Physics 218 Lecture 21

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1 Physics 218 Lecture 21 Dr. David Toback Physics 218, Lecture XXI 1

2 Checklist for Today Things due Yesterday Chapters 12 & 13 in WebCT Things that are due for today Read Chapters Things that are due tomorrow for Recitation Chapter 14 problems Read Lab hand out on webpage Things due next Monday Chapter 14 in WebCT Physics 218, Lecture XXI 2

3 The Schedule This week (4/7) Mon: Chapter 12 & 13 material due in WebCT Today: Reading: Chap Wed: Recitation on Chap 14, Lab Thurs Lecture: Chap 15, Part 1 Next Week (4/14) Monday: Chapter 14 due in WebCT Tues: Exam 3 (Chaps 10-13) Wed: Recitation on Chap 15, Lab Thurs: Lecture on Chap 15, Part 2 Week after that (4/21) Monday: Chapter 15 & 16 due in WebCT Tues: Reading for Chapter 18, Lecture on Chapter 18 Wed: Recitation on Chapter 18, Lab Thurs: Last lecture, Chapter 18 Week after that (4/28) No lectures or recitations Week after that (5/5) Final: Monday May 5 th, 1PM-3PM in this room Physics 218, Lecture XXI 3

4 Overview Chapters are about Rotational Motion While we ll do Exam 3 on Chapters 10-13, we ll do the lectures on in six combined lectures Give extra time after the lectures to Study for the exam The book does the math, I ll focus on the understanding and making the issues more intuitive Physics 218, Lecture XXI 4

5 Rotational Motion Chapters 12 through 16 in six combined lectures This is the 4 th of the 6 lectures Concentrate on the relationship between linear and angular variables Already did kinematics Move to dynamics just like earlier this semester Physics 218, Lecture XXI 5

6 Angular Quantities Last time: Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α This time we ll start by discussing the vector nature of the variables and then move forward on the others: Force Mass Momentum Energy Physics 218, Lecture XXI 6

7 Physics 218, Lecture XXI 7

8 Angular Quantities Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α Moving forward: Force Mass Momentum Energy Physics 218, Lecture XXI 8

9 Torque Torque is the analogue of Force Take into account the perpendicular distance from axis Same force further from the axis leads to more Torque Physics 218, Lecture XXI 9

10 Slamming a door We know this from experience: If we want to slam a door really hard, we grab it at the end If we try to push in the middle, we aren t able to make it slam nearly as hard Physics 218, Lecture XXI 10

11 Torque Continued What if we change the angle at which the Force is applied? What is the Effective Radius? Physics 218, Lecture XXI 11

12 Slamming a door We also know this from experience: If we want to slam a door really hard, we grab it at the end and throw perpendicular to the hinges If we try to pushing towards the hinges, the door won t even close Physics 218, Lecture XXI 12

13 Torque Torque is our slamming ability Need some new math to do Torque Write Torque as τ τ r F sinθ r r r τ r F To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist Physics 218, Lecture XXI 13

14 Vector Cross Product C r r A B r C A B Sin Θ This is the last way of multiplying vectors we will see Direction from the right-hand rule Swing from A into B! Physics 218, Lecture XXI 14

15 Vector Cross Product Cont Multiply out, but use the Sinθ to give the magnitude, and RHR to give the direction î î 0 (sin θ 0 ) î ĵ kˆ (sin θ 1 ) î kˆ ĵ (sin θ 1 ) Physics 218, Lecture XXI 15

16 Cross Product Example r A r A X î + A Y ĵ B B X î + r B Y r ĵ What is A B using Unit Vector notation? Physics 218, Lecture XXI 16

17 Torque and Force Torque problems are like Force problems 1. Draw a force diagram 2. Then, sum up all the torques to find the total torque Is torque a vector? Physics 218, Lecture XXI 17

18 Example: Composite Wheel Two forces, F 1 and F 2, act on different radii of a wheel, R 1 and R 2, at different angles Θ 1 and Θ 2. Θ 1 is a right angle. If the axis is fixed, what is the net torque on the wheel? Θ 1 Θ 2 F 2 F 1 Physics 218, Lecture XXI 18

19 Angular Quantities Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α Moving forward: Force Torque τ Mass Momentum Energy Physics 218, Lecture XXI 19

20 Analogue of Mass The analogue of Mass is called Moment of Inertia Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia Fma τια Physics 218, Lecture XXI 20

21 Calculate Moment of Inertia Calculate the moment of inertia for a ball of mass m relative to the center of the circle R Physics 218, Lecture XXI 21

22 Moment of Inertia To find the mass of an object, just add up all the little pieces of mass To find the moment of inertia around a point, just add up all the little moments I mr 2 or I r dm 2 Physics 218, Lecture XXI 22

23 Torque and Moment of Inertia Force vs. Torque Fma τ Iα Mass vs. Moment of Inertia m I mr 2 or I r 2 dm Physics 218, Lecture XXI 23

24 Pulley and Bucket A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is τ fric. Find the angular acceleration α Physics 218, Lecture XXI 24

25 Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force F T. It accelerates to an angular speed of ω in time t. What is the moment of inertia of the pulley? R Physics 218, Lecture XXI 25

26 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force F T. It accelerates to final angular speed ω in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we ll call this τ fric. What is this better estimate of the moment of Inertia? R Physics 218, Lecture XXI 26

27 Next Time More on angular Stuff Angular Momentum Energy Get caught up on your homework!!! Mini-practice exam 3 is now available Physics 218, Lecture XXI 27

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