Connection between angular and linear speed

Size: px
Start display at page:

Download "Connection between angular and linear speed"

Transcription

1 Connection between angular and linear speed If a point-like object is in motion on a circular path of radius R at an instantaneous speed v, then its instantaneous angular speed ω is v = ω R Example: A satellite is in orbit around the Earth at a height h = 800 km over the equator. If it remains above the same point all the time, what are its speed and centripetal acceleration? Earth spins about its axis at an angular speed ω = π 4 hrs = π s = rad/s Top View In order to stay above the same point all the time, satellite must travel at the same angular speed. The radius of the satellite s circular orbit is R = ( ) km Thus, the speed of the satellite is v = ω R = ( )( ) m/s = m/s The centripetal acceleration of the satellite is ac = = ω R = ( ) ( ) m/s =.8 m/s v R

2 Angular and tangential acceleration For a point-like object is in motion on a circular path of constant radius R, the angular acceleration α corresponds to a change of angular speed over a time Δt. α = Δω Δt Correspondingly, the speed of the object moving on the circular path of constant radius R changes by Δv = R Δω, i.e., there is a tangential acceleration at = Δv Δt =R Δω Δt = α R Example: A compact disk, which has a diameter D = cm, speeds up uniformly from 0.00 to 4.00 rev/s in 3.00 s. What is the tangential acceleration of a point on the disk, located halfway between the centre and the rim of the disk? D One revolution corresponds to an angle of π rads Thus, after 3s the angular speed of the disk is (4 π) rad/s Top View The angular acceleration of the disk is α = Δω Δt = 8π 3 rad/s The point of interest on the disk is at a distance D/4 from the centre Thus, the tangential acceleration of that point of the disk is at = α D/4 = 8π 0. m/s 3 4 = 0.5 m/s

3 Kinetic Energy of a rigid body Definition: The kinetic energy of a many-body system of total mass M, comprising N point-like objects of masses m, m,... mn, equals the sum of two distinct contributions K = Mv + Σi mi ui v is the velocity of the center of mass of the system ui = vi v relative velocity of the ith point with respect to the center of mass The first contribution is that of a point-like object with the mass of the whole system, moving at a velocity equal to that of the center of mass of the system. It describes the translational motion of the system as a whole, with respect to an inertial observer The second contribution is the sum of the kinetic energy of each and every a point in the system, moving as seen from an observer sitting at the center of mass Special case: Rigid Body For a rigid body, the above expression is very useful and takes on a simple form Example: hockey puck of mass M slides on ice... ω v K = Mv + I ω while spinning about axis through centre rotational kinetic energy I is the moment of inertia of the puck The two motions are generally unrelated

4 Rotation of rigid bodies Example: A dumbbell consists of two spheres of mass m =.3 kg each, connected by a rod of negligible mass. This object can spin on a plane about an axis through its centre. The two spheres are held at a distance L =. m. Side View Top View L/ =. m v The dumbbell is set on rotation at an angular speed of 6π rad/s. What are the speed and kinetic energy of each sphere? Angular speed ω = 6π rad/s each sphere does a full circle (an angle of π rads ) 3 times in s. In turn, this means that each sphere travels a distance 3 π (L/) = 3πL in s. Thus, v = 3π (.) m/s = ωl/ = 0.7 m/s The kinetic energy of each sphere is mv = mω (L /4) = mω L 8 = [.3 (36 π )(.) /8] J = 79 J The kinetic energy of the whole dumbbell is therefore I = ml moment of inertia of the rigid dumbbell mω 8 L = ( ml ) ω = Iω

5 Moment of Inertia Definition: The rotational kinetic energy of a rigid body spinning about a fixed axis going through its center of mass is generally expressed as y Krot = Iω r x I = Σi mi (xi + yi ) y = m (x + y ) + m (x + y ) mn (xn + yn ) x moment of inertia of the body with respect to specific axis rn Top View The moment of inertia is a scalar quantity, with dimensions [M][L ]. Its calculation is simple only in few cases, otherwise it requires calculus. Luckily, there are tables of moments of inertia for important objects. Important: The moment of inertia depends on the axis with respect to which it is computed. Example: Rigid dumbbell with connecting rod of negligible mass Case : axis through centre Case : axis through one end m m L I = ml /4 + ml /4 = ml / I = m 0 + ml = ml

6 Moment of Inertia (cont d) Example: What is the moment of inertia of a bicycle wheel with respect to an axis going through its centre? Assume that the wheel has a radius R = 30cm, a mass M =.75 kg (including the tire) and its spokes have negligible mass. Δm R Top View each element of mass Δm contributes to the moment of inertia the same amount, i.e., ΔmR Hence, Ih = MR = (.75)(0.3) kg m = 0.58 kg m If the wheel is spinning about its axis with a kinetic energy Krot = 4 J, what is the angular speed of the wheel? Krot = Since Iω it is ω = K rot = 8 Ih 0.58 rad /s = 59 rad /s ω = 7 rad/s Question: If a disk of equal mass and radius had the same rotational kinetic energy, what would be its angular speed? R The moment of inertia for a disk going through the centre, is Id = MR (one half of that of the hoop) Thus, Id = kg m ω = 0 rad/s and ω = K rot Id = rad /s = 0405 rad /s lower I greater ω for the same K

7 Moment of Inertia (cont d) Hoop or cylindrical shell I = MR Disk (solid cylinder) I = MR Disk (axis at rim) I = 3MR Thin rod (axis through midpoint) I = ML / Thin rod (axis through one end) I = ML /3 Hollow sphere I = MR 3 Solid sphere I = MR 5 Solid sphere (axis at rim) I = 7 MR 5 Solid plate (axis through center, in plane of plate) I =ML / Solid plate (axis perpendicular to plane of plate) I =M(L + W )/ For equal mass and characteristic size, hollow objects have higher inertia, because more of the mass is further away from the axis moments of

8 Rolling Motion In the general case, rotational and translational motions of a rigid body are disconnected and independent of one another. An important exception is constituted by rolling motion, in which translational and rotational motion are related. Rolling is made possible by the particular shape (spherical or circular) of some objects Smooth wedge If the wedge is perfectly smooth, the box slides down A ball, a cylinder, any other object would slide in the same way. Shape is immaterial on a smooth surface Rough wedge If wedge is rough (μs > tan θ), box remains at rest. A cylinder or a sphere, on the other hand, comes down rolling (without sliding). θ Rolling motion requires a rough surface. No rolling is possible on a smooth surface. Friction prevents sliding on the one hand, causing rolling on the other.

9 Rolling Motion (cont d) πr A rolling object is instantaneously spinning about an axis going through its centre, perpendicular to the direction of motion Concurrently, the object is moving forward as a result of its spin If the center of mass of the system advances by πr in a time T, then the speed of the center of mass is πr vcm = vcm = ωr rolling condition T Kinetic energy of rolling object K = Mvcm + I ω = I vcm Mvcm + R = I Mvcm ( + MR ) Rolling motion conserves mechanical energy. It must always be remembered, however, that part of the kinetic energy is going to be rotational.

10 Example: A.0-kg disk with a radius r =0 cm rolls without slipping. If the linear speed of the disk is vcm =.4 m/s, find the translational, rotational and total kinetic energy of the disk Translational: Kt = mvcm = (0.5)(.0)(.4) J =. J Total: Kt + Kr =.8 J I vcm Rotational: Kr = I ω = because r vcm = ω r for the disk, I = mr / Kr = mvcm i.e., = 0.6 J Question: What would change for a hoop with identical mass and radius? Translational kinetic energy: unchanged Rotational: Since for a hoop I = mr it is Kr = mv cm =. J same as Kt Total: Kt + Kr =.4 J Question: What would change for a solid sphere with identical mass and radius? Translational kinetic energy: unchanged Rotational: Since for a solid sphere I = mr /5 Kr = mvcm = 0.48 J 5 Total: Kt + Kr =.68 J Given two objects with the same mass and radius of rotation, more work is needed to impart a given translational speed to the one with the greater moment of inertia

11 Which object rolls down faster? Rolling objects with different moments of inertia with respect to the instantaneous axis of rotation, roll down a wedge in different times, even though mechanical energy is always conserved. Example: Solid sphere of mass M and radius R Potential energy at the top of the wedge: Mgh By the time sphere reaches the bottom, energy is entirely kinetic Since rolling sphere spins about axis through its centre h I = MR 5 Energy conservation: Mgh = yielding vcm = 0 7 gh at the bottom I Mvcm ( + MR ) = 7 0 Mvcm For a point-like object of mass M sliding frictionlessly, it would be gh Question: what if the sphere had identical mass and radius, but were hollow? For a hollow sphere, the moment of inertia with respect to the same axis is I = MR 3 Mvcm ( + 5 Energy conservation: Mgh = ) Mvcm 3 = 6 i.e., 6 vcm = gh at the bottom, solid sphere gets to the bottom sooner 5 Objects with a lower moment of inertia roll faster, for equal mass and radius

12 Non-ideal pulleys A pulley is a disk, spinning about an axis through its centre. A real pulley has a finite mass m, a radius r, and therefore a finite moment of inertia I = mr / m M M h Suppose two weights, one of mass M =. kg and the other of mass M, are initially suspended from a height h = 0 cm. The pulley is non-ideal, m = 0gr. The blocks are suddenly released. The heavy one falls while the lighter ons is lifted upward. We are interested in the speed of both blocks before the heavy one hits the floor. Energy conservation: 3Mgh is the initial potential energy Potential energy when block hits the floor: Mgh (lighter block is at height h) Kinetic energy when block hits the floor: (M + M)v + I ω = (M + M)v + v mr Angular speed of the pulley must be consistent with speed of blocks, i.e., v = ωr Hence, 3Mgh = Mgh ( 3M + + m 4 ) v (.)(9.8)(0.) J = ( ) kg v yielding v =. m /s or v =. m/s r

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

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

Phys 106 Practice Problems Common Quiz 1 Spring 2003

Phys 106 Practice Problems Common Quiz 1 Spring 2003 Phys 106 Practice Problems Common Quiz 1 Spring 2003 1. For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the rim to the speed

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

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 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

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

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

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

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

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

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

第 1 頁, 共 7 頁 Chap10 1. Test Bank, Question 3 One revolution per minute is about: 0.0524 rad/s 0.105 rad/s 0.95 rad/s 1.57 rad/s 6.28 rad/s 2. *Chapter 10, Problem 8 The angular acceleration of a wheel

More information

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc. Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity,

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

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

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

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

Mechanics II. Which of the following relations among the forces W, k, N, and F must be true?

Mechanics II. Which of the following relations among the forces W, k, N, and F must be true? Mechanics II 1. By applying a force F on a block, a person pulls a block along a rough surface at constant velocity v (see Figure below; directions, but not necessarily magnitudes, are indicated). Which

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

Gravitational potential energy

Gravitational potential energy Gravitational potential energ m1 Consider a rigid bod of arbitrar shape. We want to obtain a value for its gravitational potential energ. O r1 1 x The gravitational potential energ of an assembl of N point-like

More information

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

Name: Date: Period: AP Physics C Rotational Motion HO19

Name: Date: Period: AP Physics C Rotational Motion HO19 1.) A wheel turns with constant acceleration 0.450 rad/s 2. (9-9) Rotational Motion H19 How much time does it take to reach an angular velocity of 8.00 rad/s, starting from rest? Through how many revolutions

More information

1 Problems 1-3 A disc rotates about an axis through its center according to the relation θ (t) = t 4 /4 2t

1 Problems 1-3 A disc rotates about an axis through its center according to the relation θ (t) = t 4 /4 2t Slide 1 / 30 1 Problems 1-3 disc rotates about an axis through its center according to the relation θ (t) = t 4 /4 2t etermine the angular velocity of the disc at t= 2 s 2 rad/s 4 rad/s 6 rad/s 8 rad/s

More information

Slide 1 / 30. Slide 2 / 30. Slide 3 / m/s -1 m/s

Slide 1 / 30. Slide 2 / 30. Slide 3 / m/s -1 m/s 1 Problems 1-3 disc rotates about an axis through its center according to the relation θ (t) = t 4 /4 2t Slide 1 / 30 etermine the angular velocity of the disc at t= 2 s 2 rad/s 4 rad/s 6 rad/s 8 rad/s

More information

Rotational Motion About a Fixed Axis

Rotational Motion About a Fixed Axis Rotational Motion About a Fixed Axis Vocabulary rigid body axis of rotation radian average angular velocity instantaneous angular average angular Instantaneous angular frequency velocity acceleration acceleration

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

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

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

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

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

Suggested Problems. Chapter 1

Suggested Problems. Chapter 1 Suggested Problems Ch1: 49, 51, 86, 89, 93, 95, 96, 102. Ch2: 9, 18, 20, 44, 51, 74, 75, 93. Ch3: 4, 14, 46, 54, 56, 75, 91, 80, 82, 83. Ch4: 15, 59, 60, 62. Ch5: 14, 52, 54, 65, 67, 83, 87, 88, 91, 93,

More information

Rotation. PHYS 101 Previous Exam Problems CHAPTER

Rotation. PHYS 101 Previous Exam Problems CHAPTER PHYS 101 Previous Exam Problems CHAPTER 10 Rotation Rotational kinematics Rotational inertia (moment of inertia) Kinetic energy Torque Newton s 2 nd law Work, power & energy conservation 1. Assume that

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

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

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

More information

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc. Chapter 10 Rotational Kinematics and Energy 10-1 Angular Position, Velocity, and Acceleration 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions: 10-1 Angular Position, Velocity,

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

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

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

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

More information

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

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

UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics

UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics Physics 111.6 MIDTERM TEST #2 November 15, 2001 Time: 90 minutes NAME: STUDENT NO.: (Last) Please Print (Given) LECTURE SECTION

More information

Name (please print): UW ID# score last first

Name (please print): UW ID# score last first Name (please print): UW ID# score last first Question I. (20 pts) Projectile motion A ball of mass 0.3 kg is thrown at an angle of 30 o above the horizontal. Ignore air resistance. It hits the ground 100

More information

Angular Motion, General Notes

Angular Motion, General Notes Angular Motion, General Notes! When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same

More information

Physics of Rotation. Physics 109, Introduction To Physics Fall 2017

Physics of Rotation. Physics 109, Introduction To Physics Fall 2017 Physics of Rotation Physics 109, Introduction To Physics Fall 017 Outline Next two lab periods Rolling without slipping Angular Momentum Comparison with Translation New Rotational Terms Rotational and

More information

Physics 101 Lecture 12 Equilibrium and Angular Momentum

Physics 101 Lecture 12 Equilibrium and Angular Momentum Physics 101 Lecture 1 Equilibrium and Angular Momentum Ali ÖVGÜN EMU Physics Department www.aovgun.com Static Equilibrium q Equilibrium and static equilibrium q Static equilibrium conditions n Net external

More information

Do not fill out the information below until instructed to do so! Name: Signature: Student ID: Section Number:

Do not fill out the information below until instructed to do so! Name: Signature: Student ID:   Section Number: Do not fill out the information below until instructed to do so! Name: Signature: Student ID: E-mail: Section Number: Formulae are provided on the last page. You may NOT use any other formula sheet. You

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

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

Angular Displacement (θ)

Angular Displacement (θ) Rotational Motion Angular Displacement, Velocity, Acceleration Rotation w/constant angular acceleration Linear vs. Angular Kinematics Rotational Energy Parallel Axis Thm. Angular Displacement (θ) Angular

More information

Rotation review packet. Name:

Rotation review packet. Name: Rotation review packet. Name:. A pulley of mass m 1 =M and radius R is mounted on frictionless bearings about a fixed axis through O. A block of equal mass m =M, suspended by a cord wrapped around the

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

Description: Using conservation of energy, find the final velocity of a "yo yo" as it unwinds under the influence of gravity.

Description: Using conservation of energy, find the final velocity of a yo yo as it unwinds under the influence of gravity. Chapter 10 [ Edit ] Overview Summary View Diagnostics View Print View with Answers Chapter 10 Due: 11:59pm on Sunday, November 6, 2016 To understand how points are awarded, read the Grading Policy for

More information

Slide 1 / 133. Slide 2 / 133. Slide 3 / How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m?

Slide 1 / 133. Slide 2 / 133. Slide 3 / How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? 1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 1 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 2 / 133 3 A ball rotates

More information

Slide 2 / 133. Slide 1 / 133. Slide 3 / 133. Slide 4 / 133. Slide 5 / 133. Slide 6 / 133

Slide 2 / 133. Slide 1 / 133. Slide 3 / 133. Slide 4 / 133. Slide 5 / 133. Slide 6 / 133 Slide 1 / 133 1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 2 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 3 / 133

More information

PHYS 111 HOMEWORK #11

PHYS 111 HOMEWORK #11 PHYS 111 HOMEWORK #11 Due date: You have a choice here. You can submit this assignment on Tuesday, December and receive a 0 % bonus, or you can submit this for normal credit on Thursday, 4 December. If

More information

Rotational Kinematics

Rotational Kinematics Rotational Kinematics Rotational Coordinates Ridged objects require six numbers to describe their position and orientation: 3 coordinates 3 axes of rotation Rotational Coordinates Use an angle θ to describe

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

Physics 6A Winter 2006 FINAL

Physics 6A Winter 2006 FINAL Physics 6A Winter 2006 FINAL The test has 16 multiple choice questions and 3 problems. Scoring: Question 1-16 Problem 1 Problem 2 Problem 3 55 points total 20 points 15 points 10 points Enter the solution

More information

1.1. Rotational Kinematics Description Of Motion Of A Rotating Body

1.1. Rotational Kinematics Description Of Motion Of A Rotating Body PHY 19- PHYSICS III 1. Moment Of Inertia 1.1. Rotational Kinematics Description Of Motion Of A Rotating Body 1.1.1. Linear Kinematics Consider the case of linear kinematics; it concerns the description

More information

On my honor, I have neither given nor received unauthorized aid on this examination.

On my honor, I have neither given nor received unauthorized aid on this examination. Instructor(s): Profs. D. Reitze, H. Chan PHYSICS DEPARTMENT PHY 2053 Exam 2 April 2, 2009 Name (print, last first): Signature: On my honor, I have neither given nor received unauthorized aid on this examination.

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Common Quiz Mistakes / Practice for Final Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A ball is thrown directly upward and experiences

More information

Moment of Inertia Race

Moment of Inertia Race Review Two points, A and B, are on a disk that rotates with a uniform speed about an axis. Point A is closer to the axis than point B. Which of the following is NOT true? 1. Point B has the greater tangential

More information

Exercise Torque Magnitude Ranking Task. Part A

Exercise Torque Magnitude Ranking Task. Part A Exercise 10.2 Calculate the net torque about point O for the two forces applied as in the figure. The rod and both forces are in the plane of the page. Take positive torques to be counterclockwise. τ 28.0

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

Physics I (Navitas) FINAL EXAM Fall 2015

Physics I (Navitas) FINAL EXAM Fall 2015 95.141 Physics I (Navitas) FINAL EXAM Fall 2015 Name, Last Name First Name Student Identification Number: Write your name at the top of each page in the space provided. Answer all questions, beginning

More information

PHYSICS 221 SPRING EXAM 2: March 30, 2017; 8:15pm 10:15pm

PHYSICS 221 SPRING EXAM 2: March 30, 2017; 8:15pm 10:15pm PHYSICS 221 SPRING 2017 EXAM 2: March 30, 2017; 8:15pm 10:15pm Name (printed): Recitation Instructor: Section # Student ID# INSTRUCTIONS: This exam contains 25 multiple-choice questions plus 2 extra credit

More information

Exam 3 Practice Solutions

Exam 3 Practice Solutions Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at

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

Review. Checkpoint 2 / Lecture 13. Strike (Day 8)

Review. Checkpoint 2 / Lecture 13. Strike (Day 8) Physics 101: Lecture 14 Parallel Axis Theorem, Rotational Energy, Conservation of Energy Examples, and a Little Torque Review Rotational Kinetic Energy K rot = ½ I w 2 Rotational Inertia I = S m i r i2

More information

Relating Translational and Rotational Variables

Relating Translational and Rotational Variables Relating Translational and Rotational Variables Rotational position and distance moved s = θ r (only radian units) Rotational and translational speed d s v = dt v = ω r = ds dt = d θ dt r Relating period

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

Chapter 10 Practice Test

Chapter 10 Practice Test Chapter 10 Practice Test 1. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration of 0.40 rad/s 2 has an angular velocity of 1.5 rad/s and an angular position of 2.3 rad. What

More information

PHYSICS 221 SPRING 2014

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

More information

Physics 201 Midterm Exam 3

Physics 201 Midterm Exam 3 Name: Date: _ 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

More information

Chapter 8. Rotational Motion

Chapter 8. Rotational Motion Chapter 8 Rotational Motion 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 Rotational Work and Energy

More information

Study Questions/Problems Week 7

Study Questions/Problems Week 7 Study Questions/Problems Week 7 Chapters 10 introduces the motion of extended bodies, necessitating a description of rotation---something a point mass can t do. This chapter covers many aspects of rotation;

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 Mechanics Part III Dynamics. Pre AP Physics

Rotational Mechanics Part III Dynamics. Pre AP Physics Rotational Mechanics Part III Dynamics Pre AP Physics We have so far discussed rotational kinematics the description of rotational motion in terms of angle, angular velocity and angular acceleration and

More information

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

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

More information

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

Physics. TOPIC : Rotational motion. 1. A shell (at rest) explodes in to smalll fragment. The C.M. of mass of fragment will move with:

Physics. TOPIC : Rotational motion. 1. A shell (at rest) explodes in to smalll fragment. The C.M. of mass of fragment will move with: TOPIC : Rotational motion Date : Marks : 120 mks Time : ½ hr 1. A shell (at rest) explodes in to smalll fragment. The C.M. of mass of fragment will move with: a) zero velocity b) constantt velocity c)

More information

Quiz Number 4 PHYSICS April 17, 2009

Quiz Number 4 PHYSICS April 17, 2009 Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given

More information

Phys101 Lectures 19, 20 Rotational Motion

Phys101 Lectures 19, 20 Rotational Motion Phys101 Lectures 19, 20 Rotational Motion Key points: Angular and Linear Quantities Rotational Dynamics; Torque and Moment of Inertia Rotational Kinetic Energy Ref: 10-1,2,3,4,5,6,8,9. Page 1 Angular Quantities

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

Work and kinetic Energy

Work and kinetic Energy 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 Work and kinetic Energy Problem 66. M=4.5kg

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Two men, Joel and Jerry, push against a wall. Jerry stops after 10 min, while Joel is

More information

AP Physics 1: Rotational Motion & Dynamics: Problem Set

AP Physics 1: Rotational Motion & Dynamics: Problem Set AP Physics 1: Rotational Motion & Dynamics: Problem Set I. Axis of Rotation and Angular Properties 1. How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? 2. How many degrees are

More information

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

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

More information

Advanced Higher Physics. Rotational Motion

Advanced Higher Physics. Rotational Motion Wallace Hall Academy Physics Department Advanced Higher Physics Rotational Motion Solutions AH Physics: Rotational Motion Problems Solutions Page 1 013 TUTORIAL 1.0 Equations of motion 1. (a) v = ds, ds

More information

Rotational kinematics

Rotational kinematics Rotational kinematics Suppose you cut a circle out of a piece of paper and then several pieces of string which are just as long as the radius of the paper circle. If you then begin to lay these pieces

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

Use the following to answer question 1:

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

More information

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

DYNAMICS OF RIGID BODIES

DYNAMICS OF RIGID BODIES DYNAMICS OF RIGID BODIES Measuring angles in radian Define the value of an angle θ in radian as θ = s r, or arc length s = rθ a pure number, without dimension independent of radius r of the circle one

More information

PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)

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

More information

Advanced Higher Physics. Rotational motion

Advanced Higher Physics. Rotational motion Wallace Hall Academy Physics Department Advanced Higher Physics Rotational motion Problems AH Physics: Rotational Motion 1 2013 Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration

More information

TOPIC D: ROTATION EXAMPLES SPRING 2018

TOPIC D: ROTATION EXAMPLES SPRING 2018 TOPIC D: ROTATION EXAMPLES SPRING 018 Q1. A car accelerates uniformly from rest to 80 km hr 1 in 6 s. The wheels have a radius of 30 cm. What is the angular acceleration of the wheels? Q. The University

More information

Chapter 9 [ Edit ] Ladybugs on a Rotating Disk. v = ωr, where r is the distance between the object and the axis of rotation. Chapter 9. Part A.

Chapter 9 [ Edit ] Ladybugs on a Rotating Disk. v = ωr, where r is the distance between the object and the axis of rotation. Chapter 9. Part A. Chapter 9 [ Edit ] Chapter 9 Overview Summary View Diagnostics View Print View with Answers Due: 11:59pm on Sunday, October 30, 2016 To understand how points are awarded, read the Grading Policy for this

More information

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 12 Lecture RANDALL D. KNIGHT Chapter 12 Rotation of a Rigid Body IN THIS CHAPTER, you will learn to understand and apply the physics

More information