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A uniform 1.6 kg rodAB is welded at its midpoint G to avertical shaft GD. Knowing that the shaft rotates with an
angular velocity of constant magnitude = 1200 rpm,determine the angular momentum HGof the rod about G.
Single rotation about a fixed point
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x
y
( )
[ ]
22
cos sin , sin cos
21200 12!.66
60
sin cos "2.#$ 11$.1
1.6
1 1.6 0.610, 0."#61, 0, 0
12 12
0 0 0
0 0."#61 00 0 0
x x y y z z x y y z z x
x y z
i i j j i j
rad s rad s
j i j i j
m
I I ml I I I I
= = +
= =
= = + = +
=
= = = = = = = =
=
I
Find I and in inclined rotating frame
0%ew unit vectors 20
cos sin , sin cos
cos sin , sin cos
i i j j i j
i i j j i j
= = = +
= = +
Locate principal axes of the rod
Orient rotating coordinates along
principal axes
Find the transformations from
otating to ground frame and vice
versa
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x
y
( ) [ ] { }
0 0 0 "2.#$
0 0."#61 0 11$.1
0 0 0 00."#61 11$.1 !$.!#
x y z x y zx y z
j j
= =
= =
GH I
0%ew unit vectors 20
cos sin , sin cos
cos sin , sin cos
i i j j i j
i i j j i j
= = = +
= = +
Find H in inclined rotating frame.
j
Transform to inertial frame
( ) ( )
cos sin , sin cos
!$.!# !$.!# sin cos 20.0" !!.06xyz
i i j j i j
j i j i j
= = += = + = +
GH
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thin homogeneous dis! of mass m and radius r is mounted
on the hori"ontal axleAB. The plane of the dis! forms an
angle #$$% &ith the vertical. 'no&ing that the axle rotates
&ith an angular velocity( determine the angle formed by theaxle and the angular momentum of the dis! about G.
)etermine the rate of change HG of the angular momentum HGof the dis! for an arbitrary value of *( !no&ing that the dis! has
an angular velocity + , +i and an angular acceleration - , -i.
homogeneous ./ !g dis! is mounted on the hori"ontal shaft
AB. The plane of the dis! forms a #$0 angle &ith the yz plane
as sho&n. 'no&ing that the shaft rotates &ith a constantangular velocity + of magnitude 1$ rad2s( determine the
dynamic reactions at pointsA and B.
'no&ing that the assembly is initially at rest 3+ , $4 &hen a
couple of moment M$, 3#5.5 6.m4i is applied to the shaft(
determine 3a4 the resulting angular acceleration of the
assembly( 3b4 the dynamic reactions at pointsA and Bmmediately after the couple has been applied.
Single rotation about a fixed axis
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
2 2 2
, ,
%ew unit vectors
cos sin , sin coscos sin , sin cos
cos sin
1 1 1, , , 0
2 " "x x y y z z xy yz zx
i i j j i ji i j j i j
i i j
I mr I mr I mr I
= + = + = = +
= =
= = = =
thin homogeneous dis! of mass m and radius r is mountedon the hori"ontal axleAB. The plane of the dis! forms an
angle #$$% &ith the vertical. 'no&ing that the axle rotates
&ith an angular velocity( determine the angle formed by theaxle and the angular momentum of the dis! about G.
xes chosen
for constant I.
7alculations
are easier
[ ]{ }
2
2 2 2
2
10 0
2 cos1 1 1
0 0 sin cos sin" 2 "
01
0 0"
G
mr
i
H mr j mr i mr j
mr
= = =
I
( ) ( )2 2
2 2 2 2 2
1 1cos cos sin sin sin cos
2 "
1 1 1cos sin sin cos2 " "
GH mr i j mr i j
mr i mr i mr j
= + +
= + +
Find I and in inclinedrotating frame.
Find H8in rotating frame
Transform to
inertial frame
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y
xxes chosen
for constant I.
7alculations
are easier
2 2 2 21 1 1 1 1cos sin cos sin sin cos2 " 2 " "
GH mr i j mr i i j = = + +
)etermine the rate of change HG of the angular momentum HGof the dis! for an arbitrary value of *( !no&ing that the dis! has
an angular velocity + , +i and an angular acceleration - , -i.
( ) ( ) ( )G G G xyzxyz x y zH H H = + & &
Sitting in xy" frame &e &ill not see Ichange but &e &ill
certainly see the disc spinning faster at
and angular momentum change as
( ) [ ]{ } [ ] { } [ ] { } [ ] { } { }
( )
2
2 2
2
0
10 0
2 cos1 1 1
0 0 sin cos sin" 2 "
01
0 0"
G x y zx y z x y zx y zx y z
G x y z
d d d
H dt dt dt
mr
i
H mr j mr i j
mr
= = + = +
= =
I I I I &
&
cos sini i j = =
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
xxes chosen
for constant I.
7alculations
are easier
)etermine the rate of change HG of the angular momentum HGof the dis! for an arbitrary value of *( !no&ing that the dis! has
an angular velocity + , +i and an angular acceleration - , -i.
( ) ( ) ( )G G G xyzxyz x y zH H H = + & &
( )2
2 2 2 2 2
1 1
cos sin2 "
1 1 1cos sin sin cos
2 " "
G x y zH mr i j
mr i mr i mr j
=
= + +
&
Transform firstcomponent to inertial
frame coordinates
9otating frame rotates at
( ) 2 2 2 2 21 1 1 1
cos sin sin cos sin cos2 " " "
G xyzH i mr i i j mr k
= + + =
This is not true for all probi lems =
Second component
( ) ( ) ( )
( )2 2 2 21 1 1 1cos sin sin cos sin cos
2 " " "
G G G xyzxyz x y z
G xyz
H H H
H mr i i j k
= +
= + + +
& &
&
6et rate of change of H8as seen from ground
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
xxes chosen
for constant I.
7alculations
are easier
( )
( )
( )
2
2
2 2
2 2
1 1cos sin
2 "
1 1cos sin
2 "
1 1 1 1cos sin cos sin2 " 2 "
1 1 1 1cos sin cos
2 " 2
G
G
H mr i j
d dH mr i j
dt dt
d dmr i j mr i jdt dt
d dimr i j mr
dt dt
=
=
= + = +
( ) ( )
( ) ( )
2 2
2
2
2 2
sin"
1 1 1 1cos sin cos sin
2 " 2 "
1 1cos cos sin sin sin cos
2 "
1 1cos cos sin sin sin cos
2 "
1 1cos
2
dj
dt
mr i j mr i i i j
mr i j i j
mr i i j i i j
mr i
= + = + +
+ + +
= + 2 2 2
2 2 2 2
1 1 1sin sin cos cos sin sin cos
" " 2 "
1 1 1 1cos sin sin cos sin cos
2 " " "
i j mr k k
mr i i j k
+ + = + + +
lternate method
2 2 2 21 1 1 1cos sin sin cos sin cos2 " " "
GH mr i i j k = + + +
&
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
homogeneous ./ !g dis! is mounted on the hori"ontal shaft
AB. The plane of the dis! forms a #$0 angle &ith the yz planeas sho&n. 'no&ing that the shaft rotates &ith a constant
angular velocity + of magnitude 1$ rad2s( determine the
dynamic reactions at pointsA and B.
( ) ( )0, 0X! X!
= =G G
v a'inematics of center of mass
8 in the inertial frame
6e&ton:s la&
( )
( )0 0
, 0
y z y z
y y z z
dm
dt
dA j A k B j B k m"j
dt
A B m" A B
=
+ + + = =
+ = + =
ext GF v
Az
mg9evolute ;oints
apply only x and
y reaction forces
and no moments
Ay
Bz
By
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
2 21sin cos
"GH mr k =&
( ) ( )0, 0X! X!
= =G G
v a9ate of change of H8
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
Az
mg9evolute ;oints
apply only x and
y reaction forces
and no moments
Ay
Bz
By
2 2
2 2
2 2
2 2
2 2
, 0
1sin cos , 0
"
1 1sin cos
2 $
1 1sin cos
2 $
0
0
&tatic reactions
1
2
1
2
'ence dynamic reactions
1sin cos
$
1
sin$
y y z z
y y z z
y
y
z
z
y
y
y
y
A B m" A B
mrB A A B
l
mrA m"
l
mrB m"
l
A
B
A m"
B m"
mrA
l
mr
B l
+ = + =
= =
=
= +
==
=
=
=
= cos
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
( ) ( )0, 0X! X!
= =G G
v a'inematics of center of mass
8 in the inertial frame
6e&ton:s la&
( )
( )0 0
, 0
y z y z
y y z z
dm
dt
dA j A k B j B k m"j
dt
A B m" A B
=
+ + + = =
+ = + =
ext GF v
Az
mg9evolute ;oints
apply only x and
y reaction forces
and no moments
Ay
Bz
By
'no&ing that the assembly is initially at rest 3+ , $4 &hen a
couple of moment M$, 3#5.5 6.m4i is applied to the shaft(
determine 3a4 the resulting angular acceleration of the
assembly( 3b4 the dynamic reactions at pointsA and B
immediately after the couple has been applied.
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y
x
2 2 2 21 1 1 1cos sin sin cos sin cos2 " " "
GH mr i i j k = + + +
&
( ) ( )0, 0X! X!
= =G G
v a
9ate of change of H8
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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y
x
Az
mg9evolute ;oints
apply only x and
y reaction forces
and no moments
Ay
Bz
By
( ) ( )
2 2 2 2 0 00 2 2 2 2 2
2
2 2
2 2 2 2
2
, 0
1 1 " "cos sin
2 " 2cos sin 1 cos
1sin cos
"
1sin cos
"
1 1 1 1sin cos , sin cos
2 $ 2 $
1sin cos ,
$
y y z z
z z
y y
y y
z
A B m" A B
# ## mr mr
mr mr
A B mr
B A mr
mr mr A m" B m"
l l
A mr
+ = + =
= + = =+ +
=
=
= = +
=
( ) ( )
2
2 2 2 2
2 0 0
2 2
1sin cos
$&tatic reactions
1 1,
2 2
'ence dynamic reactions
1 1sin cos , sin cos
$ $
1 sin cos sin cossin cos ,
$ 2 1 cos 2 1 cos
z
y y
y y
z z
B mr
A m" B m"
mr mr A B
l l
# #A mr B
=
= =
= =
= = = + +
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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5=.5 !g advertising panel of length #a , #.# m and &idth #b
, 1.= m is !ept rotating at a constant rate 1about itshori"ontal axis by a small electric motor attached atA to frame
ACB. This frame itself is !ept rotating at a constant rate #
about a vertical axis by a second motor attached at C to thecolumn CD. 'no&ing that the panel and the frame
complete a full revolution in / s and 1# s( respectively(
express( as a function of the angle ( the dynamic reactionexerted on column CD by its support at D.
T&o rotations along intersecting axes
&how that (a) the dynamic reaction atD is independent
of the length 2a of the panel, (b) the ratio#1#2 of the
magnitudes of the couples e*erted by the motors atA and
$, respectively, is independent of the dimensions and
mass of the panel and is e+ual to # 1at any giveninstant.
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T&o rotations along intersecting axes
( )
( )
1 2 1 2 2 2 1
2 2 2 2
, ,
%ew unit vectors
cos sin , sin cos
cos sin , sin cos
sin cos sin cos
1 1 1, , , 0
x x y y z z xy yz zx
i i j j i j
i i j j i j
k j k i j i j k
I m a b I ma I mb I
= + = + = = +
= + = + + = + +
= + = = =
( ) [ ] { }
( )
( )
2 2
2
2
2
12
2 2 2 2
2 2 1
10 0
sin1
0 0 cos
10 0
1 1 1
sin cos
G x y z x y z x y z
m a b
i
H ma j
k
mb
m a b i ma j mb k
+
= =
= + + +
I
( ) ( )
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 1
2 2 2 2
2 2 1
2 2 2 2 2 2 2 2 2
2 2 2 2 1
2
2
1 1 1sin cos
1 1 1sin cos sin cos sin cos
1 1 1 1 1sin cos cos sin sin cos
1sin cos
G x y zH m a b i ma j mb k
m a b i j ma i j mb k
m a b i ma i m a b j ma j mb k
mb
= + + +
= + + + + +
= + + + + +
= 2 2 2 22 2 11 1 1
sin
i mb j ma j mb k + + +
Find I and in inclined rotating frame.
Find H8in rotating frame
Transform to inertial frame
x
y
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T&o rotations along intersecting axes
( )
( )
1 2 1 2 2 2 1
2 2 2 2
, ,
%ew unit vectors
cos sin , sin cos
cos sin , sin cos
sin cos sin cos
1 1 1, , , 0
x x y y z z xy yz zx
i i j j i j
i i j j i j
k j k i j i j k
I m a b I ma I mb I
= + = + = = +
= + = + + = + +
= + = = =
( ) [ ] { } [ ] { } [ ] { }
( ) ( )2 2 2 2
2 2
2 2
2 2
12 2
1 10 0 0 0
sin sin1 1
0 0 cos 0 0 c
1 10 0 0 0
G x y z x y z x y z x y z x y z x y zx y z
d d dH
dt dt dt
m a b m a b
i id d
ma j madt dt
k
mb mb
= = +
+ +
= +
I I I &
( )
( ) ( )
1
2 2
1 2
2
1 2 1 2 1
2
2 2 2
1 2 1 2
os
10 0
cos1
0 0 0 sin , , 0, 0
01
0 0
1 1cos sin
G
x y z
j
k
m a b
i
ma j
mb
H m a b i ma j
+
= + = = =
= +
& & &Q
&
( ) ( )
( ) ( ) ( )
2 2 2
1 2 1 2
2 2 2
1 2 1 2
2 2 2 2
1 2 1 2 1 2
1 1cos sin
1 1cos cos sin sin sin cos
1 1 1cos cos sin
G x y zH m a b i ma j
m a b i j ma i j
ma i mb i mb j
= +
= + + +
= + +
&
Find I and in inclined rotating frame.
Find H8> in rotating frame
Transform to inertial frame
x
y
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T&o rotations along intersecting axes
( ) ( ) ( )
( )
( ) ( )
2 2 2 2
1 2 1 2 1 2
2 2 2 2 2
2 1 2 2 2 1
2 2 2 2
1 2 1 2 1 2
2 2 2
2 1 2
2
1 2
1cos cos sin
1
sin cos sin
1 1 1cos cos sin
1 1sin cos
1sin
G G G xyzxyz x y zH H H
m a i b i b j
j k m b i b j a j b k
ma i mb i mb j
mb k mb i
mb
= +
= + +
+ + + + +
= + +
+
+
& &
2 2 2
1 2 1 2
2 2 2 2 2
1 2 1 2 2
1 1cos sin
2 2 1
cos cos sin sin cos
j mb i ma i
mb i mb j mb k
= +
x
y ( )2 1j k = +
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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homogeneous dis! of mass m = / !g rotates at the constantate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded to ahaft DCE rotating at the constant rate #, ? rad2s. )eterminehe angular momentum HAof the dis! about its centerA.
)etermine the angular momentum HDof the dis! about point D.
etermine the rate of change of HAof the disk
homogeneous dis! of mass m , !g rotates at the constant
ate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded tohaft DCE rotating at the constant rate #, , ? rad2s.
)etermine the dynamic reactions at D and E.
It is assumed that at the instant sho&n shaft DCE has an
angular velocity #, ? rad2s i and an angular acceleration#, / rad2s i. 9ecalling that the dis! rotates &ith a constantangular velocity 1, 1/ rad2sj( determine 3a4 the couplethat must be applied to shaft DCE to produce the given
angular acceleration( 3b4 the corresponding dynamicreactions at D and E.
T&o rotations along non intersecting axes
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"
y
x
homogeneous dis! of mass m = / !g rotates at the constantate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded to ahaft DCE rotating at the constant rate #, ? rad2s. )eterminehe angular momentum HAof the dis! about its centerA.
2 2 2
, ,
2 1 2 1
, ,
1 1 1, ,
is the net angular ve
, 0
loci
" 2 "
x x y y z z xy yz zx
i i j j k k
I mr I mr I mr I
i j i j
= = =
= = = =
= + = +
[ ]{ }
2
2
2 2 2
1 2 1
2
10 0
"1 1 1
0 02 " 2
01
0 0"
G
mr
i
H mr j mr i mr j
mr
= = = +
I
2 2
2 1
1 1
" 2AH mr i mr j = +
Find I and in rotating
frame. In this case aninclined frame is not
re@uired as principal axes
are already along xy"
Find Hin rotating frame
Transform to
inertial frame
is the cm of the dis!
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"
y
x
2 2 2
, ,
2 1 2 1
, ,
1 1 1, ,
is the net angular ve
, 0
loci
" 2 "
x x y y z z xy yz zx
i i j j k k
I mr I mr I mr I
i j i j
= = =
= = = =
= + = +
[ ]{ }
( ) ( ) ( )
2
2
2 2 2 2 2
1 2 1 2 1
2
2 2
2 1 2
2 2 2
2 1 2
10 0
"
1 1 1 1 10 0
2 " 2 " 20
10 0
"
1 1
" 2
1 1
" 2
A
D A A D A D$ BA $B BA $B
D BA $
mr
i
H mr j mr i mr j mr i mr j
mr
H H m mr i mr j l i l j l k m l k l j
H mr i mr j m l i l
= = = + = +
= + = + + + +
= + + +
I
r v
( )2B D$ BA D$ $Bi l l j l l k +
Find I and in rotatingframe. In this case an
inclined frame is not
re@uired as principal axes
are already along xy"
)etermine the angular momentum HDof the dis! about point D.
is the cm of the dis!
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( )
2 2
2
2 2 2
2 2 2
2
2 2
2 2
2 2 2 2
2
2 2'ere 0,
A A D D$ BA $Bxyz
A BA $Bxyz
A A D A DX!
D$ BA $B BA $B
BA $B BA $B
A $B BA BA $BX!
A BAX!
i i l i l j l k
l k l j
i i i
i l i l j l k i l k l j
l k l j l j l k
l l j l l k
l j
= = +
= +
= +
= + + +
= + + +
= + +
= = +
v r
v
a r r
a
a2
2 $Bl k
'inematics of center of
mass in the inertial
frame
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"
y
x
( ) 2 2 2 22 1 2 11 1 1 1
" 2 " 2
A x y zH mr i mr j mr i mr j = + = + ( ) ( ) ( )A A A xyzxyz x y zH H H = +
& &
( ) [ ]{ } [ ] { } [ ] { } [ ] { } { }
( )
2 1 2 1 2 1
2
2
2 2
1 2 1
2
0
10 0
"
1 1 10 0
2 " 20
10 0
"
A x y zx y z x y zx y zx y z
x y z
A x y z
d d d
H dt dt dt
i j i j i j
mr
i
H mr j mr i j
mr
= = + = + = + = + = +
= = + =
I I I I
&
& 22 1
1 1
" 2mr i j
+
etermine the rate of change of HAof the disk
2i=
( )
( )
2 2 2
2 1 2 2 1
2
2 1 1 2
1 2
2
1 2
1 1 1 1
" 2 " 2
1 1 1
" 2 2
/n this problem 0
1
2
A xyz
A
xyz
H mr i j i mr i mr j
mr i j k
H mr k
= + + +
= + +
= =
=
&
&
is the cm of the dis!
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"
y
x
is the cm of the dis!
homogeneous dis! of mass m , !g rotates at the constant
ate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded toshaft DCE rotating at the constant rate #, , ? rad2s.)etermine the dynamic reactions at D and E.
Dz
Dy
Ez
Ey
( ) 2 1 21
2A xyz
H mr k =&
( ) ( )
( ) ( ) ( )
( )
2
2 2
2 2 2 2
2 2
2 2 2'ere 0,
A BA $Bxyz
A $B BA BA $BX!
A BA $BX!
l k l j
l l j l l k
l j l k
= +
= + +
= = +
v
a
a
'inematics of center of mass
in the inertial frame
9ate of change of H
6e&ton:s la&( )
2 2
2 2
2 2
2 2,
A A
y z y z BA $B
y y BA z z $B
d
m mdt
D j D k % j % k m l j m l k
D % m l D % m l
= = + + + = +
+ = + =
extF v a
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"
y
x
is the cm of the dis!
Dz
Dy
Ez
Ey
( ) 2 1 21
2A xyz
H mr k =&9ate of change of H
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"
y
x
is the cm of the dis!
Dz
Dy
Ez
Ey
22 2 2
1 2 2
2 2
2 2
2 2 2 2
2 2 1 2 2
22 2 2
2 2
1,
2
,
1
2
D$ $By D$ BA z
%D %D
y y BA z z $B
y BA y BA D$ BA
%D
D$ $Bz $B z $B
%D
m m l l % r l l %
l l
D % m l D % m l
mD m l % m l r l l
l
m l lD m l % m l
l
= =
+ = + =
= =
= =
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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"
y
x
is the cm of the dis!
Dz
Dy
Ez
Ey
( ) 2 2 1 1 21 1 1
" 2 2A xyz
H mr i j k = + +
&
( ) ( )
( ) ( ) ( )
2
2 2
2 2 2 2
A BA $Bxyz
A $B BA BA $Bxyz
l k l j
l l j l l k
= += + +
v
a
'inematics of center of mass in the inertial frame
9ate of change of H
6e&ton:s la&( )
( ) ( )
( ) ( )
2 2
2 2 2 2
2 2
2 2 2 2,
A A
y z y z $B BA BA $B
y y $B BA z z BA $B
dm m
dt
D j D k % j % k m l l j m l l k
D % m l l D % m l l
= =
+ + + = + +
+ = + = +
extF v a
It is assumed that at the instant sho&n shaft DCE has an
angular velocity #, ? rad2s i and an angular acceleration#, / rad2s i. 9ecalling that the dis! rotates &ith a constantangular velocity 1, 1/ rad2sj( determine 3a4 the couple
that must be applied to shaft DCE to produce the givenangular acceleration( 3b4 the corresponding dynamic
reactions at D and E.
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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"
y
x
is the cm of the dis!
Dz
Dy
Ez
Ey
9ate of change of H
8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)
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"
y
x
is the cm of the dis!
Dz
Dy
Ez
Ey
( )
( ) ( )
( )
02 2 2
2 2 20 02 1 2 12 2 2 2
2 201 2 22 2
2
2
"
"
" 1 " 0
2" "
1 "
2 "
,
$A
z BA D$ $B D$ BA D$ $B D$
%D %D$A $A
y D$ $B D$ BA
%D $A
y y BA z
#
m r l
m # m # % l l l l r l l l l
l lm r l m r l
m #% r l l l l
l m r l
D % m l D
=+
= + = + =
+ + = +
+ + =
Q
2
2
2 2 2 2
2 2 1 2 2
22 2 2
2 2
1
2
z $B
y BA y BA D$ BA
%D
D$ $Bz $B z $B
%D
% m l
mD m l % m l r l l
l
m l lD m l % m l
l
+ =
= =
= =
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