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Experimental Physics - Mechanics - Basics of rotations 1
Experimental Physics EP1 MECHANICS
- Rotation. Basics -
Rustem Valiullinhttps://bloch.physgeo.uni-leipzig.de/amr/
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Experimental Physics - Mechanics - Basics of rotations 2
Some basics
vdtds =
q
dsqd
0>w
0<w
r
qrdds =
wqº=rv
dtd
# Along line Rotational
1 v = v0+at w = w0+at2 x=x0+v0t+at2/2 q = q0+w0t+at2/2
3 v2=v02+2a(x-x0) w2 = w02+2a(q-q0)
4 x=x0+(v+v0)t/2 q = q0+(w +w0)t/2
5 x=x0+vt-at2/2 q = q0+wt-at2/2
- angular velocity
awqº=÷
øö
çèæ
dtd
dtd
dtd - angular acceleration
constant angular acceleration
ra=a
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Experimental Physics - Mechanics - Basics of rotations 3
Some more basics
wqº=rv
dtd
O
v!
||v^v
r!
j 2sinsinr
rvr
v jjw ==
initialfinal rrr !!!-=D ?--=D initialfinal qqq
1
2
3 3 1
2
3 2 2
One of the properties of vectors is not reproduced.abba !!!!+=+
2rvr !!! ´
=w - it is a vector! (right-hand rule)´
13
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Experimental Physics - Mechanics - Basics of rotations 4
Kinetic energy of rotation
ò=Þ=body
kk dmrEdmvdE 22212
21 w
÷ø
öçè
溺 åòi
iibody
rmdmrI 22 moment of inertiarotational inertia
221
,2
21
, mvEIE trkrotk =Û= w
2212
22
21 vMvvmE
iiM
mM
iiik
i === åå
( )òò ===RR
rot MRdrrr
Mdmv
Mv
0
420
02
2
0
22
221 wprrpw
2241 RMEk w=
221 MRI =
- the same result!dm
rRv
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Experimental Physics - Mechanics - Basics of rotations 5
Moment of inertia
Depends on the rotation axis!a
h
b
2MRIa =òò === 222 rMMdmrMdmrI
Þ=++ 222 )( Rhxy
h R
xhhRr 2222 --=
( ) 2222
2
2 221 hRdxhxhRR
dx
dxrr
hR
hRhR
hR
hR
hR +=--== òò
ò -
---
--
-
--
22 MhMRIb +=
22 MhMRI CM +=CM hThe parallel-axis theorem.
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Experimental Physics - Mechanics - Basics of rotations 6
CM
CM
CM
CM
CM
CM
CM
CM
Rotation seen from two reference systems
Rotating of an object around an arbitrary point is accompanied by the respective rotation around its center of mass.
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Experimental Physics - Mechanics - Basics of rotations 7
Perpendicular-axis theorem
22 rMdmrI == ò
h Rj
w
2cos2)cos(
22/
0
222 MRdMRdRdMI === òòò
p
jjp
jjj
( ) cd ++=ò jjjjj sincoscos 212
x
y
z
yxz III += The perpendicular-axis theorem.
dmrIz ò= 2
dmyxIz ò += )( 22
x
yz
rxy
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 8
Rolling motion
CMv
CMv
CMv
0=CMv
Rv w=
Rv w=CMCM vRvv 2=+= w
CMv
0=v
( ) 22222
21
21
21
21 MvIMRIIE CMCMyk +=+== www
22
2
21
21 Mv
RvIMgh CM +=
Tran
slatio
nRo
tatio
nRo
lling ( )2
2
/12MRI
ghvCM+
=
ghvsphere 710=
j
s
swj RdtdR
dtdsvCM ===
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Experimental Physics - Mechanics - Basics of rotations 9
Ø Rotational motion is similar to one-dimensional
translational motion.
Ø Moment of inertia is an analogue of mass, which
might be considered as resistance to rotation.
Ø For moment of inertia of planar objects
the parallel-axis and perpendicular-axis
theorems can be applied.
ØFor rolling motion velocity of a selected
point is the sum of that of the center of mass
and of the point in the center of mass frame.
To remember!
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Experimental Physics - Mechanics - Basics of rotations 10
Torque
iiiiti lmamF a== å å= 2iitii lmFl a
F!
F!
'l
a
rF tF
im
Inet at =Torquelevel arm
M
R
RT
mg
t
IRaImgR === at
2232 MRMRII CM =+=
32ga =
mgmamgT32
-=-=-
mgT31
=
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 11
Rolling motion
CMv
CMv
CMv
0=CMv
Rv w=
Rv w=CMCM vRvv 2=+= w
CMv
0=v
( ) 22222
21
21
21
21 MvIMRIIE CMCMyk +=+== www
22
2
21
21 Mv
RvIMgh CM +=
Tran
slatio
nRo
tatio
nRo
lling ( )2
2
/12MRI
ghvCM+
=
ghvsphere 710= qsin7
5 gasphere =
j
s
swj RdtdR
dtdsvCM ===
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 12
Angular momentum
r!v! wIL =prmvr
rvmrIL ==== 2w
r!v!
q r̂
prvrmmvrL !!!!´=´== )(sinq
å å === ww IrmLL iii2
dtdL
dtId
dtdIInet ====
)( wwat
If then L = const0=nettConservation of angular momentum
The net external torque = the rate of change of L.
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 13
Conservation of angular momentum
iv102 =iv
cmV
2)(22 2
11
21
1
21
,1,
iikfk
vmmm
mmmEm
E+
=+
=
i1w1I
2I 02 =iwfw
1
212
111, 221
ILIE i
iik == w ( ) ( ) ikf
ffk EIII
IIL
IIE ,21
1
21
22
21, 21
21
÷÷ø
öççè
æ+
=+
=+= w
m1096.6 8´=SunRd3.25=SunT
m100.5 3´=nsR
2KmRI =T
KmRL p22=
222
÷÷ø
öççè
æ=Þ=
Sun
nsSunns
ns
ns
Sun
Sun
RRTT
TR
TR
s1012.1 4-´»
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 14
f1w1I
2I
f2w
Conservation of angular momentum
i1w1I
2I
02 =iw
iL1!
fL1!
iti LL 1
!!= fifftf LLLLL 2121
!!!!!+-=+= if LL 12 2
!!=Þ
fL2!
10/ 12 »mms1.01 »T
)(! m1m5.0
2
1
»»
RR
1
211
2
2221 2
2)(
TRm
TRmm
=+ s1
4)(211
2221
12 »+
=RmRmmTT
There are only internal forces acting within the system. To turn the wheel aroundwe have to apply force, i.e., torque. This will be balanced by the reaction torque.
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 15
O r!
gm!
Motion of a wheel
T!
( ) vdtrdm
dtvdrm
dtvrdm !
!!!
!!
´+´=´
=
´
t! dtpdrarm!
!!!!´=´=t
w
L!
dtLd t!!= precession
jt LdrmgdtdtdL ===
j
Lrmg
dtd
p ==jw
Nuclear Magnetic Resonance
dtLd!
!=t
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Experimental Physics - Mechanics - Static Equilibrium 16
Examples of precessional motion
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 17
Static and dynamic imbalance
dtLd!
!=tL
! w!
)( vrmL !!!´=
w!
r!
v!
L!
w!
r!
L! w
!
h
hmhmvF 2
2
w==
w!
r!
v!
L! L
!
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Experimental Physics - Mechanics - System of Particles - Rotational Motion II 18
ØTorque is the product of the tangential component of the
force and the level arm.
ØThe angular momentum is the cross-product of the radius
vector and the linear momentum.
Ø It is fundamental property that the angular momentum is
always conserved.
Ø Precession is a reaction of the angular momentum
to a net torque applied perpendicularly.
Ø A body is dynamically imbalanced when the
angular velocity and momentum are not
parallel to each other.
To remember!