Rolling, Torque, Angular Momentum


 Kathryn Bishop
 1 years ago
 Views:
Transcription
1 Chapter 11 Rolling, Torque, Angular Momentum Copyright
2 11.2 Rolling as Translational and Rotation Combined Motion of Translation : i.e.motion along a straight line Motion of Rotation : rotation about a fixed axis Pure Translation Motion + Pure Rotation Motion = ROLLING MOTION Translation of COM Rotation around COM com Smooth Rolling of a Disk Although the center of the object moves in a straight line parallel to the surface, a point on the rim certainly does not.
3 11.2 Rolling as Translational and Rotation Combined We consider only objects that roll smoothly (no slip) The center of mass (com) of the object moves in a straight line parallel to the surface The object rotates around the com as it moves The rotational motion is defined by: Wheel moves through the arc length of s Wheel s COM moves through a translational motion PATHS ARE EQUAL s= R
4 11.2 Rolling as Translational and Rotation Combined The figure shows how the velocities of translation and rotation combine at different points on the wheel B D Every point on the wheel rotates about com with angular speed of Everypoint outmost part of the wheel has linear speed of ν; Where ν = R = ν com All points on the wheel move to the right with same linear velocity, ν com At the bottom of the wheel (point P), the portion of the wheel is stationary The portion at the top (point T) moving at a speed 2ν com At points B and D, the speeds are smaller than the point T.
5 11.3 The Kinetic Energy of Rolling Rolling can be considered as pure rotation around contact point P. (V p =0) com R (V p = 0) Rotational inertia about COM: I com Rotational inertia about point P: I P = I com + MR 2 M : mass of the wheel, I com : rotational inertia about an axis through its center of mass R : the wheel s radius, at a perpendicular distance h). I P = I com + MR 2 K = ½ I com 2 + ½ M R 2 2 inserting v com (v com = R), then; (Using the parallelaxis theorem ) K.E of ROLLING: Rotational kinetic energy + Translational kinetic energy
6 11.4 The Forces of Rolling: Friction and Rolling A wheel rolls horizontally without sliding while accelerating with linear acceleration a com. A static frictional force f s acts on the wheel at P, opposing its tendency to slide. The magnitudes of the linear acceleration a com, and the angular acceleration a can be related by; R :the radius of the wheel. α : angular acceleration Rolling is possible when there is friction between the surface and the rolling object. The frictional force provides the torque to rotate the object.
7 11.4 The Forces of Rolling: Friction and Rolling If the wheel slides when the net force acts on it, the frictional force that acts at P in Figure is a kinetic frictional force, f k. The motion then is not smooth rolling, and the above relation (a COM =ar) does not apply to the motion. Sliding of Wheel Motion is not Smooth Rolling f k
8 11.4 The Forces of Rolling: Rolling Down a Ramp A round uniform body of radius R rolls (smoothly f s ) down a ramp. We want to find an expression for the body s acceleration a com,x down the ramp. We do this by using Newton s second law in its linear version (F net =Ma) and its angular version (t net =Ia). For smooth rolling down a ramp: 1. F g : The gravitational force is vertically down 2. F N : The normal force is perpendicular to the ramp 3. f s :The force of friction points up the slope (opposing the sliding) 4. a com =ar where a com is ( x) and a is (+, ccw) (see 4)
9 11.4 The Forces of Rolling: Rolling Down a Ramp We can use this equation to find the acceleration of such a body Note that the frictional force produces the rotation Without friction, the object will simply slide K i + U i = K f + U f Mgh = ½ I COM ω 2 + ½ Mv COM 2 v cm = ωr
10 11.4 The Forces of Rolling: Rolling Down a Ramp Sample problem: Rolling Down a Ramp A uniform ball, of mass M=6.00 kg and radius R, rolls smoothly from rest down a ramp at angle =30.0. a) The ball descends a vertical height h=1.20 m to reach the bottom of the ramp. What is its speed at the bottom? Calculations: Where I com is the ball s rotational inertia about an axis through its center of mass, v com is the requested speed at the bottom, and is the angular speed there. Substituting v com /R for, and 2/5 MR 2 for I com, b) What are the magnitude and direction of the frictional force on the ball as it rolls down the ramp? Calculations: First we need to determine the ball s acceleration a com,x : We can now solve for the frictional force:
11 11.6 Torque Revisited Previously, torque was defined only for a rotating body and a fixed axis. Now we redefine it for an individual particle that moves along any path relative to a fixed point. The path need not be a circle; torque is now a vector. Direction determined with righthandrule. Figure (a) A force F, lying in an xy plane, acts on a particle at point A. (b) This force produces a torque t = r x F on the particle with respect to the origin O. By the righthand rule for vector (cross) products, the torque vector points in the positive direction of z. Its magnitude is given by in (b) and by in (c).
12 11.6 Torque Revisited The general equation for torque is: We can also write the magnitude as: Or, using the perpendicular component of force or the moment arm of F:
13 11.6 Torque Revisited Sample problem: Torque on a particle due to a force In Figure, three forces, each magnitude 2.0 N, act on a particle. The particle is in the xz plane at a point A given by position vector r, where r = 3.0 m and = 30 o. Force F 1 is parallel to the x axis, force F 2 is parallel to the z axis, and F 3 is parallel to the y axis. What is the torque, about the origion O, due to each force? Calculations: Because we want the torques with respect to the origin O, the vector required for each cross product is the given position vector r. To determine the angle q between the direction of r and the direction of each force, we shift the force vectors of Fig.a, each in turn, so that their tails are at the origin. Figures b, c, and d, which are direct views of the xz plane, show the shifted force vectors F1, F2, and F3. respectively. In Fig. d, the angle between the directions of r and t is 90. Now, we find the magnitudes of the torques to be
14 11.6 Torque Revisited
15 11.7 Angular Momentum Here we investigate the angular counterpart to linear momentum We write: Note that the particle need not rotate around O to have angular momentum around it. The unit of angular momentum is kg m 2 /s, or J s To find the direction of angular momentum, use the righthand rule to relate r and v to the result To find the magnitude, use the equation for the magnitude of a cross product: Angular momentum has meaning only with respect to a specified origin Which can also be written as:
16 11.7 Angular Momentum Sample problem: Angular Momentum
17 11.8 Newton s 2 nd Law in Angular Form We rewrite Newton's second law as: Newton s 2 nd Law for translation PROOF: Newton s 2 nd Law for ANGULAR form The torque and the angular momentum must be defined with respect to the same point (usually the origin).
18 11.8 Newton s 2 nd Law in Angular Form Sample problem: Torque, Penguin Fall Calculations: The magnitude of l can be found by using The perpendicular distance between O and an extension of vector p is the given distance D. The speed of an object that has fallen from rest for a time t is v =gt. Therefore, To find the direction of we use the righthand rule for the vector product, and find that the direction is into the plane of the figure. The vector changes with time in magnitude only; its direction remains unchanged. (b) About the origin O, what is the torque on the penguin due to the gravitational force? Calculations: Using the righthand rule for the vector product we find that the direction of t is the negative direction of the z axis, the same as l.
19 11.9 The Angular Momentum of a System of Particles The total angular momentum L of the system is the (vector) sum of the angular momenta l of the individual particles (here with label i): With time, the angular momenta of individual particles may change because of interactions between the particles or with the outside. The rate of change of the net angular momentum is: Therefore, the net external torque acting on a system of particles is equal to the time rate of change of the system s total angular momentum L.
20 11.9 The Angular Momentum of a System of Particles Note that the torque and angular momentum must be measured relative to the same origin. If the center of mass is accelerating, then that origin must be the center of mass. a) A rigid body rotates about a z axis with angular speed. A mass element of mass Dm i within the body moves about the z axis in a circle with radius. The mass element has linear momentum p i and it is located relative to the origin O by position vector r i. Here the mass element is shown when is parallel to the x axis. b) The angular momentum l i, with respect to O, of the mass element in (a). The z component l iz is also shown.
21 11.9 The Angular Momentum of a System of Particles We can find the angular momentum of a rigid body through summation: The sum is the rotational inertia I of the body. Therefore this simplifies to:
22 11.11 Conservation of Angular Momentum Since we have a new version of Newton's second law, we also have a new conservation law: If the net external torque acting on a system is zero, the angular momentum L of the system remains constant, no matter what changes take place within the system. The law of conservation of angular momentum states that, for an isolated system, (net initial angular momentum) = (net final angular momentum) Depending on the torques acting on the system, the angular momentum of the system might be conserved in one or two directions but not in all directions. This means that if external torque along an axis is zero then L is constant.
23 11.11 Conservation of Angular Momentum If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system. (a) The student has a relatively large rotational inertia about the rotation axis and a relatively small angular speed. (b) By decreasing his rotational inertia, the student automatically increases his angular speed. The angular momentum of the rotating system remains unchanged.
24 11.11 Conservation of Angular Momentum Examples: Angular momentum conservation A student spinning on a stool: rotation speeds up when arms are brought in, slows down when arms are extended. A springboard diver: rotational speed is controlled by tucking her arms and legs in, which reduces rotational inertia and increases rotational speed. A long jumper: the angular momentum caused by the torque during the initial jump can be transferred to the rotation of the arms, by windmilling them, keeping the jumper upright.
25 11.11 Conservation of Angular Momentum Sample problem
26 11 Table 11.1
27 11 Table 10.3
28 Example:
29 11 Summary Rolling Bodies Eq. (112) Eq. (115) Eq. (116) Torque as a Vector Direction given by the righthand rule Eq. (1114) Angular Momentum of a Particle Newton's Second Law in Angular Form Eq. (1118) Eq. (1123)
30 11 Summary Angular Momentum of a System of Particles Angular Momentum of a Rigid Body Eq. (1131) Eq. (1126) Eq. (1129) Conservation of Angular Momentum Eq. (1132) Precession of a Gyroscope Eq. (1146) Eq. (1133)
31 11 Solved Problems 1. In Figure, a solid cylinder of radius 10 cm and mass 12 kg starts from rest and rolls without slipping a distance L=6.0 m down a roof that is inclined at the angle ϴ=30 o. (a) What is the angular speed of the cylinder about its center as it leaves the roof? (b) The roof's edge is at height H=5.0 m. How far horizontally from the roof's edge does the cylinder hit the level ground?
32 11 Solved Problems
33 11 Solved Problems 2. At time t=0, a 3.0 kg particle with velocity v=(5.0 m/s)i (6.0 m/s)j is at x=3.0 m, y=8.0 m. It is pulled by a 7.0 N force in the negative x direction. About the origin, what are (a) the particle's angular momentum, (b) the torque acting on the particle, and (c) the rate at which the angular momentum is changing?
34 11 Solved Problems 3. In Figure, a small 50 g block slides down a frictionless surface through height h=20 cm and then sticks to a uniform rod of mass 100 g and length 40 cm. The rod pivots about point O through angle ϴ before momentarily stopping. Find ϴ.
35 11 Solved Problems
36 11 Solved Problems
37 11 Solved Problems 4. In Figure, a constant horizontal force F app of magnitude 12 N is applied to a uniform solid cylinder by fishing line wrapped around the cylinder. The mass of the cylinder is 10 kg, its radius is 0.10 m, and the cylinder rolls smoothly on the horizontal surface. (a) What is the magnitude of the acceleration of the center of mass of the cylinder? (b) What is the magnitude of the angular acceleration of the cylinder about the center of mass? (c) In unit vector notation, what is the frictional force acting on the cylinder?
38 11 Solved Problems (a) What is the magnitude of the acceleration of the center of mass of the cylinder? (b) What is the magnitude of the angular acceleration of the cylinder about the center of mass? (c) In unit vector notation, what is the frictional force acting on the cylinder?
39 11 Solved Problems 5. Figure shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration (a) of the falling block, the angular acceleration (α) of the disk, and the tension (T ) in the cord. The cord does not slip, and there is no friction at the axle. (I = 1/2 MR 2 )
40 18 Scientific Notation Additional Materials
41 11.5 The YoYo As a yoyo moves down a string, it loses potential energy mgh but gains rotational and translational kinetic energy
42 11.12 Precession of a Gyroscope (a) A nonspinning gyroscope falls by rotating in an xz plane because of torque t. (b) A rapidly spinning gyroscope, with angular momentum, L, precesses around the z axis. Its precessional motion is in the xy plane. This rotation is called precession (c) The change in angular momentum, dl/dt, leads to a rotation of L about O.
43 11.12 Precession of a Gyroscope The angular momentum of a (rapidly spinning) gyroscope is: Eq. (1143) The torque can only change the direction of L, not its magnitude, because of (1143) The only way its direction can change along the direction of the torque without its magnitude changing is if it rotates around the central axis. Therefore it precesses instead of toppling over.
44 11.12 Precession of a Gyroscope The precession rate is given by: True for a sufficiently rapid spin rate. Independent of mass, (I is proportional to M) but does depend on g. Valid for a gyroscope at an angle to the horizontal as well (a top for instance).
45 11.11 Conservation of Angular Momentum Sample problem Its angular momentum is now  L wh.the inversion results in the student, the stool, and the wheel s center rotating together as a composite rigid body about the stool s rotation axis, with rotational inertia I b = 6.8 kg m 2. With what angular speed b and in what direction does the composite body rotate after the inversion of the wheel? Figure a shows a student, sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia I wh about its central axis is 1.2 kg m 2. (The rim contains lead in order to make the value of I wh substantial.) The wheel is rotating at an angular speed wh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum L wh of the wheel points vertically upward. The student now inverts the wheel (Fig b) so L wh that, as seen from overhead, it is rotating clockwise.
46 112 Rolling as Translation and Rotation Combined Learning Objectives Identify that smooth rolling can be considered as a combination of pure translation and pure rotation Apply the relationship between the centerofmass speed and the angular speed of a body in smooth rolling. Figure 112
47 113_4 Forces and Kinetic Energy of Rolling Learning Objectives Calculate the kinetic energy of a body in smooth rolling as the sum of the translational kinetic energy of the center of mass and the rotational kinetic energy around the center of mass Apply the relationship between the work done on a smoothly rolling object and its kinetic energy change For smooth rolling (and thus no sliding), conserve mechanical energy to relate initial energy values to the values at a later point Draw a freebody diagram of an accelerating body that is smoothly rolling on a horizontal surface or up or down on a ramp.
48 113_4 Forces and Kinetic Energy of Rolling Apply the relationship between the centerofmass acceleration and the angular acceleration For smooth rolling up or down a ramp, apply the relationship between the object s acceleration, its rotational inertia, and the angle of the ramp.
49 115 The YoYo Learning Objectives Draw a freebody diagram of a yoyo moving up or down its string Identify that a yoyo is effectively an object that rolls smoothly up or down a ramp with an incline angle of For a yoyo moving up or down its string, apply the relationship between the yoyo's acceleration and its rotational inertia Determine the tension in a yoyo's string as the yoyo moves up or down the string.
50 116 Torque Revisited Learning Objectives Identify that torque is a vector quantity Identify that the point about which a torque is calculated must always be specified Calculate the torque due to a force on a particle by taking the cross product of the particle's position vector and the force vector, in either unitvector notation or magnitudeangle notation Use the righthand rule for cross products to find the direction of a torque vector.
51 117 Angular Momentum Learning Objectives Identify that angular momentum is a vector quantity Identify that the fixed point about which an angular momentum is calculated must always be specified Calculate the angular momentum of a particle by taking the cross product of the particle's position vector and its momentum vector, in either unitvector notation or magnitudeangle notation Use the righthand rule for cross products to find the direction of an angular momentum vector.
52 118 Newton's Second Law in Angular Form Learning Objectives Apply Newton's second law in angular form to relate the torque acting on a particle to the resulting rate of change of the particle's angular momentum, all relative to a specified point.
53 119 Angular Momentum of a Rigid Body Learning Objectives For a system of particles, apply Newton's second law in angular form to relate the net torque acting on the system to the rate of the resulting change in the system's angular momentum Apply the relationship between the angular momentum of a rigid body rotating around a fixed axis and the body's rotational inertia and angular speed around that axis If two rigid bodies rotate about the same axis, calculate their total angular momentum.
54 1111 Conservation of Angular Momentum Learning Objectives When no external net torque acts on a system along a specified axis, apply the conservation of angular momentum to relate the initial angular momentum value along that axis to the value at a later instant.
55 119 Precession of a Gyroscope Learning Objectives Identify that the gravitational force acting on a spinning gyroscope causes the spin angular momentum vector (and thus the gyroscope) to rotate about the vertical axis in a motion called precession Calculate the precession rate of a gyroscope Identify that a gyroscope's precession rate is independent of the gyroscope's mass.
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 informationWebreview 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 0.30mradius automobile
More informationEndofChapter Exercises
EndofChapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. Figure 11.21 shows four different cases involving a
More informationRolling, 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 informationCHAPTER 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 informationPHYSICS 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 information1 MR SAMPLE EXAM 3 FALL 2013
SAMPLE EXAM 3 FALL 013 1. A merrygoround 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 informationRotational 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 informationRotation. 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 informationAP 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 informationWork 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 informationPHYSICS 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 informationSuggested 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 informationChapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity
Chapter 8 Rotational Equilibrium and Rotational Dynamics 1. Torque 2. Torque and Equilibrium 3. Center of Mass and Center of Gravity 4. Torque and angular acceleration 5. Rotational Kinetic energy 6. Angular
More informationExam 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 informationRolling Motion and Angular Momentum
P U Z Z L E R One of the most popular early bicycles was the penny farthing, introduced in 1870. The bicycle was so named because the size relationship of its two wheels was about the same as the size
More informationTest 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 information6. 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 informationPHYSICS 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 righthanded Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this
More informationChapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics
Chapter 12: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational 2 / / 1/ 2 m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv 2 /
More informationHandout 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 informationChapter 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 informationRotation 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 informationDescription: 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 informationRotation. Rotational Variables
Rotation Rigid Bodies Rotation variables Constant angular acceleration Rotational KE Rotational Inertia Rotational Variables Rotation of a rigid body About a fixed rotation axis. Rigid Body an object that
More informationChap. 10: Rotational Motion
Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics  Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Newton s Laws for Rotation n e t I 3 rd part [N
More informationPhysics 131: Lecture 22. Today s Agenda
Physics 131: Lecture 22 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 201: Lecture 10, Pg 1 An Unfair Race A frictionless block and a rolling (without slipping) disk are released
More informationTutorBreeze.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 informationCHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY
CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY OUTLINE 1. Angular Position, Velocity, and Acceleration 2. Rotational
More informationPhysics 131: Lecture 22. Today s Agenda
Physics 131: Lecture Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 01: Lecture 10, Pg 1 An Unfair Race A frictionless block and a rolling (without slipping) disk are released at
More informationPhys101 Second Major173 Zero Version Coordinator: Dr. M. AlKuhaili Thursday, August 02, 2018 Page: 1. = 159 kw
Coordinator: Dr. M. AlKuhaili Thursday, August 2, 218 Page: 1 Q1. A car, of mass 23 kg, reaches a speed of 29. m/s in 6.1 s starting from rest. What is the average power used by the engine during the
More informationBig 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 informationChapter 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 informationPhysics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium
Physics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium Strike (Day 10) Prelectures, checkpoints, lectures continue with no change. Takehome quizzes this week. See Elaine Schulte s email. HW
More informationRotational 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 informationCircular 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 informationName: Date: Period: AP Physics C Rotational Motion HO19
1.) A wheel turns with constant acceleration 0.450 rad/s 2. (99) 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 informationTranslational 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 informationIII. Angular Momentum Conservation (Chap. 10) Rotation. We repeat Chap. 28 with rotatiing objects. Eqs. of motion. Energy.
Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics  Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Toward Exam 3 Eqs. of motion o To study angular
More informationName Student ID Score Last First. I = 2mR 2 /5 around the sphere s center of mass?
NOTE: ignore air resistance in all Questions. In all Questions choose the answer that is the closest!! Question I. (15 pts) Rotation 1. (5 pts) A bowling ball that has an 11 cm radius and a 7.2 kg mass
More informationBig 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 50kg boy and a 40kg girl sit on opposite ends of a 3meter seesaw. How far from the girl should the fulcrum be placed in order for the
More information= o + t = ot + ½ t 2 = o + 2
Chapters 89 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 informationTorque. Introduction. Torque. PHY torque  J. Hedberg
Torque PHY 207  torque  J. Hedberg  2017 1. Introduction 2. Torque 1. Lever arm changes 3. Net Torques 4. Moment of Rotational Inertia 1. Moment of Inertia for Arbitrary Shapes 2. Parallel Axis Theorem
More informationChapter 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 informationName (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 informationChapter 910 Test Review
Chapter 910 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 informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice 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 informationPhys101 Third Major161 Zero Version Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1
Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1 Q1. A water molecule (H 2O) consists of an oxygen (O) atom of mass 16m and two hydrogen (H) atoms, each of mass m, bound to it (see Figure
More informationConcept 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 informationI 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 informationPHY131H1S  Class 20. Preclass reading quiz on Chapter 12
PHY131H1S  Class 20 Today: Gravitational Torque Rotational Kinetic Energy Rolling without Slipping Equilibrium with Rotation Rotation Vectors Angular Momentum Preclass reading quiz on Chapter 12 1 Last
More informationPhys 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 informationA uniform rod of length L and Mass M is attached at one end to a frictionless pivot. If the rod is released from rest from the horizontal position,
A dentist s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 10 4 rev/min. (a) Find the drill s angular acceleration. (b) Determine the angle (in radians)
More informationCHAPTER 8 TEST REVIEW MARKSCHEME
AP PHYSICS Name: Period: Date: 50 Multiple Choice 45 Single Response 5 MultiResponse Free Response 3 Short Free Response 2 Long Free Response MULTIPLE CHOICE DEVIL PHYSICS BADDEST CLASS ON CAMPUS AP EXAM
More informationIt will be most difficult for the ant to adhere to the wheel as it revolves past which of the four points? A) I B) II C) III D) IV
AP Physics 1 Lesson 16 Homework Newton s First and Second Law of Rotational Motion Outcomes Define rotational inertia, torque, and center of gravity. State and explain Newton s first Law of Motion as it
More informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice 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 informationName Date Period PROBLEM SET: ROTATIONAL DYNAMICS
Accelerated Physics Rotational Dynamics Problem Set Page 1 of 5 Name Date Period PROBLEM SET: ROTATIONAL DYNAMICS Directions: Show all work on a separate piece of paper. Box your final answer. Don t forget
More information1 of 5 7/13/2015 9:03 AM HW8 due 6 pm Day 18 (Wed. July 15) (7426858) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1. Question Details OSColPhys1 10.P.028.WA. [2611790] The specifications for
More informationChapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics
Chapter 1: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational / / 1/ m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv / / 1/ I
More informationPlane Motion of Rigid Bodies: Forces and Accelerations
Plane Motion of Rigid Bodies: Forces and Accelerations Reference: Beer, Ferdinand P. et al, Vector Mechanics for Engineers : Dynamics, 8 th Edition, Mc GrawHill Hibbeler R.C., Engineering Mechanics: Dynamics,
More informationPhysics 121, Final Exam Do not turn the pages of the exam until you are instructed to do so.
, Final Exam Do not turn the pages of the exam until you are instructed to do so. You are responsible for reading the following rules carefully before beginning. Exam rules: You may use only a writing
More information16. Rotational Dynamics
6. Rotational Dynamics A Overview In this unit we will address examples that combine both translational and rotational motion. We will find that we will need both Newton s second law and the rotational
More informationPhysics 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 informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym
FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 13 pages. Make sure none are missing 2.
More informationAP practice ch 78 Multiple Choice
AP practice ch 78 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 informationUnit 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 informationPhys101 Third Major161 Zero Version Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1
Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1 Q1. A water molecule (H 2 O) consists of an oxygen (O) atom of mass 16m and two hydrogen (H) atoms, each of mass m, bound to it (see
More informationRotation and Translation Challenge Problems Problem 1:
Rotation and Translation Challenge Problems Problem 1: A drum A of mass m and radius R is suspended from a drum B also of mass m and radius R, which is free to rotate about its axis. The suspension is
More informationLecture 6 Physics 106 Spring 2006
Lecture 6 Physics 106 Spring 2006 Angular Momentum Rolling Angular Momentum: Definition: Angular Momentum for rotation System of particles: Torque: l = r m v sinφ l = I ω [kg m 2 /s] http://web.njit.edu/~sirenko/
More informationAP 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 informationChapter 11 Rolling, Torque, and Angular Momentum
Prof. Dr. I. Nasser Chapter11I November, 017 Chapter 11 Rolling, Torque, and Angular Momentum 111 ROLLING AS TRANSLATION AND ROTATION COMBINED Translation vs. Rotation General Rolling Motion General
More informationChapter 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 information112 A General Method, and Rolling without Slipping
112 A General Method, and Rolling without Slipping Let s begin by summarizing a general method for analyzing situations involving Newton s Second Law for Rotation, such as the situation in Exploration
More informationPhysics 130: Questions to study for midterm #1 from Chapter 8
Physics 130: Questions to study for midterm #1 from Chapter 8 1. If the beaters on a mixer make 800 revolutions in 5 minutes, what is the average rotational speed of the beaters? a. 2.67 rev/min b. 16.8
More informationChapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:
linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)
More information31 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 informationForces 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: Midgrades posted in PAW Quiz today I will be at the March APS meeting the week of 1519 th. Prof. Rich Kurtz will help me. Class Website: http://www.phys.lsu.edu/classes/spring2010/phys21013/
More informationWe 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 information1. An object is dropped from rest. Which of the five following graphs correctly represents its motion? The positive direction is taken to be downward.
Unless otherwise instructed, use g = 9.8 m/s 2 Rotational Inertia about an axis through com: Hoop about axis(radius=r, mass=m) : MR 2 Hoop about diameter (radius=r, mass=m): 1/2MR 2 Disk/solid cyllinder
More informationSummer Physics 41 Pretest. Shorty Shorts (2 pts ea): Circle the best answer. Show work if a calculation is required.
Summer Physics 41 Pretest Name: Shorty Shorts (2 pts ea): Circle the best answer. Show work if a calculation is required. 1. An object hangs in equilibrium suspended by two identical ropes. Which rope
More informationMomentum. The way to catch a knuckleball is to wait until it stops rolling and then pick it up. Bob Uecker
Chapter 11 , Chapter 11 , Angular The way to catch a knuckleball is to wait until it stops rolling and then pick it up. Bob Uecker David J. Starling Penn State Hazleton PHYS 211 Chapter 11 , motion
More informationUse 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 informationPhysics 131: Lecture 21. Today s Agenda
Physics 131: Lecture 21 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 201: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia
More informationAngular 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 informationRotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia
Rotation Kinematics Rigid Bodies Kinetic Energy featuring moments of Inertia Torque Rolling Angular Motion We think about rotation in the same basic way we do about linear motion How far does it go? How
More informationAngular 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 informationPY105 Assignment 10 ( )
1 of 5 2009/10/30 8:27 AM PY105 Assignment 10 (1031274) 0/20 Tue Nov 17 2009 10:15 PM EST Question Points 1 2 3 4 5 6 7 0/2 0/6 0/2 0/2 0/2 0/5 0/1 Total 0/20 Description This assignment is worth 20 points.
More informationChapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.
Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 101 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 101 Angular Position, Velocity,
More informationSlide 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 informationSlide 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 informationMechanics 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 informationChapter 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 MFMcGrawPHY 45 Chap_10HaRotationRevised
More informationMoment 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 informationChapter 10: Rotation
Chapter 10: Rotation Review of translational motion (motion along a straight line) Position x Displacement x Velocity v = dx/dt Acceleration a = dv/dt Mass m Newton s second law F = ma Work W = Fdcosφ
More informationWritten Homework problems. Spring (taken from Giancoli, 4 th edition)
Written Homework problems. Spring 014. (taken from Giancoli, 4 th edition) HW1. Ch1. 19, 47 19. Determine the conversion factor between (a) km / h and mi / h, (b) m / s and ft / s, and (c) km / h and m
More informationReview 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