Physics 218, Lecture XVIII1 Physics 218 Lecture 18 Dr. David Toback.

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


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


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


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


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


Rotational Motion

•Here we’re talking about stuff that goes around and around

•Start by envisioning:

A spinning object like a car

tire


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?


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


Axis of Rotation: Definitions

Pick a simple place to rotate around

Call point Othe “Axis of

Rotation”Same as

picking an origin


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


Velocity and Acceleration

22

2

secradians/ dtd

or dtd

onaccelerati angular the as Define

secradians/ dtd

or t

velocity angular the as Define


Motion on a Wheel

What is the linear speed of a point rotating around in a circle with angular speed , and constant radius R?


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


Angular Velocity and Acceleration

Are and vectors?

and clearly have magnitude

Do they have direction?


Right-Hand RuleYes!Define the direction to point along the axis of rotation

Right-hand Rule

This is true for and


Uniform Angular Acceleration

Derive the angular equations of motion for constant angular

acceleration

t

t21

t

0

200


Rotation and Translation

Objects can both translate and rotate at the same time. They do both around their center of mass.


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?


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


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?


Show

Show for constant acceleration that:

220

2f


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


Uniform Circular Motion - Velocity

•Velocity vector = |V| tangent to the circle

•Is this ball accelerating?

–Yes! why?


Centripetal Acceleration

• “Center Seeking”

• Acceleration vector= V2/R towards the center

• Acceleration is perpendicular to the velocity

)r̂(Rv

a2

R

direction r̂


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


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?


The Trick To Solving Problems

)r̂(Rv

m

amF2


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?


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?


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


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


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


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?


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


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?


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


More definitions

•Frequency = Revolutions/sec

radians/sec f = /2

•Period = 1/freq = 1/f


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?

Physics 218, Lecture XVIII1 Physics 218 Lecture 18 Dr. David Toback.

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