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PROBLEM 19.60 (Continued)
(b) When the disk is riveted at A, it rotates at an angular acceleration .
α
Equation of motion. 2
eff
1
(): sin , ,sin
2
BB t
MM mgl IlmaImr
θ
αθθ
Σ=Σ − =+ = ≈
22
10
2mr ml mgl
θθ
⎛⎞
+
+=
⎜⎟
⎝⎠
2
gl
Am
PROBLEM 19.61
Two uniform rods, each of mass m and length l, are welded together to form
the T-shaped assembly shown. Determine the frequency of small oscillations
of the assembly.
SOLUTION
Let the assembly be rotated counterclockwise through the small angle
θ
about the fixed Point A.
ll
2217
ππ
SO
Det
Let
Th
e
LUTION
ermine locati
o
e
n total mass
o
n of the cent
r
PRO
A ho
m
suppo
r
down
B, 8 s
r
oid G.
ρ
=
m
=
BLEM 19.
6
m
ogeneous wi
r
r
t at A. Kno
w
20 mm and r
e
later.
mass per un
i
=
(2 )rr
ρ
π
=
+
=
6
2
r
e bent to for
m
w
ing that r
=
e
leased, dete
r
i
t length
(2 )r
ρπ
=
+
m
the figure s
h
220 mm
=
an
d
r
mine the ma
g
h
own is attac
h
d
that Point
B
g
nitude of the
h
ed to a pin
B
is pushed
velocity of
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.62 (Continued)
Frequency.
()()
2
22
33
22
2 (2)(9.81)
(0.220)
23.418 s 4.8392 rad/s
n
nn
g
r
ωππ
ωω
−
==
++
==
sin( )
mn B
tyr
θ
θωφ θ
=+=
At 0,t= 20 mm, 0
BB
yy==
0( ) cos(0 ), 2
BBmn
yy
π
ωφφ
== + =
20 mm ( ) sin 0 , ( ) 20 mm
2
BBm Bm
yy y
π
⎛⎞
== + =
⎜⎟
⎝⎠
(20 mm)sin 4.8392 rad/s
2
Bnn
yt
π
ωω
⎛⎞
=+=
⎜⎟
⎝⎠
20 cos (20 mm) sin
2
Bn nn
yt t
π
ω
ωωω
⎛⎞
=+=−
⎜⎟
⎝⎠
At 8 s,t= (20)(4.8392)sin[(4.8392)(8)] (96.78)(0.8492)
B
y=− =−
82.2 mm/s=− 82.2 mm/s
B
v=
SO
Eq
u
LUTION
u
ation of moti
o
o
n.
PRO
B
A hor
i
connec
t
found t
of unk
n
center
d
torsion
a
inertia
o
M
Σ
B
LEM 19.6
3
i
zontal platf
o
t
ed to a verti
c
o be 2.2 s wh
n
own momen
t
d
irectly abov
e
a
l constant
K
o
f object
A
.
:
G
M
I
K
α
=−
3
o
rm
P
is he
l
c
al wire. The
p
en the platfor
m
t
of inertia is
e
the center o
f
27 N m/r
a
=⋅
K
I
θθ
=
l
d by sever
a
p
eriod of osc
i
m
is empty a
n
placed on t
h
f
the plate. Kn
a
d,
determine
a
l rigid bars
i
llation of the
n
d 3.8 s when
h
e platform w
i
owing that th
e
the centroidal
which are
platform is
an object
A
i
th its mass
e
wire has a
moment of
S
O
To
r
Eq
u
Na
O
LUTION
r
sional spring
u
ation of mot
i
tural frequen
c
P
R
A u
n
equ
a
8°
wit
h
dis
k
constant.
i
on.
c
y and period.
OBLEM 1
9
n
iform disk o
a
l length wit
h
angle when
a
h
a period of
k
, (
b
) the peri
o
k
k
=
=
0
M
Σ
=
2
n
ω
=
9
.64
f radius
1r
=
h
fixed ends a
t
a
500-mN m
⋅
1.3 s when t
h
o
d of vibratio
n
(
)
180
0.5 N
m
(8)
3.581 N m/r
a
T
π
θ
⋅
=
=
=
⋅
0eff
():M
=
Σ
−
K
I
=
20 mm
is w
e
t
A
and
B
. K
n
couple is a
p
h
e couple is r
e
n
if one of the
)
m
a
d
KI
θ
θθ
−
=
+
e
lded at its ce
n
n
owing that t
h
p
plied to the
d
e
moved, dete
r
rods is remov
0
K
I
θ
+
=
n
ter to two el
a
h
e disk rotate
s
d
isk and that
r
mine (
a
) the
ed.
a
stic rods of
s
through an
it oscillates
mass of the
PROBLEM 19.65
A 5-kg uniform rod CD of length 0.7 ml
=
is welded at C to two
elastic rods, which have fixed ends at A and B and are known to have a
combined torsional spring constant 24 N m/rad.K=⋅ Determine the
period of small oscillation, if the equilibrium position of CD is
(a) vertical as shown, (b) horizontal.
SOLUTION
(a) Equation of motion.
22
t
ll
a
α
θαθ
===
:()sin ()
22
Ct
ll
M
ImadKmg I ma
αθθα
Σ=+ −− =+
11
2
2
l
ml
θθ
+
+=
⎜⎟
⎝⎠
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.65 (Continued)
Data: 24 N m/rad, 5 kg, 0.7 mKml=⋅ = =
2
(3)(24) (3)(9.81) 0
(2)(0.7)
(5)(0.7)
50.409 0
θθ
θθ
⎡⎤
++ =
⎢⎥
⎣⎦
+
=
Frequency. 250.409 7.1 rad/s
nn
ωω
==
Period. 22
7.1
n
π
π
τω
== 0.885 s
n
τ
=
(b) If the rod is horizontal, the gravity term is not present and the equation of motion is
2
30
K
ml
θθ
+=
2
22
3(3)(24)
29.388
(5)(0.7)
n
K
ml
ω
== =
22
5.4210 rad/s 5.4210
n
n
π
π
ωτ
ω
===
1.159 s
n
τ
=
Copyrig
h
ht
© McGra
w
w
-Hill Educ
a
PRO
A uni
f
vertic
a
oscill
a
about
horiz
o
a
tion. Permis
s
BLEM 19.
6
f
orm equilate
r
a
l wires of t
h
a
tions of the
p
a vertical ax
i
o
ntal displace
m
s
ion require
d
6
6
r
al triangular
p
h
e same len
g
p
late when (
a
i
s through its
m
ent in a dire
c
d
for reprodu
p
late of side
b
g
th
l
. Deter
m
a
) it is rotate
mass center
G
c
tion perpend
i
n
ction or disp
b
is suspende
d
m
ine the peri
o
d through a
s
G
, (
b
) it is gi
v
i
cular to
AB
.
n
l
ω
lay.
d
from three
o
d of small
s
mall angle
v
en a small
g
PROBLEM 19.66 (Continued)
(b)
g
