PHY2053 Lecture 17 - Department of Physics · PHY2053, Lecture 17 Rotational Form of Newton’s...

PHY2053 Lecture 17 Ch. 8.4, 8.6 (skip 8.5) Rotational Equilibrium,Rotational Formulation of Newton’s Laws

PHY2053, Lecture 17 Rotational Form of Newton’s Laws

Linear vs Rotational VariablesTransformation Table

2

Linear Variable → Rotational Variable

distance angle

velocity angular velocity

acceleration angular acceleration

mass [inertia] moment of inertia

momentum p = mv angular momentum, L = Iωforce, F = ma torque, τ = IαKtrans = ½mv2 Krotation = ½Iω2


“Right Hand Rule” for rotational systems

• change of angle (∆θ) is a vector along the axis of rotation, and its direction is determined by the right hand rule:

• orient your right hand so that the fingers point in the direction in which the angle is changing; your thumb points in the direction of the rotational vector

3

∆θ

∆θ


Reminder: from angle to angular acceleration

• instantaneous angular velocity:

• instantaneous angular acceleration:

4


Torque• The equivalent of force, for rotational systems

(somewhat hand-waving derivation of equivalence)

5

• not quite so trivial, need to account for the orientation of force wrt “lever arm” of object i, “useful force”

torque = momentum of inertia × angular acceleration


Lever Arm

6

F

r

F⊥= F sinθ

θ F

r

θ

r⊥ = r sinθ

θ

“Perpendicular force”

“lever arm”

• for the same force, the torque is directly proportional to the lever arm (projection of r perpendicular to F)

F

∆θ

∆s

r

m


Work done by torque• Start from definition of work, for rotational systems:

7

• Hooke’s law, for rotationalsystems (torsion spring):

PHY2053, Lecture 16, Rotational Energy and Inertia

Unusual Atwood’s Machine

8

m

m

M M

R

r

h

In the machine below, the small weight (mass m) is connected via an ideal pulley to the massless horizontal wheel of radius r. Ignore the moments of inertia of the connecting rods; the two masses M are both R away from the axis of rotation. What is the velocity of mass m as it falls h from its initial position, where it was at rest?


Discussion: Unusual Atwood’s Machine, via energy conservation

9


Discussion: Unusual Atwood’s Machine, via force, tension and torque

10

3t\.

4I

3A-

a ll

HB -t

as

-lt

r-1

i\p

fN

'l'r

"'T

ls

qP

l p

-r

Pl

l+

3J

,

bL}

3ro

7 G

+.

<-

sq

-l

Gs

LL

td

-l

5t

<.

J

t:

rF

l

iE

?)

3 op

J + lN ;l s

rln

vI

N

)1^

R il rle

/f /

/

F L cn-

G r0 c i6' e

r--,

*

fvl

oT

IF

Ib

F

)-

b,?

(-.-.-

l '+ oo

rl lr'{

/f

,l pP

lq TJ!

.T' R

6''

1

/_1

rl lN 'rl a nt

3t\.

4I

3A-

a ll

HB -t

as

-lt

r-1

i\p

fN

'l'r

"'T

ls

qP

l p

-r

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l+

3J

,

bL}

3ro

7 G

+.

<-

sq

-l

Gs

LL

td

-l

5t

<.

J

t:

rF

l

iE

?)

3 op

J + lN ;l s

rln

vI

N

)1^

R il rle

/f /

/

F L cn-

G r0 c i6' e

r--,

*

fvl

oT

IF

Ib

F

)-

b,?

(-.-.-

l '+ oo

rl lr'{

/f

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lq TJ!

.T' R

6''

1

/_1

rl lN 'rl a nt

3t\.

4I

3A-

a ll

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as

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ls

qP

l p

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l+

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,

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3ro

7 G

+.

<-

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LL

td

-l

5t

<.

J

t:

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l

iE

?)

3 op

J + lN ;l s

rln

vI

N

)1^

R il rle

/f /

/

F L cn-

G r0 c i6' e

r--,

*

fvl

oT

IF

Ib

F

)-

b,?

(-.-.-

l '+ oo

rl lr'{

/f

,l pP

lq TJ!

.T' R

6''

1

/_1

rl lN 'rl a nt

E {0 fi- \ o t -l- { t\ ) o o 5 u .9- o s D c) n 6 C\ D

€P tl N s X t,

,^)

-F*. ," tl N) r 0

qst\ r 4 I

-\ +

Fc n 3 rd P

rtl^ -r-

CS .'

_t ) Ps ? c

a\6\

-f.J

ll N I

f\) ss

I

*\^l ,n

\*ll

.--.S

' + 5li

ilP

n,

'J'l*

slsl

'l, l

B'

n os

5v

r(

A%

s$

FF

e+

rb

s6

+-

5A

-$

vs

uk

--

c"$

,rc

s)

6 *0 X


Discussion: Unusual Atwood’s Machine, via force, tension and torque 2

11


Statics, Rotational Equilibrium• Condition 1: Newton’s 1st law: a stationary object

that has no net force acting on it remains stationary

12

• Condition 2: For the object not to start rotating about an axis, the net torque on the object must be zero

• Important: Conditions of static equilibrium, once established for one axis, apply to any arbitrary axis


Example: Springboard• A diver weighing 80 kg is standing on the end of a 2

m long springboard which weighs 240 kg. The springboard is supported by two springs on the other end. One spring is at the very end of the board, and the other is 0.5 m toward the center of the board. Compute the forces generated by the springs if the system is in static equilibrium. Are the springs compressed or stretched?

13

- f-6 \t/ 9 c

To E

n a_ -f\ q s, a- ft- ll N 5

n t o ?'t- (o s b /t\ \ a _5 c| o E" s --{

- t- )

J' ,9 .'l r J- 9 5\

r

r-

'+ J- e

rJ

c\

o\r

-N

+)

Y'

o

xlr

q-,

r\ sc

a\

l-

\ X 1

r-.

l' t\c\ rV

l-

u1 lN

E

rs'

- tl t\.

F o : c6 o a_ s u o o J p * o F i J- D - J- s o .( A\

V- I o I s \

te

.o t^

qp eR

$v

U

es

q.s c -l-

r-l T)

+g

,t--*

=r

,"3

A\

.

*3

N* 5

N .-

49

F(\

o st

(\

\ s.o

bE

S(,.

I \.

Jio

a *r--

X rl s) Q.1 t-

s6.

a

rvl

.?t,

tl o-

-C

1

^,

t -F^.

0 I 3A

Qu o A x

-\.AQ N

lr-.

I 3q{

) f-\ tl O

N-l lt Nl3 + F

\-, \o

N/*l

-lX tl

Nl3 +

s,

.E

lfl

\9

\vI


Discussion: Springboard

14

-r\

\

ilrIrif>,,Rb

XJN

rf

-s>rN'ooC\Jt

t.Ill

-:I-T(.)tnIa-tB

I

s\o,.n

f\\,-\J,g

bq)-.S-19

*uEsa,)f

\h-_s

r\\svPqE(J3s3€6

.?h-.crDsIJC65P\AJ_sa-IY)l/rsCtD

H:=

\sq.)s.4[*v

\-'l-u

\"g

\-}.l

It

arsb0

(-7ON

_vR\rt

ll

il

N{ls\O00+sbEN/+SDoO{9ItfCrb5+SD>.

&I

NSs{L

\J)s.o-ttr'\)SDrR1\S,-Sta.)-S-f-n

+E

tJ

!)

lJ

os

\x.

ul

lrf

l*vu'

a)-

:*

\n

1/1 !'

nF_ ^P {,

tn .-l

r-

9J

o

.ol-r-N +

lf

+^slp,["rts

tl

-,

J

tl

Lt.{

-r\

\

ilrIrif>,,Rb

XJN

rf

-s>rN'ooC\Jt

t.Ill

-:I-T(.)tnIa-tB

I

s\o,.n

f\\,-\J,g

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*uEsa,)f

\h-_s

r\\svPqE(J3s3€6

.?h-.crDsIJC65P\AJ_sa-IY)l/rsCtD

H:=

\sq.)s.4[*v

\-'l-u

\"g

\-}.l

It

arsb0

(-7ON

_vR\rt

ll

il

N{ls\O00+sbEN/+SDoO{9ItfCrb5+SD>.

&I

NSs{L

\J)s.o-ttr'\)SDrR1\S,-Sta.)-S-f-n

+E

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!)

lJ

os

\x.

ul

lrf

l*vu'

a)-

:*

\n

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nF_ ^P {,

tn .-l

r-

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o

.ol-r-N +

lf

+^slp,["rts

tl

-,

J

tl

Lt.{


Discussion: Springboard 2

15


Angular Momentum• Momentum-impulse theorem:

16

• by analogy with p = mv, define angular momentum:

• angular momentum-impulse theorem:


Example: “Angular Collision”

• A disk of mass m1 and radius R is spinning with an angular frequency ωi . A second disk, of mass 3m1 is dropped onto disk 1 so that they share the same axis of rotation. After a brief period of friction, the two disks are rotating with the same angular frequency ωf.

• What is the ratio ωf/ωi ?• What fraction of the initial kinetic energy was wasted

through friction?

17

-l 7 rD s 5C

\Q

e F- ) T o F CO s c 5 0 h -t 5 6 vl v1 -L- o J q- Y c) r \ q,

0 r0 o .*

f- )l t\ h *H FF

f-l H.\ H

Fs -F J r0 t1 t(\

r fllrnS ll UJ F Ft

J o J +

3J

+ H N) L-t F

h-tl

*[It -F

Jr ll

F3 >s oP

Nl^ l-.l

t'J

3s }}q

.+ 3 c.l q€̂

{

NIH

?t

-\

s .\.

.r<.

^)

j, LJ: n

rl sC.

f rJ .d NJ^

ll .H -i l,tJ

I -"ls'

IN J- c6 +- o a c\<$

fb-

tal .s R

l--'

s Irf--

).F

J + f-J n) L_J s

*

tl F'-) ii " -' l-- h

.+q,

\ -

N ^5 tl $ fl

, -J-.l t- t--l

pl

slq

e ..$ 1+p


Discussion: Angular Collision

18

-l 7 rD s 5C

\Q

e F- ) T o F CO s c 5 0 h -t 5 6 vl v1 -L

- o J q- Y c) r \q,

0 r0 o .*

f- )l t\ h *H FF

f-l H.\ H

Fs -F J r0 t1 t(\

r fllr

nS ll UJ F Ft

J o J +

3J

+ H N) L-t F

h-tl

*[It -F

Jr ll

F3 >s oP

Nl^ l-.l

t'J

3s }}q

.+ 3 c.l q€̂

{

NIH

?t

-\

s .\.

.r<.

^)

j, LJ: n

rl sC

.f rJ .d NJ^

ll .H -i l,t

JI -"l

s'

IN J- c6 +- o a c\<$

fb-

tal .s R

l--'

s Irf--

).F

J + f-J n) L_J s

*

tl F'-) ii " -' l-

-h

.+q,

\ -

N ^5 tl $ fl

, -J-.l t- t--l

pl

slq

e ..$ 1+p

s N lF 5 o t\>

F6\

ll

-rln t\

It\

(tl N +. + s-, o 3" ={

-.' p e* 5- q F o- o s a

^o \< .s \u s \-

5' d ? -T-\ -)

-

o )-L- 6 : s V1

Jr J,

nIt

-\.I

N

l

I

IH

\

r

TS

.t

t\

\d

tp

+

l-1L (,1

) r

t)c I

dl- I \t I

o l-r il I

-cl* n N L|.) ;a

N I-1 ti + h l-! ,*e"

qB

*ls T n

P\-

l \ tO

'N

h Nh ui .s Nlry

,

IH IPJ

o { IL rr'r,

rl \r

Lqt F rlA

Jr

p

-t-,I Pl

tsl-Ja\

t-J e \.J

Ir-

,\

.+ N

{r

I

sl*

{.-'

Nfr

h-r

I t-r

H l*

r-A

IH

' lP

e*

6.

cb .E

s\R

c6

y

6\ t$1* d' $

{ \rs \

HA

ee.

d


Discussion: Angular Collision 2

19


The vector nature of torque and angular momentum

20

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