physics 218 chapter 12-16
TRANSCRIPT
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
Dynamics of Rotational Motion 5
– 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
Dynamics of Rotational Motion 6
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
Dynamics of Rotational Motion 8
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
Dynamics of Rotational Motion 9
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
Dynamics of Rotational Motion 10
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
Dynamics of Rotational Motion 11
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?
Dynamics of Rotational Motion 12
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?
Dynamics of Rotational Motion 13
Angular Velocity and Accelerationg y
Are and vectors?Are and vectors?
and clearly have and clearly have magnitudeg
Do they have direction?y
Dynamics of Rotational Motion 14
Right-Hand RulegYes!Define the direction to
i t l point along the axis of rotationrotation
Right-hand lRule
This is true for and
Dynamics of Rotational Motion 15
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.
Dynamics of Rotational Motion 17
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
Dynamics of Rotational Motion 19
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?
Dynamics of Rotational Motion 21
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
Dynamics of Rotational Motion 23
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?
Dynamics of Rotational Motion 24
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
Dynamics of Rotational Motion 25
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?
Dynamics of Rotational Motion 29
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?
Dynamics of Rotational Motion 30
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
Dynamics of Rotational Motion 31
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?
Dynamics of Rotational Motion 32
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?
Dynamics of Rotational Motion 33
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
Dynamics of Rotational Motion 34
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
Dynamics of Rotational Motion 37
Energy
Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration Moving forward (chapter 14 today):
– Force– MassMass– Momentum– Energy
Dynamics of Rotational Motion 38
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
Dynamics of Rotational Motion 39
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
Dynamics of Rotational Motion 40
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?
Dynamics of Rotational Motion 41
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
Dynamics of Rotational Motion 42
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
Dynamics of Rotational Motion 43
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!
Dynamics of Rotational Motion 44
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
Dynamics of Rotational Motion 45
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
Dynamics of Rotational Motion 49
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 =
Dynamics of Rotational Motion 50
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
Dynamics of Rotational Motion 51
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
Dynamics of Rotational Motion 52
Torque and Moment of Inertiaq
• Force vs. Torque Force vs. Torque F=ma = I
• Mass vs. Moment of Inertia
or mrIm 2
dmrI 2
Dynamics of Rotational Motion 53
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
Dynamics of Rotational Motion 54
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?
Dynamics of Rotational Motion 55
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?
Dynamics of Rotational Motion 56
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
Dynamics of Rotational Motion 57
Angular Quantitiesg Q• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Acceleration Angular Acceleration Moving forward:
– Force Torque – MassMass– Momentum– Energy
Dynamics of Rotational Motion 58
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
Physics 218, Lecture XXII 60
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
Physics 218, Lecture XXII 63
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
Physics 218, Lecture XXII 64
then pull them in
Problem SolvinggFor Conservation of Angular Momentum
problems:
BEFORE and AFTERBEFORE and AFTER
Physics 218, Lecture XXII 65
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
Physics 218, Lecture XXII 68
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
Physics 218, Lecture XXII 69
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
Physics 218, Lecture XXII 70
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
Physics 218, Lecture XXII 71
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
Physics 218, Lecture XXII 72
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?
Physics 218, Lecture XXII 73
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?
Physics 218, Lecture XXII 74
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?
Physics 218, Lecture XXII 75
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
Physics 218, Lecture XXII 76
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”
Physics 218, Lecture XXII 78
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
Dynamics of Rotational Motion 81
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
Dynamics of Rotational Motion 82
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
Dynamics of Rotational Motion 83
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