Experimental Physics - Mechanics - Basics of rotations 1
Experimental Physics EP1 MECHANICS
- Rotation. Basics -
Rustem Valiullinhttps://bloch.physgeo.uni-leipzig.de/amr/
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
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
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
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.
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.
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
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 ===
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!
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
=
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 ===
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.
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-´»
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.
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
Experimental Physics - Mechanics - Static Equilibrium 16
Examples of precessional motion
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
!
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!