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

2. 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

3. 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

4. Overview Chapters 12-16 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

5. 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

6. 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

7. Physics 218, Lecture XXI

8. 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

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

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

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

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

13. 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

14. 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

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

16. Cross Product Example
Physics 218, Lecture XXI

17. 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

18. 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

19. 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

20. 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

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

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
Physics 218, Lecture XXI

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

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

25. 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

26. 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

27. 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

28. End of Lecture Notes
Physics 218, Lecture XXI

29. 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

30. 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

31. 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

32. 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

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

34. Moments of Inertia
Physics 218, Lecture XXI