Physics 218 Chapter 12-16people.physics.tamu.edu/mahapatra/teaching/ch12-16.pdfPhysics 218 Chapter 12-16 Prof. Rupak Mahapatra Dynamics of Rotational Motion 1 Overview • Chapters

Physics 218Physics 218Chapter 12 16Chapter 12-16

Prof. Rupak Mahapatra

Dynamics of Rotational Motion

1

Overview• Chapters 12-16 are about Rotational

M iMotion• While we’ll do Exam 3 on Chapters 10-p

13, we’ll do the lectures on 12-16 in six combined lectures

• Give extra time after the lectures to Study for the examy

• The book does the math, I’ll focus on the understanding and making the issues the understand ng and mak ng the ssues more intuitive

Dynamics of Rotational Motion 2

Rotational Motion• Start with Fixed Axis motionStart w th F s mot on• The relationship between linearand angular variables

• Rotating and translating at the • Rotating and translating at the same time

• First kinematics, then dynamics –just like earlier this semester

Dynamics of Rotational Motion 3semester


Overview: Rotational Motion• Take our results from “linear” physics p yand do the same for “angular” physics

• We’ll discuss the analogue of We ll discuss the analogue of – PositionVelocity– Velocity

– Acceleration– Force– Mass– Momentum– Energy


– Energy

Rotational Motion• Here we’re talking about stuff that g ffgoes around and aroundSt t b i i i• Start by envisioning:

A i i bj A spinning object p g jlike a car tire a car t r


Some Buzz Phrases• Fixed axis: I.e, an object spins in th l t the same place… an ant on a spinning top goes around the same l d iplace over and over again

Another example: Earth has a fixed mp faxis, the sun

• Rigid body: I e the objects don’t Rigid body: I.e, the objects don t change as they rotate. Example: a bicycle wheelbicycle wheel

Examples of Non-rigid bodies?Dynamics of Rotational Motion 7

Overview: Rotational Motion• Take our results from “linear” physics and

do the same for “angular” physicsdo the same for angular physics• Analogue of

P siti n 1-3

– Position ←–Velocity ← Start

h ! ters

y–Acceleration ←– Force

here!

Chap

t

– Force–Mass

C

–Momentum–Energy


Energy

Axis of Rotation: DefinitionsPick a simple

l t place to rotate around

Call point OCall point Othe “Axis of

Rotation”RotationSame as picking

an origing


An Important Relation: DistancepIf we are sitting at a radius Rradius R

relative to our axis and we axis, and we

rotate through l

θRl an angle ,

then we travel

R2Circ through a distance l


stanc

Velocity and Accelerationy

velocity angular the as Define

di / d

velocity angular the as Define

secradians/ dt

or t

onaccelerati angular the as Define

22

di / d d

onaccelerati angular the as Define

22 secradians/

dtd or

dtd


Motion on a WheelWhat is the linear speed of a point rotating a point rotating around in a

l h circle with angular speed angular speed , and constant r dius R?radius R?


ExamplespConsider two points on a protating wheel. One on the inside (P) and the bthe inside (P) and the other at the end (b):Whi h h t

R1

•Which has greater angular speed?

R2

•Which has greater linear speed?linear speed?


Angular Velocity and Accelerationg y

Are and vectors?Are and vectors?

and clearly have and clearly have magnitudeg

Do they have direction?y


Right-Hand RulegYes!Define the direction to

i t l point along the axis of rotationrotation

Right-hand lRule

This is true for and


Uniform Angular AccelerationgDerive the angular equations of motion g q f

for constant angular acceleration

t1t 2 t2

t 200

t2

t0 Dynamics of Rotational Motion 16

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


Rolling without Slippingg pp g• In reality, car tires both y,rotate and translateTh d l f • They are a good example of something which rollsg(translates, moves forward, rotates) without slippingrotates) without slipping

• Is there friction? What kind?Dynamics of Rotational Motion 18

Derivation• The trick is to pick your

reference frame correctly!• Think of the wheel as sitting g

still and the ground moving past it with speed V.p p

Velocity of ground (in bike frame) = -Rframe) = R

=> Velocity of bike (in ground frame) = Rframe) = R


Try Differently: Paper Rolly y p• A paper towel unrolls

ith l it Vwith velocity V– Conceptually same thi th h lthing as the wheel

– What’s the velocity f i t of points:

– A? B? C? D? ABC ABC

D• Point C is where rolling

part separates from the D p punrolled portion

Both have same velocity

C B A– Both have same velocity

there 20Dynamics of Rotational Motion

Bicycle comes to RestyA bicycle with initial linear velocity V0 (at

t =0) decelerates uniformly (without slipping) 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?) f pc) Total angular distance traversed

by the wheel?d) The angular acceleration?e) The total time until it stops?


Uniform Circular Motion• Fancy words for moving in a circle y gwith constant speed

• We see this around us all the time• We see this around us all the time– Moon around the earth– Earth around the sunMerry go rounds– Merry-go-rounds

• Constant and Constant R


Uniform Circular Motion - Velocityy

• Velocity vector Velocity vector = |V| tangent to the circle

h b ll • Is this ball accelerating?accelerating?–Yes! why?–Yes! why?


Centripetal Accelerationp

• “Center Seeking” 2Center Seeking• Acceleration

V2/R )r(v a2

vector= V2/Rtowards the

)r(R

a

centerA l ti n i

direction r

• Acceleration is perpendicular to Rthe velocity


The Trick To Solving Problemsg

amF amF

2

)r(vm

2

)r(R

m


Banking Angleg gYou 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 CurvegA car of mass m rounds a curve on a fl t d f di R t d V flat road of radius R at a speed V. What coefficient of friction is

i d th i kiddi ?required so there is no skidding?Kinetic or static friction?


Conical PendulumA small ball of mass m

i d d b is suspended by a cord of length Land revolves in a and revolves in a circle with a radius given by given by

r = Lsin.1.What is the velocity 1.What is the velocity

of the ball?2. Calculate the period p

of the ball


Circular Motion ExamplepA ball of mass m is at the end of a f f

string and is revolving uniformly in a horizontal circle (ignore gravity) a horizontal circle (ignore gravity) of radius R. The ball makes Nrevolutions in a time t.

a)What is the centripetal a)What is the centripetal acceleration?

b)What is the centripetal force?


Computer Hard DrivepA computer hard drive typically rotates at 5400 / i t5400 rev/minute

Find the: •Angular Velocity in rad/sec• Linear Velocity on the rim (R=3.0cm)y ( )• Linear AccelerationIt takes 3.6 sec to go from rest to 5400 It takes 3.6 sec to go from rest to 5400 rev/min, with constant angular acceleration.

•What is the angular acceleration?What is the angular acceleration?


More definitions• Frequency = Revolutions/secFr qu ncy o ut ons/s c

radians/sec f = radians/sec f = /2/2

• Period = 1/freq = 1/f• Period = 1/freq = 1/f


Motion on a Wheel cont…A point on a pcircle, with

t t di Rconstant radius R, is rotating with gsome speed and

lan angular acceleration .acceleration .What is the linear

Dynamics of Rotational Motion 35acceleration?


Angular Quantitiesg QLast time:

P i i A l • Position Angle • Velocity Angular Velocity

A l i A l A l i• Acceleration Angular Acceleration This time we’ll start by discussing the

t t f th i bl d th vector nature of the variables and then move forward on the others:

F– Force– Mass– Momentum– Energy


Energy

Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration Moving forward (chapter 14 today):

– Force– MassMass– Momentum– Energy


Torqueq• Torque is the analogue of Force• Take into account the perpendicular

distance from axis– Same force further from the axis leads to more Torqueto more Torque


Slamming a doorgWe know this from experience:We know this from experience–If we want to slam a door really hard, we grab it at the endend

–If we try to push in the If we try to push in the middle, we aren’t able to

k l l h dmake it slam nearly as hard


Torque Continuedq

• What if we What if we change the angle at which angle at which the Force is

li d?applied?• What is the What is the “Effective Radius?”Radius?


Slamming a doorgWe also know this from experience:If t t l d – If we want to slam a door really hard, we grab it at the y , gend and “throw” perpendicular to the hingesto the hinges

– If we try to pushing towards y p gthe hinges, the door won’t even close


even close

Torqueq• Torque is our “slamming” ability• Need some new math to do Torque

Write Torque as Write Torque as

sin|F||r|||

Fr||||||

• To find the direction of the torque, wrap

Fr To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist


Vector Cross Product

SinB ACB A C

SinB AC

This is the last way of multiplying vectors we p y gwill see

•Direction from the “ i ht h d l ”“right-hand rule”

•Swing from A into B!


Vector Cross Product Cont…Multiply out, but use the p ySin to give the magnitude and RHR to magnitude, and RHR to give the directionˆˆ

)1( i kji

)0(sin 0ii

)1( i jki

)1(sin kji

)1(sin jki


Cross Product Examplep

ˆˆj A i A A YX

j B i B B

i BA i Whj B i B B YX

using BA is What

notation? Vector Unit Dynamics of Rotational Motion 46

Torque and ForceqTorque problems are like Force q p

problems1 D f di1. Draw a force diagram2 Then sum up all the torques 2. Then, sum up all the torques

to find the total torque

Is t rque vect r?Is torque a vector?Dynamics of Rotational Motion 47

Example: Composite Wheelp pTwo forces, F1 and

F F2F2, act on different radii of a wheel R and R

F2

wheel, R1 and R2, at different angles 1 and 2 1 is a 1 and 2. 1 is a right angle.

If the axis is fixed If the axis is fixed, what is the net torque on the

torque on the wheel?

F1Dynamics of Rotational Motion 48

1

Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration Moving forward:

– Force Torque – MassMass– Momentum– Energy


Analogue of MassgThe analogue of g fMass is called Moment of Moment of Inertia

Example: A ball of mass m moving in a gcircle of radius Raround a point has a pmoment of inertiaF=ma =


F=ma =

Calculate Moment of Inertia

Calculate the Calculate the moment of inertia for a ball f mass ball of mass m relative to m relative to the center of the circle R


Moment of Inertia•To find the mass of an object, just add up all the li l i f little pieces of massT fi d th m m t f To find the moment of inertia around a point just inertia around a point, just add up all the little moments

dmrI or mrI 22

p


Torque and Moment of Inertiaq

• Force vs. Torque Force vs. Torque F=ma = I

• Mass vs. Moment of Inertia

or mrIm 2

dmrI 2


Pulley and BucketyA heavy pulley, with y p yradius R, and known moment of inertia Imoment of inertia Istarts at rest. We attach it to a bucket attach it to a bucket with mass m. The f i ti t i friction torque is fric.

Find the angular F gacceleration


Spherical Heavy Pulleyp y yA heavy pulley, with Rradius R, starts at rest. We pull on an

R

attached rope with a constant force FT. It accelerates to an angular speed of in time t.

What is the moment of m m finertia of the pulley?


Less Spherical Heavy Pulleyp y yA heavy pulley, with radius

R W ll RR, starts at rest. We pull on an attached rope with constant force FT It

Rconstant force FT. It accelerates to final angular speed in time t.

A better estimate takes into account that there is friction in the system This friction in the system. This gives a torque (due to the axel) we’ll call this fric.fric

What is this better estimate of the moment of Inertia?


Next TimeMore on angular “Stuff”M gu uff–Angular Momentum g–EnergyG h •Get caught up on your homework!!!homework!!!

•Mini-practice exam 3 is Mini practice exam 3 is now available


Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration Moving forward:

– Force Torque – MassMass– Momentum– Energy


MomentumMomentum vs. Angular Momentum:

ILvmp ILvmp

Newton’s Laws:

LdpdF

dtdt

F

Physics 218, Lecture XXII 59

Angular MomentumgFirst way to define the Angular Momentum L:

Ld)L(d)I(ddII

Lddtdtdtdt

II

dtLdI


Angular Momentum DefinitiongAnother Another definition:

p r L p r L


Angular Motion of a ParticlegDetermine the angular momentum L momentum, L, of a particle, with mass mand speed v and speed v, moving in i l ti circular motion

with radius rPhysics 218, Lecture XXII 62

w u

Conservation of Angular Momentumg

Ld

dtLd

Const L 0 if B N ’ l h l f By Newton’s laws, the angular momentum of

a body can change, but the angular momentum for a system cannot change

C ns rv ti n f An ul r M m ntumConservation of Angular MomentumSame as for linear momentum


Same as for linear momentum

Ice Skater• This one you’ve

IL y

seen on TV• Try this at home

IL Try this at home in a chair that rotatesrotates

• Get yourself spinning with your arms and legs gstretched out, then pull them in


then pull them in

Problem SolvinggFor Conservation of Angular Momentum

problems:

BEFORE and AFTERBEFORE and AFTER



Before



After


Clutch DesigngAs a car engineer, you model a car clutch as model a car clutch as two plates, each with radius R, and masses ,MA and MB (IPlate = ½MR2). Plate A spins with speed and plate with speed 1 and plate B is at rest. you close them so they spin y ptogether

Find the final angular l it f th tvelocity of the system


Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration • Force Torque

M M t f I ti • Mass Moment of Inertia Today we’ll finish:

– Momentum Angular Momentum LEnergy– Energy


Rotational Kinetic Energygy

KEtrans = ½mv2KEtrans ½mv KE t t = ½I2 KErotate = ½I

Conservation of Energy must take rotational kinetic energy into accountrotational kinetic energy into account


Rotation and Translation• Objects can both Rotate and

TranslateTranslate

• Need to add the two• Need to add the two

KEtotal = ½ mv2 + ½I2

• Rolling without slipping is a special case where you can special case where you can relate the twoV = rV = r


Rolling Down an InclinegYou take a solid ball of mass m and radius R and

h ld i l i h h i h Z Y hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping.

Wh t ill b th d f th b ll t th b tt ?What will be the speed of the ball at the bottom?What would be the speed if the ball didn’t roll and

there were no friction?there were no friction?

Note: / 2Isphere = 2/5MR2

Z


A bullet strikes a cylinderyA bullet of speed Vand mass m strikes a and mass m strikes a solid cylinder of mass M and inertia I=½MR2, at radius Rand sticks. The cylinder is anchored cylinder is anchored at point 0 and is initially at rest.y

What is of the system after the

lli i ?collision?Is energy Conserved?


Rotating RodgA rod of mass uniform

density mass m and density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and of inertia I=ml/3 and starts at rest at a right angle. You let it :it go:

What is when it reaches the bottom?m

What is the velocity of the tip at the bottom?bottom?


Less Spherical Heavy Pulleyp y yA heavy pulley, with radius

R, starts at rest. We pull RR, starts at rest. We pull on an attached rope with constant force FT. It

l t t fin l n l

R

accelerates to final angular speed in time t.

A better estimate takes into A better estimate takes into account that there is friction in the system. This i (d h gives a torque (due to the

axel) we’ll call this fric.What is this better estimate What is this better estimate

of the moment of Inertia?


Person on a DiskA person with mass m

stands on the edge of a disk with radius R and moment ½MR2. Neither is moving.

The person then starts moving on the gdisk with speed V.

Find the angular Find the angular velocity of the disk


Same Problem: ForcesSame problem but with Forces


Next TimeExam 3!!!• Covers Chapters 10-13G t ht • Get caught up on your homework!!!homework!!!

• Mini-practice exam 3 is now il blavailable

Thursday:Thursday:- Finish up angular “Stuff”


p g



Example of Cross ProductpThe location of a body zis length r from the

origin and at an z

angle from the x-axis. A force F acts on the body purely in the y direction.

What is the Torque on the body?

yy

x


Calculate Moment of Inertia1.Calculate the

moment of inertia for a ball of mass for a ball of mass m relative to the center of the center of the circle R

2.What about lots of points? For f p Fexample a wheel


Rotating RodgA uniform rod of mass m, length l, and moment of

inertia I = ml2/3 rotates around a pivot It is inertia I = ml /3 rotates around a pivot. It is held horizontally and released.

Find the angular acceleration and the linear Find the angular acceleration and the linear acceleration a at the end. Where, along the rod, is a = g?g


Two weights on a bargFind the middled t emoment of i tiinertia for thefor the two different A

Dynamics of Rotational Motion 84Axes

Schedule ChangesgPlease see the handout for schedule

changes

N E 3 D tNew Exam 3 Date:Exam 3

Tuesday Nov 26thTuesday Nov. 26thDynamics of Rotational Motion 85

Moments of Inertia


Documents

Physics 218 Chapter 12-16people.physics.tamu.edu/mahapatra/teaching/ch12-16.pdfPhysics 218 Chapter 12-16 Prof. Rupak Mahapatra Dynamics of Rotational Motion 1 Overview • Chapters