General Physics I Rotational Motion

Slides:



Advertisements
Similar presentations
Classical Mechanics Review 3, Units 1-16
Advertisements

Physics 101: Lecture 13 Rotational Kinetic Energy and Inertia
Chapter 11 Angular Momentum
Comparing rotational and linear motion
Rotational Motion and Equilibrium
Physics 111: Lecture 19, Pg 1 Physics 111: Lecture 19 Today’s Agenda l Review l Many body dynamics l Weight and massive pulley l Rolling and sliding examples.
Physics 207: Lecture 16, Pg 1 Lecture 16Goals: Chapter 12 Chapter 12  Extend the particle model to rigid-bodies  Understand the equilibrium of an extended.
Lecture 37, Page 1 Physics 2211 Spring 2005 © 2005 Dr. Bill Holm Physics 2211: Lecture 37 l Work and Kinetic Energy l Rotational Dynamics Examples çAtwood’s.
Physics 101: Lecture 15, Pg 1 Physics 101: Lecture 15 Rolling Objects l Today’s lecture will cover Textbook Chapter Exam III.
Lecture 36, Page 1 Physics 2211 Spring 2005 © 2005 Dr. Bill Holm Physics 2211: Lecture 36 l Rotational Dynamics and Torque.
Physics 207: Lecture 17, Pg 1 Lecture 17 Goals: Chapter 12 Chapter 12  Define center of mass  Analyze rolling motion  Introduce and analyze torque 
Physics 2211: Lecture 38 Rolling Motion
Physics 101: Lecture 18, Pg 1 Physics 101: Lecture 18 Rotational Dynamics l Today’s lecture will cover Textbook Sections : è Quick review of last.
Chapter 11 Rolling, Torque, and Angular Momentum In this chapter we will cover the following topics: -Rolling of circular objects and its relationship.
Lecture 34, Page 1 Physics 2211 Spring 2005 © 2005 Dr. Bill Holm Physics 2211: Lecture 34 l Rotational Kinematics çAnalogy with one-dimensional kinematics.
Physics 151: Lecture 20, Pg 1 Physics 151: Lecture 20 Today’s Agenda l Topics (Chapter 10) : çRotational KinematicsCh çRotational Energy Ch
Department of Physics and Applied Physics , F2010, Lecture 19 Physics I LECTURE 19 11/17/10.
Classical Mechanics Review 4: Units 1-19
Chap. 11B - Rigid Body Rotation
Physics 1501: Lecture 20, Pg 1 Physics 1501: Lecture 20 Today’s Agenda l Announcements çHW#7: due Oct. 21 l Midterm 1: average ~ 45 % … l Topics çMoments.
Lecture Outline Chapter 8 College Physics, 7 th Edition Wilson / Buffa / Lou © 2010 Pearson Education, Inc.
Rotational Dynamics Just as the description of rotary motion is analogous to translational motion, the causes of angular motion are analogous to the causes.
ROTATIONAL MOTION AND EQUILIBRIUM
Physics 201: Lecture 19, Pg 1 Lecture 19 Goals: Specify rolling motion (center of mass velocity to angular velocity Compare kinetic and rotational energies.
Physics 207: Lecture 14, Pg 1 Physics 207, Lecture 14, Oct. 23 Agenda: Chapter 10, Finish, Chapter 11, Just Start Assignment: For Wednesday reread Chapter.
Chapter 10 Chapter 10 Rotational motion Rotational motion Part 2 Part 2.
Physics 1501: Lecture 19, Pg 1 Physics 1501: Lecture 19 Today’s Agenda l Announcements çHW#7: due Oct. 21 l Midterm 1: average = 45 % … l Topics çRotational.
Physics 203 – College Physics I Department of Physics – The Citadel Physics 203 College Physics I Fall 2012 S. A. Yost Chapter 8 Part 1 Rotational Motion.
Moment Of Inertia.
Physics 111: Lecture 17, Pg 1 Physics 111: Lecture 17 Today’s Agenda l Rotational Kinematics çAnalogy with one-dimensional kinematics l Kinetic energy.
Rotational Motion. 6-1 Angular Position, Velocity, & Acceleration.
Rotational Motion About a Fixed Axis
Physics 207: Lecture 16, Pg 1 Lecture 16Goals: Chapter 12 Chapter 12  Extend the particle model to rigid-bodies  Understand the equilibrium of an extended.
4.1 Rotational kinematics 4.2 Moment of inertia 4.3 Parallel axis theorem 4.4 Angular momentum and rotational energy CHAPTER 4: ROTATIONAL MOTION.
1 Rotation of a Rigid Body Readings: Chapter How can we characterize the acceleration during rotation? - translational acceleration and - angular.
Chapter 11 Angular Momentum. The Vector Product and Torque The torque vector lies in a direction perpendicular to the plane formed by the position vector.
Physics 101: Lecture 13, Pg 1 Physics 101: Lecture 13 Rotational Kinetic Energy and Inertia Exam II.
Wednesday, Nov. 10, 2004PHYS , Fall 2004 Dr. Jaehoon Yu 1 1.Moment of Inertia 2.Parallel Axis Theorem 3.Torque and Angular Acceleration 4.Rotational.
Physics 1501: Lecture 21, Pg 1 Physics 1501: Lecture 21 Today’s Agenda l Announcements çHW#8: due Oct. 28 l Honors’ students çsee me after class l Midterm.
Chapter 8 Lecture Pearson Physics Rotational Motion and Equilibrium Prepared by Chris Chiaverina © 2014 Pearson Education, Inc.
Mechanics Lecture 17, Slide 1 Physics 211 Lecture 17 Today’s Concepts: a) Torque Due to Gravity b) Static Equilibrium Next Monday 1:30-2:20pm, here: Hour.
PHYS 1443 – Section 001 Lecture #19
F1 F2 If F1 = F2… …no change in motion (by Newton’s 1st Law)
PHYS 1443 – Section 003 Lecture #18
Physics 101: Lecture 13 Rotational Kinetic Energy and Inertia
PHYS 1441 – Section 002 Lecture #21
PHYS 1443 – Section 001 Lecture #14
Dynamics of Rotational Motion
Physics 101: Lecture 15 Rolling Objects
Physics 101: Lecture 15 Rolling Objects
Rotational Dynamics Chapter 9.
Rotational Motion – Part II
Rotational Inertia & Kinetic Energy
Aim: How do we explain the rolling motion of rigid bodies?
PHYS 1443 – Section 002 Lecture #18
Wednesday: Review session
Rotational Motion AP Physics.
Figure 10.16  A particle rotating in a circle under the influence of a tangential force Ft. A force Fr in the radial direction also must be present to.
Rolling, Torque, and Angular Momentum
Equilibrium and Dynamics
Lecture 17 Goals Relate and use angle, angular velocity & angular acceleration Identify vectors associated with angular motion Introduce Rotational Inertia.
Lecture 17 Goals: Chapter 12 Define center of mass
Physics 207, Lecture 17, Nov. 1 Work/Energy Theorem, Energy Transfer
Spring 2002 Lecture #15 Dr. Jaehoon Yu Mid-term Results
Chapter 11 - Rotational Dynamics
Chapter 11 Angular Momentum
Lecture 17 Goals: Chapter 12
Rotational Dynamics.
Rotational Motion – Part II
Physics 111: Lecture 18 Today’s Agenda
Presentation transcript:

General Physics I Rotational Motion

Rotation Up until now we have gracefully avoided dealing with the rotation of objects. We have studied objects that slide, not roll. We have assumed pulleys are without mass. Rotation is extremely important, however, and we need to understand it! Most of the equations we will develop are simply rotational analogues of ones we have already learned when studying linear kinematics and dynamics.

System of Particles Until now, we have considered the behavior of very simple systems (one or two masses). But real life is usually much more interesting! For example, consider a simple rotating disk. An extended solid object (like a disk) can be thought of as a collection of parts. The motion of each little part depends on where it is in the object!

System of Particles: Center of Mass How do we describe the “position” of a system made up of many parts? We define the Center of Mass The center of mass is where the system is balanced! Building a mobile is an exercise in finding centers of mass. m1 m2 + m1 m2 +

System of Particles: Center of Mass We can use intuition to find the location of the center of mass for symmetric objects that have uniform density: It will simply be at the geometrical center ! + CM + + + + +

Rotation & Kinetic Energy Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods). The kinetic energy of this system will be the sum of the kinetic energy of each piece: r1 r2 r3 r4 m4 m1 m2 m3 

Rotation & Kinetic Energy... So: but vi = ri r1 r2 r3 r4 m4 m1 m2 m3  v4 v1 v3 v2 Which we write as: ...the moment of inertia about the rotation axis I has units of kg m2.

Moment of Inertia So where Notice that the moment of inertia I depends on the distribution of mass in the system. The further the mass is from the rotation axis, the bigger the moment of inertia. For a given object, the moment of inertia will depend on where we choose the rotation axis. We will see that in rotational dynamics, the moment of inertia I appears in the same way that mass m does when we study linear dynamics!

Some Sample Moments of Inertia Thin hoop (or cylinder) of mass M and radius R, about an axis through its center, perpendicular to the plane of the hoop. R Thin rod of mass M and length L, about a perpendicular axis through its center. L Thin rod of mass M and length L, about a perpendicular axis through its end. L

Some More Samples... Solid sphere of mass M and radius R, about an axis through its center. R R Solid disk or cylinder of mass M and radius R, about a perpendicular axis through its center; a pulley!

Rolling... v 2v v Linear Motion Only Linear Motion + Rotation  v Rotation Only Where v = R 

Connection with Linear motion... So for a solid object which rotates about its center and which is also moving with linear velocity:  VCM

Example of Rolling Motion Objects of different I rolling down an inclined plane: K = - U = Mgh v = 0 = 0 K = 0 R M h v = R

Consider the conservation of energy of the system!! Rotations Two uniform cylinders are machined out of solid aluminum. One has twice the radius of the other. If both are placed at the top of the same ramp and released (they roll without slipping), which is moving faster (linear velocity) at the bottom? (a) bigger one (b) smaller one (c) same Consider the conservation of energy of the system!!

Rotations - Solution OK…let’s starts with our conservation of energy equation and go from there….

Torque If an applied force causes a mass to rotate (circular motion) then we have defined a torque:  = rF = r F sin  = r sin  F F F  Fr r

Rotational Dynamics: What makes it spin? Suppose a force acts on a mass constrained to move in a circle. Consider its acceleration in the direction at some instant: a = r Now use Newton’s 2nd Law in the  direction: F = ma = mr rF = mr2 r a  F m  F Multiply by r :

Rotational Dynamics: What makes it spin? So…. rF = mr2 &  = rF & Thus: Torque has a direction: + if it tries to make the system spin CCW. - if it tries to make the system spin CW. r a  F m  F

Rotational Dynamics: What makes it spin?  NET = I This is the rotational analogue of FNET = ma Torque is the rotational analogue of force: The amount of “twist” provided by a force. Moment of inertia I is the rotational analogue of mass. If I is big, more torque is required to achieve a given angular acceleration. Torque has units of kg m2/s2 = (kg m/s2) m = Nm.

Atwoods Machine with Massive Pulley A pair of masses are hung over a massive disk-shaped pulley as shown. What are the equations necessary to solve for the unknowns in the problem? Three objects have mass so we will have three equations. What does that imply about the number of unknowns? First step is to complete the free-body diagram! M R m1 m2

Atwood’s Machine with Massive Pulley... For the hanging masses use F = ma M For the pulley use  = I R m2 m2 m1 m1

What About Statics? In general, we can use the two equations to solve any statics problem. When choosing axes about which to calculate torque, we can be clever and make the problem easy....

Using Torque in Static Cases: Now consider a plank of mass M suspended by two strings as shown. We want to find the tension in each string: L/2 L/4 M x cm T1 T2 Mg y x

Statics A 1 kg ball is hung at the end of a rod 1 m long. The system balances at a point on the rod 0.25 m from the end holding the mass. What is the mass of the rod? (a) 0.5 kg (b) 1 kg (c) 2 kg 1 m 1 kg

Statics A (static) mobile hangs as shown below. The rods are massless and have lengths as indicated. The mass of the ball at the bottom right is 1kg. What is the total mass of the mobile? (a) 5 kg (b) 6 kg (c) 7 kg 1 m 2 m 1 kg 1 m 3 m

Angular momentum of a rigid body about a fixed axis: We define the rotational analogue of momentum p to be angular momentum Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle: Using vi =  ri , we get: v1 m2 y r2 r1 m1 I  x v2 Analogue of p = mv!!  r3 m3 v3

Example: Rotating Table... f Ii If Li Lf

End of Rotational Motion Lecture