Chapters 10 & 11 – Rotational motion, torque, and angular momentum

PHY 113 C Fall 2013-- Lecture 12 110/08/2013

PHY 113 C General Physics I11 AM - 12:15 PM MWF Olin 101

Plan for Lecture 12:

Chapters 10 & 11 – Rotational motion, torque, and angular momentum

1. Torque

2. Angular momentum



Angular motion

angular “displacement” q(t)

angular “velocity” angular “acceleration”

dtd(t) ωα

dtd(t) θω

“natural” unit == 1 radian

Relation to linear variables: sq = r (qf-qi)

vq = r w

aq = r a

so180 radians 3.14159 radians


Special case of constant angular acceleration: a = a0:

w(t) = wi + a0 t

q(t) = qi + wi t + ½ a0 t2

( w(t))2 wi2 + 2 a0 (q(t) - qi )

angular “displacement” q(t)

angular “velocity” angular “acceleration”

dtd(t) ωα

dtd(t) θω

q(t)w(t)

Rotational motion


Webassign Assignment #10:

A centrifuge in a medical laboratory rotates at an angular speed of 3800 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.

Special case of constant angular acceleration: a = a0:

w(t) = wi + a0 t

q(t) = qi + wi t + ½ a0 t2

( w(t))2 wi2 + 2 a0 (q(t) - qi )

Recall:

( )( )( )

( ) revrad 2

s 60min 1

rev 482rev/min 3800

2

0ωFor 222

qqwa

i

i

t

(t)


Review of rotational energy associated with a rigid body

( )

iii

iii

iii

iiirot

rmI

Irm

rmvmK

2

222

22

here w21

21

21

21

:energy Rotational

ww

w


Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal

ji

k

iclicker exercise:Which moment of inertia is the smallest? (A) i (B) j (C) k

d dm m

IB=2md2 IC=2md2IA=0


( )( ) ( )( ) ( )( ) ( )( )( ) 222222 24222223 mkgxmIi

iiyy

From Webassign:


Digression – use of rotational energy in energy storage http://www.beaconpower.com/products/about-flywheels.asp

http://www.beaconpower.com/products/about-flywheels.asp


Beacon Power company (went bankrupt in 2011)

Continuing efforts – http://www.scientificamerican.com/article.cfm?id=energy-storage-role-in-electric-grid

http://www.scientificamerican.com/article.cfm?id=energy-storage-role-in-electric-grid




22

21

21

:object rolling ofenergy kinetic Total

CM

CMrottotal

MvI

KKK

w

CMvRdtdR

dtds

dtd

wq

qw

: thatNote

( )

22

222

21

21

21

CM

CM

CMrottotal

vMRI

MvRRI

KKK

w

CM CM

Physics of rolling --


Kinetic energy associated with rotation:

i

iirot rmIIK 2221 w

Distance to axis of rotation

Rolling: rotcomtot KKK

222

1 1

:slipping no is thereIf

comtot

com

vMR

IMK

Rv

w

Rolling motion reconsidered:


Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?

AB C

22 0.1 MRMRI A 22 5.0

21 MRMRIB

22 4.052 MRMRIC

( )2

22

/12

01210

MRIghv

vMR

IMMgh

UKUK

CM

CM

ffii


How can you make objects rotate?

Define torque:

t = r x F

t = rF sin q

r

F

q

αarτFraF

Imm

qsinr

q

F sin q


Another example of torque:


clockwise)(counter :T from Torque

)(clockwise :T from Torque

222

2

111

1

TR

TR

t

t

( )clockwise)(counter 52

(1)(5)(0.5)(15)

: torqueTotal15N 0.5m,

5N 1m, :Example

1122

22

11

Nm .Nm

TRTR

TRTR

t


Example form Webassign #11

X

t1

t3

t2

iclicker exerciseWhen the pivot point is O, which torque is zero?

A. t1?B. t2?C. t3?


Newton’s second law applied to rotational motion

dtdM

dtdm CM

totali

ii

ii

vFvF

Newton’s second law applied to center-of-mass motion

mi Fi

ri

dtdm

dtdm i

iiiii

iivrFrvF

( )

axis) principalabout rotating(for αωτ

rωrτ

rωvFrτ

IdtdI

dtdm

total

iiii

ii

iii

i

iidmI 2

di


An example:

A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s.

K = ½ I w2 t I a w wi at = atIn this case I = ½ m R2 and t = FR

R F

( ) JsN

Nmg

tFgRItFt

IFRIIK

Rg

mgItI

FRtIFR

625.275)3(80050m/s8.9

/21

21

21

21

22

222

2

2222

2

w

awa


Re-examination of “Atwood’s” machine

T1

T2

T2T1

IT1-m1g = m1a

T2-m2g = -m2a

t =T2R – T1R = I a = I a/R

R

212

12

212

12

τ

a

I/Rmmmm

RIg

I/Rmmmmg


Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m .

m1 m2v1 v2

hh/2

( ) ( )

( ) ( )( ) ( ) ( )( ) sm

RImmmmv

hgmhgm

vvmvmghm

UKUK

RI

ffii

/19.02.0/112

12

/

0

:energy ofon Conservati

2

221

21

21

221

1

2212

2212

121

1 2


Tl

2

21 MRI

IrTrT lu

a

Example from Webassign #10


Newton’s law for torque:

αωτ IdtdItotal

F

fs( )

FfMRI

IMRFf

IRfaRIaIRf

MafF

s

s

sCMCMs

CMs

312

21

2

2

cylinder, solid aFor

/11

/

a

Note that rolling motion is caused by the torque of friction:


Bicycle or automobile wheel:

t

fs

( )

τ/Rf

MRI

I/MRτ/Rf

RIaIfMaf

s

s

CMs

CMs

21

For

1

/αR-τ

2

2


iclicker exercise:What happens when the bicycle skids?

A. Too much torque is appliedB. Too little torque is appliedC. The coefficient of kinetic friction is too smallD. The coefficient of static friction is too smallE. More than one of these


Vector cross product; right hand rule

qsinBACBAC

ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

i i j j k ki j j i kj k k j ik i i k j


ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

i i j j k ki j j i kj k k j ik i i k j

For unit vectors:

i

k

j


y z x yx zx y z

y z x yx zx y z

A A A AA AA A A

B B B BB BB B B

ˆ ˆ ˆˆ ˆ ˆ

i j kA B i j k

More details of vector cross products:

( ) ( ) ( )y z z y x z z x x y y xA B A B A B A B A B A Bˆ ˆ ˆ A B i j k


iclicker exercise:What is the point of vector products

A. To terrify physics studentsB. To exercise your right handC. To define an axial vectorD. To keep track of the direction of rotation


( ) ( )prvrvrarτFr

aF

dtd

dtmd

dtdmm

m

From Newton’s second law:

( ) ( )prvr dtd

dtmd

ConsiderIs this

A. Wrong?B. Approximately right?C. Exactly right?

iclicker exercise:


From Newton’s second law – continued – conservation of angular momentum:

( )

(constant)

0 0 If

:Define

L

Lτ

prL

prτFr

dtd

dtd


Torque and angular momentum

Define angular momentum:

For composite object: L = Iw

constant 0 then 0 If

LLτ

Lωτ

dtd

dtd

dtdI

total

total

prL

Newton’s law for torque:

In the absence of a net torque on a system,

angular momentum is conserved.


A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted?

iclicker exercise:

(a) The student will remain at rest.

(b) The student will rotate counterclockwise.

(c) The student will rotate clockwise.

counterclockwise


More details:

wheelbf

wheelwheelbf

wheelibiwheelfbf

LL

LLL

LLLL

2

0


Other examples of conservation of angular momentum

mm

d1 d1

mm

d2 d2

I1=2md12 I2=2md2

2

I1w1=I2w2 w2=w1 I1/I2

w1 w2


Webassign problem:A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf?

( )

21

1

211

III

III

if

fi

ww

ww


Summary – conservation laws we have studied so far

Conserved quantity Necessary conditionLinear momentum p Fnet = 0Angular momentum L tnet = 0

Mechanical energy E No dissipative forces

Documents

Chapters 10 & 11 – Rotational motion, torque, and angular momentum