Physics 218, Lecture XVIII 1 Physics 218 Lecture 18 Dr. David Toback
Dec 20, 2015
Physics 218, Lecture XVIII
1
Physics 218Lecture 18
Dr. David Toback
Physics 218, Lecture XVIII 2
Checklist for Today•Things that were due Monday:
–Chapter 9 HW on WebCT•Things that are due yesterday for
Recitation–Chapter 10 problems
•Things due Today:–Read Chapters 12 & 13
•Things due next Monday–Chapter 10 in WebCT
Physics 218, Lecture XVIII 3
The ScheduleRest of this Week: (3/24)• Lecture on Chaps 12 & 13Next Week (3/31) • Mon: Chapter 10 & 11 due material in WebCT• Tues: Second lecture on Chaps 12 & 13• Wed: Recitation on Chapters 12 & 13, Lab• Reading for Thurs: Chapters 14-16• Thurs Lecture: Chap 14 Week after that (4/7)• Mon: Chapter 12 & 13 due material in WebCT• Tues: Second Lecture on Chap 14• Wed: Recitation on Chap 14, Lab• Thurs Lecture: Chap 15, Part 1Week of 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
Physics 218, Lecture XVIII 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 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 XVIII 5
Rotational Motion
•Start with Fixed Axis motion•The relationship between linear
and angular variables•Rotating and translating at the
same time•First kinematics, then dynamics
–just like earlier this semester
Physics 218, Lecture XVIII 6
Physics 218, Lecture XVIII 7
Overview: Rotational Motion
• Take our results from “linear” physics and do the same for “angular” physics
• We’ll discuss the analogue of – Position– Velocity– Acceleration– Force– Mass– Momentum– Energy
Physics 218, Lecture XVIII 8
Rotational Motion
•Here we’re talking about stuff that goes around and around
•Start by envisioning:
A spinning object like a car
tire
Physics 218, Lecture XVIII 9
Some Buzz Phrases•Fixed axis: I.e, an object spins
in the same place… an ant on a spinning top goes around the same place over and over againAnother example: Earth has a fixed axis, the sun
•Rigid body: I.e, the objects don’t change as they rotate. Example: a bicycle wheel
Examples of Non-rigid bodies?
Physics 218, Lecture XVIII 10
Overview: Rotational Motion
• Take our results from “linear” physics and do the same for “angular” physics
• Analogue of –Position ←–Velocity ←–Acceleration ←–Force–Mass–Momentum–Energy
Start here!
Ch
ap
ters
1-3
Physics 218, Lecture XVIII 11
Axis of Rotation: Definitions
Pick a simple place to rotate around
Call point Othe “Axis of
Rotation”Same as
picking an origin
Physics 218, Lecture XVIII 12
An Important Relation: Distance
R2Circ
θRl
If we are sitting at a
radius R relative to
our axis, and we rotate
through an angle , then we travel
through a distance l
Physics 218, Lecture XVIII 13
Velocity and Acceleration
22
2
secradians/ dtd
or dtd
onaccelerati angular the as Define
secradians/ dtd
or t
velocity angular the as Define
Physics 218, Lecture XVIII 14
Motion on a Wheel
What is the linear speed of a point rotating around in a circle with angular speed , and constant radius R?
Physics 218, Lecture XVIII 15
Examples
Consider two points on a rotating wheel. One on the inside (P) and the other at the end (b):
•Which has greater angular speed?
•Which has greater linear speed?
b
R1
R2
Physics 218, Lecture XVIII 16
Angular Velocity and Acceleration
Are and vectors?
and clearly have magnitude
Do they have direction?
Physics 218, Lecture XVIII 17
Right-Hand RuleYes!Define the direction to point along the axis of rotation
Right-hand Rule
This is true for and
Physics 218, Lecture XVIII 18
Uniform Angular Acceleration
Derive the angular equations of motion for constant angular
acceleration
t
t21
t
0
200
Physics 218, Lecture XVIII 19
Rotation and Translation
Objects can both translate and rotate at the same time. They do both around their center of mass.
Physics 218, Lecture XVIII 20
Rolling without Slipping
•In reality, car tires both rotate and translate
•They are a good example of something which rolls (translates, moves forward, rotates) without slipping
•Is there friction? What kind?
Physics 218, Lecture XVIII 21
Derivation• The trick is to pick your
reference frame correctly!
• Think of the wheel as sitting still and the ground moving past it with speed V.
Velocity of ground (in bike frame) = -R
=> Velocity of bike (in ground frame) = R
Physics 218, Lecture XVIII 22
Bicycle comes to RestA bicycle with initial linear velocity V0
(at t0=0) decelerates uniformly (without slipping) to rest over a distance d. For a wheel of radius R:
a)What is the angular velocity at t0=0?b)Total revolutions before it stops?c)Total angular distance traversed
by the wheel?d)The angular acceleration?e)The total time until it stops?
Physics 218, Lecture XVIII 23
Show
Show for constant acceleration that:
220
2f
Physics 218, Lecture XVIII 24
Uniform Circular Motion
• Fancy words for moving in a circle with constant speed
• We see this around us all the time– Moon around the earth– Earth around the sun– Merry-go-rounds
• Constant and Constant R
Physics 218, Lecture XVIII 25
Uniform Circular Motion - Velocity
•Velocity vector = |V| tangent to the circle
•Is this ball accelerating?
–Yes! why?
Physics 218, Lecture XVIII 26
Centripetal Acceleration
• “Center Seeking”
• Acceleration vector= V2/R towards the center
• Acceleration is perpendicular to the velocity
)r̂(Rv
a2
R
direction r̂
Physics 218, Lecture XVIII 27
Circular Motion: Get the speed!
Speed = distance/time Distance in 1 revolution
divided by the time it takes to go around once
Speed = 2r/TNote: The time to go around once is known as the Period, or T
Physics 218, Lecture XVIII 28
Ball on a String
A ball at the end of a string is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t.
What is the centripetal acceleration?
Physics 218, Lecture XVIII 29
The Trick To Solving Problems
)r̂(Rv
m
amF2
Physics 218, Lecture XVIII 30
Banking Angle
You are a driver on the NASCAR circuit. Your car has m and is traveling with a speed V around a curve with Radius R
What angle, , should the road be banked so that no friction is required?
Physics 218, Lecture XVIII 31
Skidding on a CurveA car of mass m rounds a curve on a flat road of radius R at a speed V. What coefficient of friction is required so there is no skidding?Kinetic or static friction?
Physics 218, Lecture XVIII 32
Conical PendulumA small ball of
mass m is suspended by a cord of length L and revolves in a circle with a radius given by
r = Lsin.
1.What is the velocity of the ball?
2.Calculate the period of the ball
Physics 218, Lecture XVIII 33
Exam 2• Class average for the 2nd exam (including
the 5 points) was 80.1%
– Average for first two exams is a 78.7%
• Straight scale for curve for now• Many have asked “should I q-drop?”
– Talk to your advisor and read my FAQ!
– Generic advice: Drop if you can’t keep up with the homework by yourself
Physics 218, Lecture XVIII 34
Next Week
•Monday: Chapter 10 & 11 due material in WebCT
•Tuesday: Second lecture on Chaps 12 & 13
•Wednesday Recitation: Recitation on Chapters 12 & 13
•Wednesday Lab: Elastic Collisions
Physics 218, Lecture XVIII 35
Physics 218, Lecture XVIII 36
Circular Motion Example
A ball of mass m is at the end of a string and is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t.
a)What is the centripetal acceleration?
b)What is the centripetal force?
Physics 218, Lecture XVIII 37
Next Time
• Exam 2 is Thursday!• Bonus Points for getting a 100 on the
mini-practice exam BEFORE the in-class exam
Next week:• Chapter 8HW due Monday morning• Lecture next Thursday will cover
Chapters 9 and 10:– Reading questions due: Q10.7 &
Q10.26
Physics 218, Lecture XVIII 38
Computer Hard Drive
A computer hard drive typically rotates at 5400 rev/minute
Find the: •Angular Velocity in rad/sec•Linear Velocity on the rim (R=3.0cm)•Linear AccelerationIt takes 3.6 sec to go from rest to 5400 rev/min, with constant angular acceleration.
•What is the angular acceleration?
Physics 218, Lecture XVIII 39
Next Time
• Read Chapter 10– More on angular “Stuff”– Angular kinematics– Torque– Reading questions: Q10.7 & Q10.26
• HW7 Due Monday (released this afternoon)
• Exam 2 next Thursday on Chapters 4-7
Physics 218, Lecture XVIII 40
More definitions
•Frequency = Revolutions/sec
radians/sec f = /2
•Period = 1/freq = 1/f
Physics 218, Lecture XVIII 41
Motion on a Wheel cont…
A point on a circle, with constant radius R, is rotating with some speed and an angular acceleration . What is the linear acceleration?