot = 370MJ A lot of energy which can be released very fast in fusion reactors, in braking systems used in trains etc.

13 Torque We have discussed the equations describing the dynamical variables of rotation ( ) calculated the kinetic energy associated with a rotating body pointed out the similarities with translations What is missing: The 'force' of rotation, the torque

14

15

16 Note: Nm = J but J is only used for energies and Nm only for torques

17

18 Example Pivot. Point L/4 3L/4 F What is the angular acceleration? Use = /I need to get I and Moment of inertia: I = I com +M(L/4) 2 = ML 2 /12+ML 2 /16 I = 7ML 2 /48 Torque: = F L/4 (Note F is perpendicular to lever arm) = /I = (FL/ML 2 ) (48/(7*4)) = 12 * F/(7*M*L)

19 Example Pivot. Point L/4 3L/4 F 2 F 1 Assume a second force is now applied at the other end of the stick. What would be its magnitude and direction if it prevents the stick from rotating? No rotation: Net = 0 1 = F 1 L/4 = 2 = F 2 3L/4 Direction of the force will be the same (both down) Magnitude: F 2 =F 1 /3

20 Example. Notice: A uniform disk with mass M = 2.5kg and radius R=20cm is mounted on a horizontal axle. A block of mass m=1.2kg hangs from a massless cord that is wrapped around the rim of the disk. Q1: Find the acceleration of the falling block. We have two forces acting on mass m: Gravity and tension from the string We have one torque caused by the tension in the string acting on the disk The linear motion of the mass is linked to the circular motion of the disk via the cord.

21 Example. Notice: A uniform disk with mass M = 2.5kg and radius R=20cm is mounted on a horizontal axle. A block of mass m=1.2kg hangs from a massless cord that is wrapped around the rim of the disk. Q1: Find the acceleration of the falling block. We have two forces acting on mass m: Gravity and tension from the string ma = T-mg (equation of motion of the falling block) (Unknowns: a and T)

22 Example. Notice: A uniform disk with mass M = 2.5kg and radius R=20cm is mounted on a horizontal axle. A block of mass m=1.2kg hangs from a massless cord that is wrapped around the rim of the disk. Q1: Find the acceleration of the falling block. We have one torque caused by the tension in the string acting on the disk = -TR (force T perpendicular to lever arm negative, because of clockwise rotation) = I =0.5 MR 2 T = -0.5MR

23 Example A uniform disk with mass M = 2.5kg. and radius R=20cm is mounted on a horizontal axle. A block of mass m=1.2kg hangs from a massless cord that is wrapped around the rim of the disk. Q1: Find the acceleration of the falling block. Notice: ma = T-mg and T = -0.5MR The linear motion of the mass is linked to the circular motion of the disk via the cord. a = R 3 equations, 3 unknowns: T, a,

24 Example. A uniform disk with mass M = 2.5kg and radius R=20cm is mounted on a horizontal axle. A block of mass m=1.2kg hangs from a massless cord that is wrapped around the rim of the disk. ma = T-mg, T = -0.5MR a = R Some math: ma = -0.5Ma mg a = -2mg/(M+2m) = -4.8m/s 2 Angular accel.: = a/r = -4.8/0.2rad/s 2 = -24rad/s 2 Tension in the cord: T = -0.5MR = 6.0N

25 Check out Table 10.3 for analogy with translation

26 Work and kinetic Energy y F r F t Rod m F r x The radial component of F does not do any work as it does not displace the mass m The tangential component of F does displace the mass by ds and does work: dw = F t ds = F t rd = d

27 Work and kinetic Energy w f W= dw = d = f i Work done w i for constant torque during displacement Recall Work for constant force and translation: w W= dw = Fdx = F(x f -x i ) x i x f Power: P=dW/dt = (rotation about fixed axis)

28 Work and kinetic Energy y F r F t Rod m F r x The tangential component of F does increase the velocity of the mass: K f -K i = 0.5mv 2 f -0.5mv 2 i =W use v= r K f -K i = 0.5mr 2 f2-0.5mr 2 i 2 K f -K i = 0.5I f 2-0.5I i 2 =W

29 Work and kinetic Energy y F t F r Rod m F r x We just derived the relation between work done by a net torque and the change in kinetic energy for a single particle. If we assume that the particle is part of a solid body then we would have to repeat this for all particles in the body. Adding up all these changes in kinetic energy and the work would show that this also works for a solid body.

30 Work and kinetic Energy Problem 66. M=4.5kg r = 0.05m I = 0.003kgm 2 Q: What is the velocity of mass m after it dropped a distance h? (No friction) h m=0.6kg mg

31 Work and kinetic Energy Problem 66. M=4.5kg r = 0.05m I = 0.003kgm 2 Q: What is the velocity of mass m after it dropped a distance h? (No friction) Two possible approaches: a) Use forces (gravity and tension) and torques b) Energy conservation h m=0.6kg mg

32 Problem 66. Work and kinetic Energy spherical shell of M=4.5kg r = 0.05m I = 0.003kgm 2 b) Energy conservation As we are only interested in the situation at the end and energy is conserved, this should work. h m=0.6kg mg Initial: U=mgh, Final: K shell + K pulley + K mass

33 Work and kinetic Energy Problem 66. spherical shell of M=4.5kg r = 0.05m I = 0.003kgm 2 Initial: U=mgh, Final: K shell =0.5I S S 2 =0.5(2MR 2 /3) S 2 K pulley = 0.5I p P 2 K mass = 0.5mv 2 h m=0.6kg mg Also know that the 'length of the string is conserved': v= P r, v= S R Use this to replace the 's in the kinetic energies mgh = Mv 2 / I p v 2 /r mv 2 Solve for v = 1.4m/s

Physics-1 Recitation-7

Physics-1 Recitation-7 Rotation of a Rigid Object About a Fixed Axis 1. The angular position of a point on a wheel is described by. a) Determine angular position, angular speed, and angular acceleration

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

Rotational Kinematics and Dynamics

Rotational Kinematics and Dynamics Name : Date : Level : Physics I Teacher : Kim Angular Displacement, Velocity, and Acceleration Review - A rigid object rotating about a fixed axis through O perpendicular

Linear and Rotational Kinematics

Linear and Rotational Kinematics Starting from rest, a disk takes 10 revolutions to reach an angular velocity. If the angular acceleration is constant throughout, how many additional revolutions are required

Rotational Dynamics. Luis Anchordoqui

Rotational Dynamics Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is r. All points on a straight line

Phys101 Third Major Exam Term 142

Phys0 Third Major Exam Term 4 Q. The angular position of a point on the rim of a rotating wheel of radius R is given by: θ (t) = 6.0 t + 3.0 t.0 t 3, where θ is in radians and t is in seconds. What is

Solution Derivations for Capa #10

Solution Derivations for Capa #10 1) The flywheel of a steam engine runs with a constant angular speed of 172 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel

PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013

PHYSICS 111 HOMEWORK SOLUTION #10 April 10, 013 0.1 Given M = 4 i + j 3 k and N = i j 5 k, calculate the vector product M N. By simply following the rules of the cross product: i i = j j = k k = 0 i j

Rotation, Rolling, Torque, Angular Momentum

Halliday, Resnick & Walker Chapter 10 & 11 Rotation, Rolling, Torque, Angular Momentum Physics 1A PHYS1121 Professor Michael Burton Rotation 10-1 Rotational Variables! The motion of rotation! The same

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

physics 111N rotational motion

physics 111N rotational motion rotations of a rigid body! suppose we have a body which rotates about some axis! we can define its orientation at any moment by an angle, θ (any point P will do) θ P physics

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

Chapt. 10 Rotation of a Rigid Object About a Fixed Axis

Chapt. otation of a igid Object About a Fixed Axis. Angular Position, Velocity, and Acceleration Translation: motion along a straight line (circular motion?) otation: surround itself, spins rigid body:

Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

1. A radian is about: A. 25 ± B. 37 ± C. 45 ± D. 57 ± E. 90 ± ans: D Section: 10{2; Di±culty: E

Chapter 10: ROTATION 1 A radian is about: A 25 ± B 37 ± C 45 ± D 57 ± E 90 ± Section: 10{2; Di±culty: E 2 One revolution is the same as: A 1 rad B 57 rad C ¼=2rad D ¼ rad E 2¼ rad Section: 10{2; Di±culty:

9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration

Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear

Chapter 9 Rotation of Rigid Bodies

Chapter 9 Rotation of Rigid Bodies 1 Angular Velocity and Acceleration θ = s r (angular displacement) The natural units of θ is radians. Angular Velocity 1 rad = 360o 2π = 57.3o Usually we pick the z-axis

Angular velocity. Angular velocity measures how quickly the object is rotating. Average angular velocity. Instantaneous angular velocity

Angular velocity Angular velocity measures how quickly the object is rotating. Average angular velocity Instantaneous angular velocity Two coins rotate on a turntable. Coin B is twice as far from the axis

Angular acceleration α

Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

Rotation, Angular Momentum

This test covers rotational motion, rotational kinematics, rotational energy, moments of inertia, torque, cross-products, angular momentum and conservation of angular momentum, with some problems requiring

Center of Gravity. We touched on this briefly in chapter 7! x 2

Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.

Lab 7: Rotational Motion

Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125

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

Linear Centripetal Tangential speed acceleration acceleration A) Rω Rω 2 Rα B) Rω Rα Rω 2 C) Rω 2 Rα Rω D) Rω Rω 2 Rω E) Rω 2 Rα Rω 2 Ans: A

1. Two points, A and B, are on a disk that rotates about an axis. Point A is closer to the axis than point B. Which of the following is not true? A) Point B has the greater speed. B) Point A has the lesser

Chapter 8- Rotational Motion

Chapter 8- Rotational Motion Textbook (Giancoli, 6 th edition): Assignment 9 Due on Thursday, November 26. 1. On page 131 of Giancoli, problem 18. 2. On page 220 of Giancoli, problem 24. 3. On page 221

5.2 Rotational Kinematics, Moment of Inertia

5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

Physics 2101 S t ec i tti on 3

Physics 101 Section 3 March 5 rd : Ch. 10 Announcements: Grades for Exam # posted next week. Today Ch. 10.7-10, 10, 11.-3 Next Quiz is March 1 on SHW#6 Class Website: http://www.phys.lsu.edu/classes/spring010/phys101-3/

Physics 201 Homework 8

Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A lawn roller in the form of a uniform solid cylinder is being pulled horizontally by a horizontal

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics of a rigid body undergoing rotational motion. In-Class Activities: Applications

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics In-Class Activities: of a rigid body undergoing rotational motion. Check Homework

Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

Angular Momentum Problems Challenge Problems

Angular Momentum Problems Challenge Problems Problem 1: Toy Locomotive A toy locomotive of mass m L runs on a horizontal circular track of radius R and total mass m T. The track forms the rim of an otherwise

Chapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.

Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed

Linear Motion vs. Rotational Motion

Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a

ROLLING, TORQUE, AND ANGULAR MOMENTUM

Chapter 11: ROLLING, TORQUE, AND ANGULAR MOMENTUM 1 A wheel rolls without sliding along a horizontal road as shown The velocity of the center of the wheel is represented by! Point P is painted on the rim

Ch.8 Rotational Equilibrium and Rotational Dynamics.

Ch.8 Rotational Equilibrium and Rotational Dynamics. Conceptual question # 1, 2, 9 Problems# 1, 3, 7, 9, 11, 13, 17, 19, 25, 28, 33, 39, 43, 47, 51, 55, 61, 79 83 Torque Force causes acceleration. Torque

AP PHYSICS C: MECHANICS ROTATION REVIEW

AP PHYSICS C: MECHANICS ROTATION REVIEW 1. Two points, A and B, are on a disk that rotates about an axis. Point A is closer to the axis than point B. Which of the following is not true? A) Point B has

Physics 11 Fall 2012 Practice Problems 6 - Solutions

Physics 11 Fall 01 Practice Problems 6 - s 1. Two points are on a disk that is turning about a fixed axis perpendicular to the disk and through its center at increasing angular velocity. One point is on

PHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013

PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.

9 ROTATIONAL DYNAMICS

CHAPTER 9 ROTATIONAL DYNAMICS CONCEPTUAL QUESTIONS 1. REASONING AND SOLUTION The magnitude of the torque produced by a force F is given by τ = Fl, where l is the lever arm. When a long pipe is slipped

Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

s r or equivalently sr linear velocity vr Rotation its description and what causes it? Consider a disk rotating at constant angular velocity.

Rotation its description and what causes it? Consider a disk rotating at constant angular velocity. Rotation involves turning. Turning implies change of angle. Turning is about an axis of rotation. All

Physics 271 FINAL EXAM-SOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath

Physics 271 FINAL EXAM-SOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath 1. The exam will last from 8:00 am to 11:00 am. Use a # 2 pencil to make entries on the answer sheet. Enter the following id information

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

PHYS 211 FINAL FALL 2004 Form A

1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

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

HW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian

HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along

ANGULAR POSITION. 4. Rotational Kinematics and Dynamics

ANGULAR POSITION To describe rotational motion, we define angular quantities that are analogous to linear quantities Consider a bicycle wheel that is free to rotate about its axle The axle is the axis

Physics 211 Week 12. Simple Harmonic Motion: Equation of Motion

Physics 11 Week 1 Simple Harmonic Motion: Equation of Motion A mass M rests on a frictionless table and is connected to a spring of spring constant k. The other end of the spring is fixed to a vertical

Physics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd

Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle

Rotational Motion. Symbol Units Symbol Units Position x (m) θ (rad) (m/s) " = d# Source of Parameter Symbol Units Parameter Symbol Units

Introduction Rotational Motion There are many similarities between straight-line motion (translation) in one dimension and angular motion (rotation) of a rigid object that is spinning around some rotation

CHAPTER 15 FORCE, MASS AND ACCELERATION

CHAPTER 5 FORCE, MASS AND ACCELERATION EXERCISE 83, Page 9. A car initially at rest accelerates uniformly to a speed of 55 km/h in 4 s. Determine the accelerating force required if the mass of the car

Rotational Inertia Demonstrator

WWW.ARBORSCI.COM Rotational Inertia Demonstrator P3-3545 BACKGROUND: The Rotational Inertia Demonstrator provides an engaging way to investigate many of the principles of angular motion and is intended

E X P E R I M E N T 8

E X P E R I M E N T 8 Torque, Equilibrium & Center of Gravity Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics, Exp 8:

Chapter 9 Rotational Motion

In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypothesis that may be imagined,

Name: Class: Date: Exam 4--PHYS 101--F14 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel, initially at rest, rotates with a constant acceleration

Practice Exam Three Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,

Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

Version PREVIEW Practice 8 carroll (11108) 1

Version PREVIEW Practice 8 carroll 11108 1 This print-out should have 12 questions. Multiple-choice questions may continue on the net column or page find all choices before answering. Inertia of Solids

Rotational Mechanics - 1

Rotational Mechanics - 1 The Radian The radian is a unit of angular measure. The radian can be defined as the arc length s along a circle divided by the radius r. s r Comparing degrees and radians 360

EXPERIMENT: MOMENT OF INERTIA

OBJECTIVES EXPERIMENT: MOMENT OF INERTIA to familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body as mass plays in

Find the angle through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocity.

HW #9: Chapter 7 Rotational Motion Due: 8:59pm on Tuesday, March 22, 2016 To understand how points are awarded, read the Grading Policy for this assignment. Flywheel Kinematics A heavy flywheel is accelerated

Chapter 8. Rotational Motion. 8.1 Purpose. 8.2 Introduction. s r 2π (rad) = 360 o. r θ

Chapter 8 Rotational Motion 8.1 Purpose In this experiment, rotational motion will be examined. Angular kinematic variables, angular momentum, Newton s 2 nd law for rotational motion, torque, and moments

Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

Physics-1 Recitation-3

Physics-1 Recitation-3 The Laws of Motion 1) The displacement of a 2 kg particle is given by x = At 3/2. In here, A is 6.0 m/s 3/2. Find the net force acting on the particle. (Note that the force is time

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6

Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.

Torque and Rotary Motion

Torque and Rotary Motion Name Partner Introduction Motion in a circle is a straight-forward extension of linear motion. According to the textbook, all you have to do is replace displacement, velocity,

Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

Physics 41 HW Set 1 Chapter 15

Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

Circular Motion. Physics 1425 Lecture 18. Michael Fowler, UVa

Circular Motion Physics 1425 Lecture 18 Michael Fowler, UVa How Far is it Around a Circle? A regular hexagon (6 sides) can be made by putting together 6 equilateral triangles (all sides equal). The radius

PHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true?

1. Which one of the following statements concerning kinetic energy is true? A) Kinetic energy can be measured in watts. B) Kinetic energy is always equal to the potential energy. C) Kinetic energy is always

Angular Velocity. Midterm on Thursday at 7:30pm! Old exams available on website. Chapters 6 9 are covered. Go to same room as last time.

Angular Velocity Announcements: Midterm on Thursday at 7:30pm! Old exams available on website. Chapters 6 9 are covered. Go to same room as last time. You are allowed one calculator and one doublesided

Acceleration due to Gravity

Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision

3600 s 1 h. 24 h 1 day. 1 day

Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

Solution Derivations for Capa #11

Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform

A B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

1 Dot Product and Cross Products For two vectors, the dot product is a number A B = AB cos(θ) = A B = AB (1) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

VIII. Magnetic Fields - Worked Examples

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.0 Spring 003 VIII. Magnetic Fields - Worked Examples Example : Rolling rod A rod with a mass m and a radius R is mounted on two parallel rails

Physics 2101, First Exam, Fall 2007

Physics 2101, First Exam, Fall 2007 September 4, 2007 Please turn OFF your cell phone and MP3 player! Write down your name and section number in the scantron form. Make sure to mark your answers in the

Rotational Motion: Moment of Inertia

Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body

Chapter 12 Rotational Motion

Chapter 12 Rotational Motion 1) 1 radian = angle subtended by an arc (l) whose length is equal to the radius (r) A bird can only see objects that subtend an angle of 3 X 10-4 rad. How many degrees is that?

RIGID BODY MOTION: TRANSLATION & ROTATION (Sections ) Today s Objectives :

RIGID BODY MOTION: TRANSLATION & ROTATION (Sections 16.1-16.3) Today s Objectives : Students will be able to analyze the kinematics of a rigid body undergoing planar translation or rotation about a fixed

9.1 Angular Velocity and Acceleration. Angular quantities. Angular position is most conveniently describe in terms of radians defined by

Chapter 9 Rotation of rigid bodies 9.1 Angular Velocity and Acceleration Angular quantities. Angular position is most conveniently describe in terms of radians defined by π rad 360 = 1 rev (9.1) or 1rad

226 Chapter 15: OSCILLATIONS

Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion

2. A grindstone spinning at the rate of 8.3 rev/s has what approximate angular speed? a. 3.2 rad/s c. 52 rad/s b. 26 rad/s d.

Rotational Motion 1. 2 600 rev/min is equivalent to which of the following? a. 2600 rad/s c. 273 rad/s b. 43.3 rad/s d. 60 rad/s 2. A grindstone spinning at the rate of 8.3 rev/s has what approximate angular

AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2001 APPLIED MATHEMATICS HIGHER LEVEL

M3 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 00 APPLIED MATHEMATICS HIGHER LEVEL FRIDAY, JUNE AFTERNOON,.00 to 4.30 Six questions to be answered. All questions carry equal marks.

Chapter 7 - Rotational Motion w./ QuickCheck Questions

Chapter 7 - Rotational Motion w./ QuickCheck Questions 2015 Pearson Education, Inc. Anastasia Ierides Department of Physics and Astronomy University of New Mexico October 1, 2015 Review of Last Time Uniform

Page 1. Klyde. V Klyde. ConcepTest 9.3a Angular Displacement I. ConcepTest 9.2 Truck Speedometer

ConcepTest 9.a Bonnie and Klyde I Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.

Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

Forces. -using a consistent system of units, such as the metric system, we can define force as:

Forces Force: -physical property which causes masses to accelerate (change of speed or direction) -a push or pull -vector possessing both a magnitude and a direction and adds according to the Parallelogram

Chapter 7 Homework solutions

Chapter 7 Homework solutions 8 Strategy Use the component form of the definition of center of mass Solution Find the location of the center of mass Find x and y ma xa + mbxb (50 g)(0) + (10 g)(5 cm) x