Physics 218 Lecture 21 Dr David Toback Physics

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

Checklist for Today • Things due Yesterday – Chapters 12 & 13 in Web. CT • Things that are due for today – Read Chapters 14 -16 • Things that are due tomorrow for Recitation – Chapter 14 problems – Read Lab hand out on webpage • Things due next Monday – Chapter 14 in Web. CT Physics 218, Lecture XXI 2

The Schedule This week (4/7) • Mon: Chapter 12 & 13 material due in Web. CT • Today: Reading: Chap 14 -16 • Wed: Recitation on Chap 14, Lab • Thurs Lecture: Chap 15, Part 1 Next Week (4/14) • Monday: Chapter 14 due in Web. CT • 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 Web. CT • 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, 1 PM-3 PM in this room Physics 218, Lecture XXI 3

Overview • Chapters 12 -16 are about Rotational Motion • While we’ll do Exam 3 on Chapters 1013, we’ll do the lectures on 12 -16 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

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

Angular Quantities Last time: • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a 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

Physics 218, Lecture XXI 7

Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a Moving forward: – Force – Mass – Momentum – Energy Physics 218, Lecture XXI 8

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

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

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

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

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

Vector Cross Product 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

Vector Cross Product Cont… Multiply out, but use the Sinq to give the magnitude, and RHR to give the direction Physics 218, Lecture XXI 15

Cross Product Example Physics 218, Lecture XXI 16

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

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 Q 1 and Q 2. Q 1 is a right angle. If the axis is fixed, what is the net torque on the wheel? F 2 Q 1 F 1 Physics 218, Lecture XXI 18

Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a Moving forward: – Force Torque t – Mass – Momentum – Energy Physics 218, Lecture XXI 19

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 F=ma t=Ia Physics 218, Lecture XXI 20

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

Moment of Inertia • To find the mass of an object, just add up all the little pieces of mass ØTo find the moment of inertia around a point, just add up all the little moments Physics 218, Lecture XXI 22

Torque and Moment of Inertia • Force vs. Torque F=ma t = Ia • Mass vs. Moment of Inertia Physics 218, Lecture XXI 23

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 tfric. Find the angular acceleration a Physics 218, Lecture XXI 24

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

Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed w 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 tfric. What is this better estimate of the moment of Inertia? Physics 218, Lecture XXI R 26

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

Physics 218, Lecture XXI 28

Example of Cross Product The location of a body is length r from the origin and at an angle q from the x-axis. A force F acts on the body purely in the y direction. What is the Torque on the body? z y q x Physics 218, Lecture XXI 29

Calculate Moment of Inertia 1. Calculate the moment of inertia for a ball of mass m relative to the center of the circle R 2. What about lots of points? For example a wheel Physics 218, Lecture XXI 30

Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml 2/3 rotates around a pivot. It is held horizontally and released. Find the angular acceleration a and the linear acceleration a at the end. Where, along the rod, is a = g? Physics 218, Lecture XXI 31