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

Checklist for Today Things due Yesterday Chapters 12 & 13 in WebCT 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 WebCT Physics 218, Lecture XXI

The Schedule This week (4/7) Mon: Chapter 12 & 13 material due in WebCT 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 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 5th, 1PM-3PM in this room Physics 218, Lecture XXI

Overview Chapters 12-16 are about Rotational Motion While we’ll do Exam 3 on Chapters 10-13, 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

Rotational Motion Chapters 12 through 16 in six combined lectures This is the 4th 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

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

Physics 218, Lecture XXI

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

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

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

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

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

Torque Write Torque as t 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

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

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

Cross Product Example Physics 218, Lecture XXI

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

Example: Composite Wheel Two forces, F1 and F2, act on different radii of a wheel, R1 and R2, at different angles Q1 and Q2. Q1 is a right angle. If the axis is fixed, what is the net torque on the wheel? F2 Q2 Q1 F1 Physics 218, Lecture XXI

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

Analogue of Mass F=ma  t=Ia 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

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

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 Physics 218, Lecture XXI

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

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

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? R Physics 218, Lecture XXI

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? R Physics 218, Lecture XXI

More on angular “Stuff” Angular Momentum Energy 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

End of Lecture Notes Physics 218, Lecture XXI

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

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

Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml2/3 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

Find the moment of inertia for the two different Axes Two weights on a bar Find the moment of inertia for the two different Axes middle A B Physics 218, Lecture XXI

Please see the handout for schedule changes New Exam 3 Date: Exam 3 Tuesday Nov. 26th Physics 218, Lecture XXI

Moments of Inertia Physics 218, Lecture XXI