Rotation Angular Momentum

Size: px
Start display at page:

Download "Rotation Angular Momentum"

Transcription

1 Rotation Angular Momentum Lana Sheridan De Anza College Nov 28, 2017

2 Last time rolling motion

3 Overview Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum of rigid objects

4 Reminder about Force and Torque Torque relates to force: τ = r F We can write Newton s Second Law in its more general form: F net = dp dt This relates force to momentum.

5 Reminder about Force and Torque Torque relates to force: τ = r F We can write Newton s Second Law in its more general form: F net = dp dt This relates force to momentum. Is there some similar rotational expression? Relating torque to r p?

6 Reminder about Force and Torque Torque relates to force: τ = r F We can write Newton s Second Law in its more general form: F net = dp dt This relates force to momentum. Is there some similar rotational expression? Relating torque to r p? Yes!

7 Angular Momentum A new quantity, angular momentum: L = r p where r is the displacement vector of a particle relative to some axis of rotation, and p is the momentum of the particle

8 Angular Momentum A new quantity, angular momentum: L = r p where r is the displacement vector of a particle relative to some axis of rotation, and p is the momentum of the particle So that we have τ = dl dt = d(r p) dt

9 Angular Momentum A new quantity, angular momentum: L = r p where r is the displacement vector of a particle relative to some axis of rotation, and p is the momentum of the particle So that we have τ = dl dt = d(r p) dt Units: kg m 2 s 1

10 Angular Momentum This is a more general form of Newton s second law for rotations! τ net = dl dt torques cause changes in angular momentum, L.

11 Angular Momentum: Confirming Equation Confirming dl dt = τ net:

12 Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt = d(r p) dt

13 Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt d(r p) = dt = r dp dt + dr dt p = r dp dt + v p 0 because, v and p are always parallel. Noting F net = dp dt : dl dt = r F net

14 Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt d(r p) = dt = r dp dt + dr dt p = r dp dt + v p 0 because, v and p are always parallel. Noting F net = dp dt : dl dt = r F net By definition τ net = r F net, so dl dt = τ net

15 Angular Impulse We have: τ net = dl dt We can now define the angular impulse on a system as L = τ net dt A torque applied over time changes the angular momentum. Angular impulse is a vector Units: kg m 2 s 1

16 um p S : Angular Momentum (11.10) L = r p (11.11) 5 d p S /dt. Torque s linear momentum easured about the in an inertial frame. that both the magollowing the righte plane formed by s in the z direction. (11.12) hen r S is parallel to x The angular momentum L of a particle about an axis is a vector perpendicular to both the S particle s position r relative to S the axis and its momentum p. z O S r S L m S r S p S S p f y

17 Angular Momentum L = r p is a vector equation. If we only need to know the magnitude of L, then we can use the following expression: L = mvr sin φ where we used p = mv, and φ is the angle between r and p.

18 traight path that is c) mva (d) imposabout any axis displaced from Angular Momentum the path of of athe Particle particle. in Circular Motion A particle has mass, m, velocity v, and travels in a circular path of radius r about a point O. What is the magnitude of its angular momentum relative to the axis O? Circular Motion s shown in Figure relative to an axis y S v (Example 11.3) A ing in a circle of radius r ar momentum about an O that has magnitude or L S 5 r S 3 p S points e. O S r m x continued

19 omentum Angular of a Particle Momentum in Circular Motion of a Particle in Circular Motion in a circular path of radius r as shown in Figure ection of its angular momentum relative to an axis y S v entum of the ection (but not ore be tempted mentum of the this situation, s see why. Figure 11.5 (Example 11.3) A particle moving in a circle of radius r has an angular momentum about an axis through O that has magnitude mvr. The vector L S 5 r S 3 p S points out of the page. O S r continued A particle has mass, m, velocity v, and travels in a circular path of radius r about a point O. What is the magnitude of its angular momentum relative to the axis O? m x L = mrv What is the direction of the angular momentum vector L? (A) +k (B) k

20 omentum Angular of a Particle Momentum in Circular Motion of a Particle in Circular Motion in a circular path of radius r as shown in Figure ection of its angular momentum relative to an axis y S v entum of the ection (but not ore be tempted mentum of the this situation, s see why. Figure 11.5 (Example 11.3) A particle moving in a circle of radius r has an angular momentum about an axis through O that has magnitude mvr. The vector L S 5 r S 3 p S points out of the page. O S r continued A particle has mass, m, velocity v, and travels in a circular path of radius r about a point O. What is the magnitude of its angular momentum relative to the axis O? m x L = mrv What is the direction of the angular momentum vector L? (A) +k (B) k

21 Angular Momentum of a Falling Object A rock, mass m, is dropped near the surface of the Earth from a height h. After a time t its velocity is v. Let R Earth be the radius of the Earth. What is the magnitude of angular momentum of the rock about the center of the Earth? (A) mgh (B) mvh (C) mvr Earth (D) 0

22 Angular Momentum of a Falling Object A rock, mass m, is dropped near the surface of the Earth from a height h. After a time t its velocity is v. Let R Earth be the radius of the Earth. What is the magnitude of angular momentum of the rock about the center of the Earth? (A) mgh (B) mvh (C) mvr Earth (D) 0

23 Angular mtm of an object moving in a Straight Line Suppose a neutron of mass m passes by an atom with a distance of closest approach of R, traveling at a speed v. Assume no forces act. (Or they are negligibly small.) p i p p f θ r i R What is the angular momentum of the neutron about an axis through the atom?

24 Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point?

25 Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point? At t f, θ f = 90 L f = r f p = mvr sin(90 ) ( k) = mvr ( k)

26 Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point? At t f, θ f = 90 L f = r f p = mvr sin(90 ) ( k) = mvr ( k) At t i : L i = r i p = mvr i sin θ i ( k) But, r i sin θ i = R, so L i = mvr ( k) and L i = L f.

27 Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point? At t f, θ f = 90 L f = r f p = mvr sin(90 ) ( k) = mvr ( k) At t i : L i = r i p = mvr i sin θ i ( k) But, r i sin θ i = R, so L i = mvr ( k) and L i = L f.

28 Angular mtm of an object moving in a Straight Line The magnitude of the angular momentum of an object, of mass m and velocity v, traveling in a straight line about an axis O is L = mvr where R is the distance of closest approach of the axis O.

29 Angular Momentum of Rigid Object z S r S L m i 11.3 Angul In Example 11.4, w particles. Let us no Consider a rigid ob coordinate system of this object. Each an angular speed v m i about the z axis tude of the angula x The vector S L i for t We can now fin Figure 11.7 When a rigid object component) of the All parts of the object have the same angular velocity ω. v S S v i rotates about an axis, the angular momentum S L is in the same direction as the angular velocity S y

30 Angular Momentum of Rigid Object From the definition L = r p = m r v. We know v = ω r For one particle, mass m: L = m r (ω r) = m ω(r r) r 0 (r ω) (because r ω) = m r 2 ω For a rigid object made of many particles:

31 Angular Momentum of Rigid Object From the definition L = r p = m r v. We know v = ω r For one particle, mass m: L = m r (ω r) = m ω(r r) r 0 (r ω) (because r ω) = m r 2 ω For a rigid object made of many particles: L tot = L i i ( ) = m i ri 2 ω i = Iω

32 Angular Momentum of Rigid Object For a rigid object: L tot = Iω where I is the moment of inertia and ω is the angular velocity.

33 Question Quick Quiz A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. Which one has the higher angular momentum? (A) the solid sphere (B) the hollow sphere (C) both have the same angular momentum (D) impossible to determine 1 Serway & Jewett, page 343.

34 Question Quick Quiz A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. Which one has the higher angular momentum? (A) the solid sphere (B) the hollow sphere (C) both have the same angular momentum (D) impossible to determine 1 Serway & Jewett, page 343.

35 Newton s Second Law for Rotations We said τ net = dl dt is the more general form of Newton s second law for rotations. What about our previous version of Newton s second law for rotations (τ net = I α)? τ net = dl dt = d(iω) dt 0 di = dt ω + I dω dt If we consider a fixed rigid object and fixed axis of rotation, di dt = 0: τ net = I α

36 Example Rigid Object Angular Momentum A father of mass m f and his daughter of mass m d sit on opposite ends of a seesaw at equal distances from the pivot at the center. The seesaw is modeled as a rigid rod of mass M and length l, and is pivoted without friction. At a given moment, the combination rotates in a vertical plane with an angular speed ω. on the as out n a s Find an expression for the magnitude of the system s angular momentum. Figure 11.9 (Example 11.6) A father and daughter demonstrate angular momentum on a seesaw. S m f g O y u m d g S x

37 Example Rigid Object Angular Momentum Magnitude of the system s angular momentum? Use: L = Iω

38 Example Rigid Object Angular Momentum Magnitude of the system s angular momentum? Use: L = Iω So, I = 1 12 Ml2 + m f ( l 2 L = l2 4 ) 2 ( ) l 2 + m d 2 ( ) 1 3 M + m f + m d ω

39 Example Rigid Object Angular Momentum Find an expression for the magnitude of the angular acceleration of the system when the seesaw makes an angle θ with the horizontal. y s t O u x m d g S Figure 11.9 (Example 11.6) A father and daughter demonstrate angular momentum on a seesaw. S m f g of rotation in Figure The rotating system has angular (The seesaw is modeled as a rigid rod of mass M.) father and daughter and model them both as particles. The rt of the example is categorized as a substitution problem.

40 Example Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα

41 Example Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα τ net = i r i F i = m f g l 2 sin(90 θ) m dg l sin(90 + θ) 2 = g l 2 cos θ(m f m d ) α = τ I = 2g cos θ(m f m d ) l( 1 3 M + m f + m d )

42 Summary introducing angular momentum angular impulse angular momentum of rigid objects (Uncollected) Homework Serway & Jewett: Ch 11, onward from page 355. Probs: 1, 3, 11, 13, 17

Rotation Angular Momentum Conservation of Angular Momentum

Rotation Angular Momentum Conservation of Angular Momentum Rotation Angular Momentum Conservation of Angular Momentum Lana Sheridan De Anza College Nov 29, 2017 Last time Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum

More information

Rotation Work and Power of Rotation Rolling Motion Examples and Review

Rotation Work and Power of Rotation Rolling Motion Examples and Review Rotation Work and Power of Rotation Rolling Motion Examples and Review Lana Sheridan De Anza College Nov 22, 2017 Last time applications of moments of inertia Atwood machine with massive pulley kinetic

More information

Rotation Moment of Inertia and Applications

Rotation Moment of Inertia and Applications Rotation Moment of Inertia and Applications Lana Sheridan De Anza College Nov 20, 2016 Last time net torque Newton s second law for rotation moments of inertia calculating moments of inertia Overview calculating

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

Rotation Atwood Machine with Massive Pulley Energy of Rotation

Rotation Atwood Machine with Massive Pulley Energy of Rotation Rotation Atwood Machine with Massive Pulley Energy of Rotation Lana Sheridan De Anza College Nov 21, 2017 Last time calculating moments of inertia the parallel axis theorem Overview applications of moments

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum Angular Momentum Angular momentum plays a key role in rotational dynamics. There is a principle of conservation of angular momentum. In analogy to the principle of conservation

More information

Rotation Torque Moment of Inertia

Rotation Torque Moment of Inertia Rotation Torque Moment of Inertia Lana Sheridan De Anza College Nov 17, 2017 Last time rotational quantities rotational kinematics torque Quick review of Vector Expressions Let a, b, and c be (non-null)

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

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

More information

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

= 2 5 MR2. I sphere = MR 2. I hoop = 1 2 MR2. I disk

= 2 5 MR2. I sphere = MR 2. I hoop = 1 2 MR2. I disk A sphere (green), a disk (blue), and a hoop (red0, each with mass M and radius R, all start from rest at the top of an inclined plane and roll to the bottom. Which object reaches the bottom first? (Use

More information

Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia. 8.01t Nov 3, 2004

Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia. 8.01t Nov 3, 2004 Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia 8.01t Nov 3, 2004 Rotation and Translation of Rigid Body Motion of a thrown object Translational Motion of the Center of Mass Total

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

Physics 121, March 25, Rotational Motion and Angular Momentum. Department of Physics and Astronomy, University of Rochester

Physics 121, March 25, Rotational Motion and Angular Momentum. Department of Physics and Astronomy, University of Rochester Physics 121, March 25, 2008. Rotational Motion and Angular Momentum. Physics 121. March 25, 2008. Course Information Topics to be discussed today: Review of Rotational Motion Rolling Motion Angular Momentum

More information

Handout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration

Handout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration 1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps

More information

16. Rotational Dynamics

16. 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 information

Extended or Composite Systems Systems of Many Particles Deformation

Extended or Composite Systems Systems of Many Particles Deformation Extended or Composite Systems Systems of Many Particles Deformation Lana Sheridan De Anza College Nov 15, 2017 Overview last center of mass example systems of many particles deforming systems Continuous

More information

CIRCULAR MOTION AND ROTATION

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

More information

Static Equilibrium. Lana Sheridan. Dec 5, De Anza College

Static Equilibrium. Lana Sheridan. Dec 5, De Anza College tatic Equilibrium Lana heridan De Anza College Dec 5, 2016 Last time simple harmonic motion Overview Introducing static equilibrium center of gravity tatic Equilibrium: ystem in Equilibrium Knowing that

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 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

Oscillations Simple Harmonic Motion

Oscillations Simple Harmonic Motion Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and

More information

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

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

More information

Physics 141. Lecture 18. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 18, Page 1

Physics 141. Lecture 18. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 18, Page 1 Physics 141. Lecture 18. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 18, Page 1 Physics 141. Lecture 18. Course Information. Topics to be discussed today: A

More information

Kinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction)

Kinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction) Kinematics (special case) a = constant 1D motion 2D projectile Uniform circular Dynamics gravity, tension, elastic, normal, friction Motion with a = constant Newton s Laws F = m a F 12 = F 21 Time & Position

More information

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

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

More information

Dynamics. Dynamics of mechanical particle and particle systems (many body systems)

Dynamics. Dynamics of mechanical particle and particle systems (many body systems) Dynamics Dynamics of mechanical particle and particle systems (many body systems) Newton`s first law: If no net force acts on a body, it will move on a straight line at constant velocity or will stay at

More information

Physics 121, March 27, Angular Momentum, Torque, and Precession. Department of Physics and Astronomy, University of Rochester

Physics 121, March 27, Angular Momentum, Torque, and Precession. Department of Physics and Astronomy, University of Rochester Physics 121, March 27, 2008. Angular Momentum, Torque, and Precession. Physics 121. March 27, 2008. Course Information Quiz Topics to be discussed today: Review of Angular Momentum Conservation of Angular

More information

AP Physics QUIZ Chapters 10

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

More information

Ch 8. Rotational Dynamics

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

More information

Angular Momentum. Objectives CONSERVATION OF ANGULAR MOMENTUM

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

More information

Chapter 8 continued. Rotational Dynamics

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

More information

Linear Momentum Center of Mass

Linear Momentum Center of Mass Linear Momentum Center of Mass Lana Sheridan De Anza College Nov 14, 2017 Last time the ballistic pendulum 2D collisions center of mass finding the center of mass Overview center of mass examples center

More information

Chapters 10 & 11: Rotational Dynamics Thursday March 8 th

Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Review of rotational kinematics equations Review and more on rotational inertia Rolling motion as rotation and translation Rotational kinetic energy

More information

Physics 5A Final Review Solutions

Physics 5A Final Review Solutions Physics A Final Review Solutions Eric Reichwein Department of Physics University of California, Santa Cruz November 6, 0. A stone is dropped into the water from a tower 44.m above the ground. Another stone

More information

Rotational Motion Rotational Kinematics

Rotational Motion Rotational Kinematics Rotational Motion Rotational Kinematics Lana Sheridan De Anza College Nov 16, 2017 Last time 3D center of mass example systems of many particles deforming systems Overview rotation relating rotational

More information

Midterm 3 Thursday April 13th

Midterm 3 Thursday April 13th Welcome back to Physics 215 Today s agenda: Angular momentum Rolling without slipping Midterm Review Physics 215 Spring 2017 Lecture 12-2 1 Midterm 3 Thursday April 13th Material covered: Ch 9 Ch 12 Lectures

More information

Chapter 9. Rotational Dynamics

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

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum C H A P T E R O U T L I N E 11 1 The Vector Product and Torque 11 2 Angular Momentum 11 3 Angular Momentum of a Rotating Rigid Object 11 4 Conservation of Angular Momentum 11

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

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

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

More information

Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass

Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass Lana Sheridan De Anza College Nov 13, 2017 Last time inelastic collisions perfectly inelastic collisions the ballistic pendulum

More information

PHY131H1S - Class 20. Pre-class reading quiz on Chapter 12

PHY131H1S - Class 20. Pre-class reading quiz on Chapter 12 PHY131H1S - Class 20 Today: Gravitational Torque Rotational Kinetic Energy Rolling without Slipping Equilibrium with Rotation Rotation Vectors Angular Momentum Pre-class reading quiz on Chapter 12 1 Last

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

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

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

More information

Rotational Dynamics continued

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

More information

Chapter 8- 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

Lecture 20 Chapter 12 Angular Momentum Course website:

Lecture 20 Chapter 12 Angular Momentum Course website: Lecture 20 Chapter 12 Angular Momentum Another Law? Am I in a Law school? Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi IN THIS CHAPTER, you will continue discussing rotational

More information

t = g = 10 m/s 2 = 2 s T = 2π g

t = g = 10 m/s 2 = 2 s T = 2π g Annotated Answers to the 1984 AP Physics C Mechanics Multiple Choice 1. D. Torque is the rotational analogue of force; F net = ma corresponds to τ net = Iα. 2. C. The horizontal speed does not affect the

More information

Dynamics of Rotational Motion

Dynamics of Rotational Motion Chapter 10 Dynamics of Rotational Motion To understand the concept of torque. To relate angular acceleration and torque. To work and power in rotational motion. To understand angular momentum. To understand

More information

Chapter 8 - Rotational Dynamics and Equilibrium REVIEW

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

More information

= o + t = ot + ½ t 2 = o + 2

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

More information

Chapter 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

Dynamics Applying Newton s Laws Air Resistance

Dynamics Applying Newton s Laws Air Resistance Dynamics Applying Newton s Laws Air Resistance Lana Sheridan De Anza College Oct 20, 2017 Last Time accelerated frames and rotation Overview resistive forces two models for resistive forces terminal velocities

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

Lecture-XII. Angular momentum and Fixed axis rotation

Lecture-XII. Angular momentum and Fixed axis rotation Lecture-XII Angular momentum and Fixed axis rotation Angular Momentum of a System of Particles Consider a collection of N discrete particles. The total angular momentum of the system is The force acting

More information

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

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

More information

Conceptual Physics Gravity

Conceptual Physics Gravity Conceptual Physics Gravity Lana Sheridan De Anza College July 18, 2017 Last time rotational motion centripetal force vs. centrifugal force torque moment of inertia center of mass and center of gravity

More information

Chapter 9. Rotational Dynamics

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

More information

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

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

More information

Notes on Torque. We ve seen that if we define torque as rfsinθ, and the N 2. i i

Notes on Torque. We ve seen that if we define torque as rfsinθ, and the N 2. i i Notes on Torque We ve seen that if we define torque as rfsinθ, and the moment of inertia as N, we end up with an equation mr i= 1 that looks just like Newton s Second Law There is a crucial difference,

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

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

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

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

More information

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 Motion and Torque

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

More information

ROTATIONAL MOTION FROM TRANSLATIONAL MOTION

ROTATIONAL MOTION FROM TRANSLATIONAL MOTION ROTATIONAL MOTION FROM TRANSLATIONAL MOTION Velocity Acceleration 1-D otion 3-D otion Linear oentu TO We have shown that, the translational otion of a acroscopic object is equivalent to the translational

More information

Lecture II: Rigid-Body Physics

Lecture II: Rigid-Body Physics Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2 Rigid-Body Kinematics Objects as sets of points. Relative distances

More information

Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass

Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass Linear Momentum 2D Collisions Extended or Composite Systems Center of Mass Lana Sheridan De Anza College Nov 13, 2017 Last time inelastic collisions perfectly inelastic collisions the ballistic pendulum

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum Angular Momentum Angular momentum plays a key role in rotational dynamics. There is a principle of conservation of angular momentum. In analogy to the principle of conservation

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

Slide 1 / 37. Rotational Motion

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

More information

Torque. Introduction. Torque. PHY torque - J. Hedberg

Torque. 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 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

Rotational Motion. Rotational Motion. Rotational Motion

Rotational Motion. Rotational Motion. Rotational Motion I. Rotational Kinematics II. Rotational Dynamics (Netwton s Law for Rotation) III. Angular Momentum Conservation 1. Remember how Newton s Laws for translational motion were studied: 1. Kinematics (x =

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

Chapter 10: Rotation

Chapter 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 information

Physics 4A Solutions to Chapter 10 Homework

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

More information

Static Equilibrium Gravitation

Static Equilibrium Gravitation Static Equilibrium Gravitation Lana Sheridan De Anza College Dec 6, 2017 Overview One more static equilibrium example Newton s Law of Universal Gravitation gravitational potential energy little g Example

More information

Dynamics Applying Newton s Laws Air Resistance

Dynamics Applying Newton s Laws Air Resistance Dynamics Applying Newton s Laws Air Resistance Lana Sheridan De Anza College Feb 2, 2015 Last Time accelerated frames and rotation Overview resistive forces two models for resistive forces terminal velocities

More information

Physics 2A Chapter 10 - Rotational Motion Fall 2018

Physics 2A Chapter 10 - Rotational Motion Fall 2018 Physics A Chapter 10 - Rotational Motion Fall 018 These notes are five pages. A quick summary: The concepts of rotational motion are a direct mirror image of the same concepts in linear motion. Follow

More information

Dynamics Applying Newton s Laws Introducing Energy

Dynamics Applying Newton s Laws Introducing Energy Dynamics Applying Newton s Laws Introducing Energy Lana Sheridan De Anza College Oct 23, 2017 Last time introduced resistive forces model 1: Stokes drag Overview finish resistive forces energy work Model

More information

Chapter 8 continued. Rotational Dynamics

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

More information

Kinematics Motion in 1-Dimension

Kinematics Motion in 1-Dimension Kinematics Motion in 1-Dimension Lana Sheridan De Anza College Jan 15, 219 Last time how to solve problems 1-D kinematics Overview 1-D kinematics quantities of motion graphs of kinematic quantities vs

More information

Classical Mechanics. FIG. 1. Figure for (a), (b) and (c). FIG. 2. Figure for (d) and (e).

Classical Mechanics. FIG. 1. Figure for (a), (b) and (c). FIG. 2. Figure for (d) and (e). Classical Mechanics 1. Consider a cylindrically symmetric object with a total mass M and a finite radius R from the axis of symmetry as in the FIG. 1. FIG. 1. Figure for (a), (b) and (c). (a) Show that

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

Angular Momentum Conservation of Angular Momentum

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

More information

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

Kinematics Motion in 1-Dimension

Kinematics Motion in 1-Dimension Kinematics Motion in 1-Dimension Lana Sheridan De Anza College Jan 16, 2018 Last time unit conversions (non-si units) order of magnitude calculations how to solve problems Overview 1-D kinematics quantities

More information

Rotation. I. Kinematics - Angular analogs

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

More information

Lesson 8. Luis Anchordoqui. Physics 168. Thursday, October 11, 18

Lesson 8. Luis Anchordoqui. Physics 168. Thursday, October 11, 18 Lesson 8 Physics 168 1 Rolling 2 Intuitive Question Why is it that when a body is rolling on a plane without slipping the point of contact with the plane does not move? A simple answer to this question

More information

Dynamics Energy and Work

Dynamics Energy and Work Dynamics Energy and Work Lana Sheridan De Anza College Oct 24, 2017 Last Time resistive forces: Drag Equation Drag Equation, One more point What if the object is not dropped from rest? (See Ch 6, prob

More information

Rotational motion problems

Rotational motion problems Rotational motion problems. (Massive pulley) Masses m and m 2 are connected by a string that runs over a pulley of radius R and moment of inertia I. Find the acceleration of the two masses, as well as

More information

DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS

DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS OPTION B-1A: ROTATIONAL DYNAMICS Essential Idea: The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual

More information

Physics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 20: Rotational Motion. Slide 20-1

Physics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 20: Rotational Motion. Slide 20-1 Physics 1501 Fall 2008 Mechanics, Thermodynamics, Waves, Fluids Lecture 20: Rotational Motion Slide 20-1 Recap: center of mass, linear momentum A composite system behaves as though its mass is concentrated

More information

E = K + U. p mv. p i = p f. F dt = p. J t 1. a r = v2. F c = m v2. s = rθ. a t = rα. r 2 dm i. m i r 2 i. I ring = MR 2.

E = K + U. p mv. p i = p f. F dt = p. J t 1. a r = v2. F c = m v2. s = rθ. a t = rα. r 2 dm i. m i r 2 i. I ring = MR 2. v = v i + at x = x i + v i t + 1 2 at2 E = K + U p mv p i = p f L r p = Iω τ r F = rf sin θ v 2 = v 2 i + 2a x F = ma = dp dt = U v dx dt a dv dt = d2 x dt 2 A circle = πr 2 A sphere = 4πr 2 V sphere =

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

Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration

Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration Lana Sheridan De Anza College May 17, 2018 Last time pulse propagation the wave equation Overview solutions to the wave

More information

Physics 8 Monday, October 28, 2013

Physics 8 Monday, October 28, 2013 Physics 8 Monday, October 28, 2013 Turn in HW8 today. I ll make them less difficult in the future! Rotation is a hard topic. And these were hard problems. HW9 (due Friday) is 7 conceptual + 8 calculation

More information

Mechanics Oscillations Simple Harmonic Motion

Mechanics Oscillations Simple Harmonic Motion Mechanics Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 3, 2018 Last time gravity Newton s universal law of gravitation gravitational field gravitational potential energy Overview

More information

Physics 207: Lecture 24. Announcements. No labs next week, May 2 5 Exam 3 review session: Wed, May 4 from 8:00 9:30 pm; here.

Physics 207: Lecture 24. Announcements. No labs next week, May 2 5 Exam 3 review session: Wed, May 4 from 8:00 9:30 pm; here. Physics 07: Lecture 4 Announcements No labs next week, May 5 Exam 3 review session: Wed, May 4 from 8:00 9:30 pm; here Today s Agenda ecap: otational dynamics and torque Work and energy with example Many

More information