DYNAI'II
CS
PR0BLEM
NB/1
I
\r
A
A 2 n b o a r d i s p l a c e d i n a t r u c k s o t h a t o n ee n d r e s t s a g a i n s ta b l o c k o n t h e
f l o o r w h i l e t h e o t h e r e n d r e s t s a g a i n s t a v e r t i c a l w a l l . D e t e r m i n teh e max i rnurn
p o s si b l e u n i f o r m a c c e le r a t i o n o f t h e t r u c k i f t h e b o a r d i s t o r e n n i n i n the
position shown.
LIMITING
C A S E : RTACTION
AT B I S Z E R O
g
Ng; o
YT\A-
ts-
tn^
A 1so
=t)
ZMn= - ( 1 ) c o s 7 5 ( d g+) 1 s i n T S W u )
g
e
\' = cos75
sin 75
a = 2 . 6 2 ms/e c
DYNAM
I CS
PR0BLEM
NB/Z
c y l ' in d r i caI ca n s a re tra n sp o r ted f r om one el evati on to another"by the mov'i
ng
h o r i z o n t al a rmssh o w n . A ssu rn'
ngi that A= .2A, betweenthe cans and the arms,
d e t e r m i n e( a ) t h e m a g n i t u d eo f t h e u p w a r dacceler at' ion a for whi ch the c ans
s l i d e o n t h e h o r i z o n t a l a r m s , ( b ) t h e s m a l l e s t r a t i o h / d f o r w h i c h the cans
ti p befot1
rna.9
tl:y
sI 'ide.
hrto-boe gO
( a ) L F x = - , & ( N= - m a s i n 3 0
L"=
N - mg = ma cos30
N= m(g + acos3O)
.Zyl(S+ a cos30)= p{asin30
29= a sin 30 - .Z a cos30
a =
= .6129 6.0m/r..2
( b ) T i p p i n g a b o u t B w i I I i m p e n dw h e nr eacti ons Ft and N are
appliedat B as shown
.1
1
r F(i
h) = 0
EMe= ru(i_ol
!=
d
]
aa
[=
d
N
But, F = .2N
F
N
:zN-= 5
DYNAM
I CS
PROBLEM
NB/3
A zofs qubinet is mounted
o n c a s t e r s w h i c h a l w s i t to movef reely (
tl =
o n t h e f l o o r . I f a 1 0 0 Nf o r c e i s a p p i i e . a s sl ohown
d
e
r
e
r
m
i
n
e
(
a
)
,
tne
a c c eI e r a t i o n o f t h e c a b in e t , ( b ) t h e ' r a n g eo f va
I ues of h for whi ch the
c a b in e t w i I I n o t t i p .
/DO N
/t'la
rt
ttx
.1 t"r
1- -r\ onrvr*,a
/a*u
(a) fF*=
r
cr
100
ma=0
100
a
a = sn/s2
(b)
F o r t i p p i n g c l o c k w i s, e m u s ta c t a t p o i n t
ry
g M B . - L 0 0 h+ ( 3 0 ) ( g . q ) ( . s+) 2 0 ( 5 ) ( . e =
) 0
h = 1.488
F o r t i p p in g c o u n t e r col c k wsi e , N mus
t a c t a t p o in t A
f . M n = -1 0 0 h - ( z o ) ( g . B ) ( . 3+) 2 a ( 5 ) .( 9 ) = o
=
h
.3Iz
3 1 2t h * , L . 4 B g
o)
DYNAM
I CS
PROBLEM
NB/4
A 2 0 k g c a b ln e t i s rn o u n te do n caster s w h ic h a r e
I ocked
g l o u g h f l o o r ( * = . 2 5 ). I f a l o o N f o r c e i s a p p l i e d a n d s l i d e a l o n g
a s s h o w n ,determfne
( a) t h e a c c e e
l r a t i o n o f t h e c a b in e t , ( b ) t h e r a n g e
o
f
v
a l u e s o f h f o r w h ic h
t h e c a b in e t w i I I n o t t i p .
%Tc{*g
. ttnt
4F6
lN
=,'LIN
,A= ' L5
( a ) A c c eel r a t i o n
tr,
= N - mg = 0
N = 2 A ( 9 . 8 )= 1 9 6
f F x = 1 0 0 - m a- 4 N = 0
1 0 0 - ' . 2 5 ( 1 9 6 )= 2 0 ( a )
a = Z.SS^/sZ
( b ) F o r c l o c k w i s et i p p i n g ,
F f , a n dN m u s t a c t a t B .
, mq(,9)- h(t00) = 0
t M g = m g ( . 3 )+
h = L.A47n
F o r c o u n t ecr l o c k w i s et i p p i n g F a n d N
m u s ta c t a t A .
E M n= _ (. 3 ) ( m g )- 1 o o ( h )+ . 9 ( m a )= o
-(
. 3 ) ( 2 0 )( 9 . 8 ) - 1 0 0 ( h )+ . 9 ( 2 0 ) ( 2 . 5 5 )=
0
h = -.13m(impossible)
Cabn
i et will not tip for hgl .047n
k-
D.b^J
':
DYNAMICS
'A.Lz'tg
U j o c k i s p l a c e do n . a 3 k g p l a t f o r mB Dw h i c h j s h e l d i n t h e p o s i t i o n
rsrlrtuow
wrlruby
J th
L r re
r I see w ri re
l ' e s,
si D
ea
acceler
C C e l e f aations
t l O n SO
U eetter
e r mminel n e E nthe
off ttl.ie
he b
Ufolt
lOCk a
and
nd O
off tthe
he
-platform-immediately
after''.ryi!"e
ABhas been'cut; Assume-(;i-ihii'in..Ui..f.
i s r i g i d ] y a t t a c h e d! o A D ,( b ) t h a t , 4 =0 b e t w e e n ' t hbel o c ki o j : g p .
|
bd_
(a)
fZlq.
L
lr_,1 Jhlr
/
3o"
F * ,= 1 5 ( g . B ) c o s 6 0 = 1
- -5- ta
x
t
=
di
4.'9 rn/sec'
L
(b)
.:
I
Vt'Yl3
t*
SincelA= 0, Fr = 0; The block
m u s t f f i o v ev e r t l c a l l y v
d,tock '=I
t U E ' - m g + N = *uy
N = ( 9 . 8 ) ( L ? ) - LZa bl ock
N = ( 9 . 8 ) ( 1 2 )-
(1)
oup
A
V e r t i c a l m o t io n o f b l o c k a n d
platform size equal
t7
' r7 \
1:
up = ubtoc_K
, . - ;( u p ) y =
T
a
PI atform
q
1
I
- r
F * = ( 3 )( 9 . 8 ) s i n 3 o + N s i n 3 0 =
2 9 . 4+ N
Uuo
U s i n g E Q . ( 1) t l N = 1 1 7 . 6
29.{ + LI7.6
uuo uuo
a p = 1 2 . 2 5n / s 2
u b l o c k=- 6 ' I25 nl sZ
uuo
ap
Block
T
L
qP'
\ A
o
DYNAMI
CS
PROBLEM
NB/6
,'
,2
E a c ho f t h e d o u b e
l p u l l e y s s h o w nh a s a m a s sm o m e n o
t f. inertia..of 10kg-m'
and is initially at rest.. fhe outside radius is 400mm
and th'e inner-radius
i s 200mm.Determi
ne'(a ) th g angu' laracceler "ationof: each pujl.ey.,( b) the
a 5 g ual r v e lo c it y o f e a c hp u l l e y a t , t = 2 s , ( c ) t h ' e a n g u l a rv e ] o c i t l r o f e a c h
p u iI e y a f t e r p o in t A 'on the coY'khas moved2n,
@
For each case Ip( =
r. = ,2m r =..4m
10
Point'0 is center of
Ao
C a s e1
I
+
qBIN
,e=
r o o T t . (r)
(r)
,l
-f . 1o4 ^= 9 8 1( . 2 ) = 1 0 4 (
( a) d ,= l g . 6 2 r a d / s e c z1
( b ) w , =* t = ( 1 9 . 6 2 () Z )
w = 3 9 . 2 4r a d / s t
G
9.-
rM
?
(c) W-= 24 g = 2(196
. 2 . () 1 0)
=(
vo
="
10rad, a = rol
n
C a s e3
3oog 200g) (
(loo
$r
n\
/l
ro(. .2A
\rc*
Loo&
( b) w
f M o = 1 0 0 ( 9 . 8 ).(? )
2 0 d ,( . 2 ) f I a d = I 4 a
,
( a ) s \ = 1 9 6 . 2 1 4 . 0 1r a d / { \
(b) w = dt
IT\A-
- l0g[,fa'1
't eo o\
=1
= (6.53)(2)
t 3 . 0 6 r a d / s\
2 a ( , 0= ( 2 ) ( 6 . 5 3 ) ( 1 0 )
=
W
J.1 . 43rad/s T
(c)
28.0 rad/s
4 =,-.4(a
C a s e4
Zae = 2Q4.01)(10)
w = L 6 . 7 4r a d / s T
'drt
W
lF-=
(c) *2
'1Co<
= 30"'t
1 00 ( 9 .Br) /\t /.
196
( a) d = -36-= 6 .5 3 r a d/ s 2 T
n ; '= 2 n =
g
l0rad
%
a
0 0 )( .2o{ )
Lr
'
w= 19.Blrad/secf
C a s e2
tr
'M
h
-.0
=.) -tr
5r"ad
= ( 9 . 8 ) 5 0 ( . 4 ) = I 0 1+ L } d
( a ) r r= 1 0 , 8 ' 8/rraZdT
( b ) w = A t = 1 0 . 8 8 ( 2 )= ? l . 8 8 r a d
/.\
o
L
\CJ W
?^ g = 2(t0.88XS
)
w = 1 0 4r3a d / s5
DYNAMI
CS
PROBL
EMNB/t
T h eu n i f o r md i s k . A a t r e s t
j t i s p r a c e di n c o n t a c tw i t h a
it
movingat a constant
conve y orbel
rpi.ir.'"r,r.grecting
"!gn
e x p r e s s i o fno r t h e a n g i , r i J " ' u . . J i . r a t i othe
of the l i n k A B , d e r iv e a n t
iJ i - t iweight
i . ' d i s k w h i r es l i p p i n g 0 c c u r s .
trne
tr=
v
0
ft, = N - h r = 0
N=mg
Fr =/ng
f,Mn = Ffr = I.<
/4{ngr- *nr?o< = 0
2//g
6r - T=
D YN AC
MI
S
1
NB/B
A c y l i n d e r o f r a d iu s r a n d m
' a s sm i s p l a c e dw i t h n o
a b e l t a s s h o w n . D e n o tn' ig b y A t h e c o e f f i c i e n t i f
a n d a s s u nnr ig t h a t i s I e s s t h a n
. o n e ;d e t e r m i n et h e
a l p h ao f t h ; c . y l i n d e r .
tlFrF
. 1 ,
V
tu
= N g - 'r, ,*.A
/r
= NA -
,FFX
,t
A
rr [\l
l,{,tB
. t
A,
NA
1 1 ' tB
- m g = \,,
Ig
1-y'((
N
D
' t n
N A= / N g
)
=
L M e= r ( U N g )- r ( l t N n ) = r a
r(4--!r9=)
1
y'(-
- rayZ_\L1
= + mr?x
-1
(
Ht
:
r
2gA
t,
t -
^ ---)
e-*:'
'a*/'
l--:-in
a
DYNAMI
CS
PROBLEM
N B /9
Th e t w o f r i c t ' l O n C I I S K S ' Aa n d B a r e b r o u g h t t o gether by a p p l y in g t he 40N
f o r c e s h o w n . D i sks A w e ,'i
gh s 3 k g a n d h a d a n i n 1 t i a l a ng u l a r v ' e l oict y o f
1z}}rpm cl ockw i s e ; d i s k B w e jg h s B k g a n d w a s i ni t i a l ' 1 y a t r e s t . Knowing
t h a t / - L = . 3 0 b e tw e e nth. e d i s k s a n d n e gI e c t i n g beari ng f r i c t i o n ; determj n e
( a ' ) ,t h e a n g u la r a c c e e
l r a t i o n o f g Q c hd i s k , ( b ) the fi na l a n g ual r veI oc'ity
a
t
a
t
a
a
.\
of e a c h d i s k . r€----.:-.*F
T. +
4oN -l
ItD
.{L
.
f,.M
^= 4 0 ( 1 0 ) . - N ( 2 0 )= 0
C
KINErqAT_r
C!
U n for
i m acceler ati o n m o t io n s
A: ' ,F.y\
DIS.K
I
N
=20N
'
a
F f = / l N ^ =. 3 ( z o )r 6 N
o
(wn
)o = 1200rpm=
+A
t^ln = (r^r1)o
DISK A
I25. Z r a d / i
A
l=7
l^l^ = 1 2 5 . 7 F . 5 7 t
A
2o
+*vR
'A^A
vrr
l.t
V ' = 8 7 9 . 9- 4 * 0 f
A
DISK
fMn = F f t = I n d A
( 6 )( 7 )
a(A
*tt)ez)
b J g= ( w g ) o 4 r t = o +
,
An
t9 Vg= rBWB
A
. 5 7 rad/szT A
B7g,g 3.99t = 1.5t
t = 160 sec
Subst. into (1) and (2)
l,,lA= 34.Lradls )
/,
W g= 2 0 r a d / t I
0
1 .5 t
S l i p p i n g s t o p s w h e nvA
D I S KB
zMg =
F
' fl ' B
6(12)
dB=
=
T
^Ba(
B
1 r . r \ / r , ' r Z/
6.
Z\o)\LL)
B
. L 2 5 r a d/ r ? 1 A
1 25 t
.A
vsj
PROBLEM
NB/rO
DYNAMI
CS
T h e u n i f o r ms l e n d e rr o d A Bw e i g h s4 k g a n d i s a t r e s t o n a f r f c t i o n l e s , sh o r t z o n t a l
surface. A force P of magnitude
1 0 Nj s a p p li e d a t A i n a h o r iz o n t a l d i r e c t i o np e r p e n dciu la r t o t h e .r o d . D e t e r mnie ( a ) t h e a n g u l a ra: c c e l e r a t i o no f t h e r o d ,
( b ) t h e a c c e l e r a t i o no f t h e c e n t e r o f t h e r o d , ( c ) t h e p o 'ni t g f : t h e r o d w h i c h
'
h a s n o a c c e el r a t i o n .,
,|{
r\
S M
sl'ht't1
o
'
. t t -
u
Dl
T
1 nt?A
v
x - Ixf6 P
For L = lm;
d =tffi
m
t(,
4kg; P = loN
= 15ra
d/szt
-r
ct
(b) IF* = p = ma
0=a'f
^P
q.-
;CI
m4
2,5 n/sZ
aQ,lG
c
d=
,d=
,dA
=,
t7m
2.5
15
.,
m
.
DYNAMI
CS
PROBLE
NM
B/11
B y p u l1i n g o n th e co rd ,o f yo - yo j u s t f a s t e n o u g h a
, m a nm a n a g etso m a , k e
t h e y o - y o s p in c o u n t e r col c k wsi e, w h iI e r e m a i ' n ig a t a c o n s t a n th e i g h t a b o v e
t h e f I o or . D e ntio n g 'th e w e ,i
ght of the yo-yo by t^l, the radj us of tiie j nner
d r u mo n w h i c h t h e c o r d i s w o u n d!y, r , and the r adi us of gyl.atior i of the y o- y o
b y k , d e te rmin e ( a ) th e te n sion i n t h e c o r d , ( b ) t h e a n g u i a ra c c e l e r a t j o n
:.
of the yc'-yo.
(=
rng
ma=0
..r
mg
I- M
o = Tr
rc[
Unilr = ttUza
/ =gI
Arz
K
DYNAMI
CS
PROBLEM
NB/12
A u n if o rm s I e n d e r b a r A B o f massm i s s u s p e n d e d . f r ot m
wo
I f s p r i n g B Cb r e a k s ,d e t e r m i n ea t t h a t i n s ! a n ! ( a ) t h e a nsgpi rr it na gr -saacscsehl eo rwani t.o n
o f t h e b a r , ( b ) t h e a c c e l e r a t i o no f p o i n t A , ( c ) t h e a c c e ] e r a t i o n
of point B.
BEFORES P R I NB
GR E A K S :
= T R Cs i n 3 0 + T g Cs i n 3 0 r r n g= 0
ft,
TRc+Tgc=zTg
fr=
\-X
TRc co s3 0- T g Cco s3 0= 0
Tnc Tsc
TRc
a
aa
Tgc = mg
i m m e d i a t e l ya f t e r B Cb r e a k s , t h e
e lo n g a to
i n of the spring ACis
u n c h a n g e dJ;, T A Ci s s t i l l T R C= m g
ACCELERATION
OFAANDOFB
(b) aA= tc + Io/n
Tae
uA=uG+
+t-,}*
A
uA:
.59*
gf
oA
t
( a ) { i l n = T R cs i n 3 oL( / z ) = r d
mssin30(
L/z) =
b nLZ<
4=P ) I
TRCcos30 = *u*
rngcos30 = ru*
u* = . 8669
tr,
= mg - TRCsin30 = *uy
uY = '59
(c)
+
+
. 866g-r
1.3235 *
aB = a " + a
tr
.'5g*
t
4 9 .L o A
B/G
+ . 8669-r-
2g* + .9669_>
a B = 2 . 1 B g ** 66. 60
+ tfll rfr+
ry4
DYNAMI
CS
PROBLE
NM
B/TS
A u ni f o rm s Je n d e r ro d , o f I e ngth L = $ 0 0 m m
a n d m a s sm = 4 k g , i s s u p p o r t e d
as shown. A horizontal for P-of magnitude
7
5 Ni s a p p l i e d ; l . n a - 8 . ' D a i . r T i l e ( a ) t h e d i s t a n c eF f o r w h i c h t h e h o r i z o n t a l c g m p o n e notf t h e r e a c t i o n
at
c i s z e r o , ( b ) t h e c o r r e s p o n d i nagn g u l a r
a c c e l e r a t i o ho f t h e r o o .
A
a- =trd
r,J = rvt3
ts
EM . = P(L /Z + r) - ma 'r= I a <
P(r+ r) =#ntk
a<
+ m(rr<)F
(1)
( b ) S u b s t i t u t er' 6 = I i n t o E Q N .( 1 )
L,
L
./_ P z-r6A'
mtre
E'r 36-
A = 6.L
ML
For P = 75Nr[n= 4, and L = .9m
LF* = tt* - P + ma = 0
P=ma=mTd
I
01=(6),u,ifo)_
For C* = 0
(a)
2
l'-
b t
f=$+
'
. o IZ
?
L_
E
=
Dc
L . f, J .
r
E
F) F
t-
-l-f.
2'.
.9m
r= L
6r-il=
r = 1 5 0 mm 4
!+F
15m
r
A = L25r ad/s2
./
'PROBLElvt
NB/14
DYNAMI
CS
A u n i f o r m s l e n d e r r o d o f l e n g t h I a n d m assm r otates a b o u t a v e r t ' ic a l a x i s A A '
a t a c o n s t a n t a n g u l a r v e l o c i t y o m e g a . Determi
ne the tensi on i n the r od at a
d'is ta nce x f rom the axi s of rotati on .
A
M A S S / U NLITE N GT=Hn /(, 4 = 0
r=
Cl-
( 1 ,+ x )
m t = [(L
*
?1
+
rw' =
;( {, x ) w 2
= T = mtE-= m
v
' Fx
b
' -z fr*2(r 2 -
T1
( , (
r
x2)
- x)
x)F(L +-)u,1
a
DYNAMI
CS
f,
PROBLEI'I
NB/t S
A u n i f o r mb e a mo f l e n g t h L a n d w e i g h thl i s s u p p o r t e dA S s h o w n . I f t . h e
c a b le s u d d e n l yb r e a k s, d e t e r mni e ( a) t h e r e a c t io n a t t h e p j n ' s u p p o r t ;
( b ) t h e a c c e l e r r a t j oonf p o i n t B .
,
t
\/.-l
lq
I
,Lr
t4l
(+.
*.(+
mLzA
m g f ,=
I
k*\2n
..
rzg
d = 7L
;D
I
(l) fF'
,l
= C ' - mg = -ma = -In(;d
)
c+
l
-mL
C - m g = T rlSsr
,7L'
+
'
:
.
.
.
.
/re' \ 4
-l
rng
-
I
*a G + A
uB = 0
BlC
lx
r#r
uB =L4
?
uB 7n
G
tA
t4
DYNAMI
CS
Tworods AB and u g r . . r n a m
: sp
. e r q n i t l e n g t ha r e c o n n e c t eads s h o w n . tao d i s k
w h ic h i s ma d eto r o E
t aatree]i n a v e rrttliccaatl p l aanneea t a c o nnssttaanntat n gquullaarrv e lloocciittyv .
d;.
'o
em
x eergt U
d F o r th e p o s i t i o n s h o w n , d e t e r m i n e . t h e c o m p o n e n ttsh.eo f t h e . f o r c e s
f
--l at A andB on rod AB.
L'(
rrL.
. Yt ,rr.
, J -
|
I n s t a n t c e n t e r . o f AB
' w R g=
.e
VR
:Gl
.:
T
d;n
ltI
1
ilRs = / o
hl
"BC
=
O( AB
-BC
f
'd,^r
0
=0
=
bI
*G2
-r
A
2
=3 w l I
o
2
aA = r w
o
q
A
qno
\f 6_
, <-"r'
m g c:
RODAB
fn'l.
I
Gl
r
J
moc,1
:
fr
"l
f t zMc= BXr =
r
r
I 4 s c + m"'BC
" G22
-l
i
B r = 1 ( m r ) r2 t-:
l;w- l +
X
L2
rt o l
2
B = 1 m r2w
'o
X
\
a
'r,
+
+-LF
'*fI 7
n,'.(i
r
A
t1
B
x=MRs%r
A
x-z mr-wo' = zmr(f; '*o2 )
XX
1
I
A
n
z
R
IJ
fro
Yield
X
X
-:n
\,r
1
?
= 3 m r?-?w
+
^A
22
=T mr-wo'
-+
I
0
2
'F y=0
Ar= B,= mgr'lt A
m
DYNAM
I CS
PROBLEM
NB/TO
t
Two u ni form r od s, e a c h o f m a s sm , a r e a t t acch e d a s shownto smaI 1 gea,rsof
n es 1i si b l e maS s . . I f t h e r o d s a r e r e l e a s e
d dfrom r e s t ' i n t h e p o s j t i o n s h o w n ,
d ete rmi n e t h e a ng u l a r a c ce l e r a t i o n o f r o d AB imnr
ed!:a
te'ly a f l e r r e l e a s € d ,
A SS Um in s ( a ) g
0, (b) s ; 30'.
...
' '.,
CO
NSI DER EAC
H RO
D SEPARAT
ELY
:
'
e = 0:
( a)
P ' = F o r c ea t g e a r t e e t h
.....
RqD .AB
:
+
EMe= F r = * u t * l l +
I
.
iI
I
Td
': .(l !,1fu. !2a
#rn
Fr
/
Fr = + m ( - 4
J.
-t
{
( t \I
-;
? .
\
1
RO
D CD
a
@
2*(
g= 300
a
+ !-)a
RODAB
= -ma
$)
*(i L)'d
+Fr
fM. = -mgi.{i)
'1
ns!-- F r =
'
bna
0
Zot
F r = 1 * , o 2A' , .
3il,r
tTI Ms= F r - mstall= ma(
{l
A d d( 2 ) a n d ( 1 ) :
n zI r n q ( = 2 m'L
d'
3
2J
6(=i
/0,
Fr
1
=
T ms,[ ^t*l?< + b
Fr
t4 ,n!- =
g
!-
= . 7 5 g Il . j
a- {
+Id
h! 2 n
!-za
(3)
R O DC D
...
+ ) xl{U- = mg(
(4)
Add ( 3 ) a n d ( 4 )
6
n
) - Fr = nraqJ
%nn-Fr=
msf
^Ak
Y+
,T;.,
*(i
lt_)L
n !..2x
o (= .275g/g-T
DYNAM
I CS
PROBLEM
NB/l7
Klg*ilg that the coefficient of friction between
t h e rod and the f] oor
' 3 0 , d e i e r m i n et h e r a n g e o f v a l u e s o f b e t a f o r
w
h
ichthe rod will slip
i m m e d i a t e l ya f t e r b e i n g r e l e a s e df r o m r e s t .
lAlEASSUI4E
THATRODDOESN O TS L IP
-L.
Iuo = *g(f cosH
t
2
mgl cosp =
6
Itt
3N=
= N - mg= -macas4
fr,
N - m g= - * ( f 4 ) c o sp
(b)
+it
*(r")
t* cosp
mL2o(
(1)
F = m a sinr€.
,F X
F.u'[
l-
= *utf) +r4
^(1
") s i n B
(3 )
E l i m in a t e N b e t w e e n( Z ) a n d ( 3
)
*(bla(cosb+13sin
6 - ms= 0
Substi tute for d from ( 1)
r(f) (* fl cosd( cosyg+ sinp) r ms= 0
+
3€orZA* l$sin|cosB - 4 = 0
3 ,or% r 4 = -10 sinp cosg
(g ,orz/ - qz = loo sini cosrp
9 cos4p - 24 ror2o+ 16 = 1oo(1r c o s p
' ) c os z p
109,otfo - rz4corz1+ 16= o
'
So lve fo r co s2p
cos'6 = .,gggz and cos,p
= . 14g4
c o s p = . 9 9 4 6 a n d co s = .3952
/)
ts = 6.00 and,a
R o dw i l l s l i p
6 . o o1 , a I 6 7 . 3 0
= 67.30
{
' 'v.
DYNAM
I CS
:
PROBLE
M18
NB/
A h o m o g e n e o ucsy l i n d e r C a n d a s e c t i o n o f p i p e P are 'in contact w h e nt h e
a r e r e l e a s e d f r o m r e s t . K n s w i n gt h a t b o t h t h e cyl i n.derand the pip;;;ii
s I i p p i n g r d e t e r m i n et h e c l e a r d i s t a n c e betw.eent h e m a f t e r 2 . 5 s.
Tl
-2
l -
1
2
mr
sinP=
+
g sin4
I
'sinp
-ac- y, }
Y
=u.yl-upipe=$nsinP
/pi pe
For p = 100
" c y 1/ p i p e
s i np
=
Ll
2\
and
+ s s in p
* s s i n/ 3
t = Z.5s
a. y 1 / p i p^)
o
e t *=*,EnsinloE
v
ZL6ss
J
* . Y 1 / P i P e= ' B B 7 r n 4
( z . s ) ?= . o g o 4 g
I
DYNAMI
CS
PROBLEM
NB/T{.
A s m a l l b l o c k o f m a s sm i s a t t a c h e da t B t o a h o o p
of mass andradius r.
Knowi
ng that whenthe syste mis r eleasedfr om r est i t s t a r t m
s to roll without
s l ! d i n g , d e t e r r n i n e( a ) t h e a n g u l a ra c c e l e r a t i o no f
the
h
o
o
p
,(b) the acceJeration of B.
? 2= m r
I
uA = fo(
Rolling:
t ) fM, = mgr
But
I A + m a A r+ n r ( a g l * r+ m ( a g ) y ,
;A + Irlo = [uo{ + [r. *] = x.J +
L
aB
mg r
*rk
(a) A =+
!{
.J-
(b)
vfl
I
uA=YA
uB=
+ m f t a) r + m ( r X) r + m ( r a (
)r
Ja
=t g->
=gn
[+n-']+
F,q
.+\.^
q6o
L'**l
CS
DYNAMI
ofZ
T h e t w o .b a r s A B a n d B C a r e r e l e a s e d f r o m r e s t i n t h e
ne
. b ar i s 6 0 0 m ml o n g a n d h ' a sg m a s - so f 4 k g . Determi
o f e a c h b a r , ( b ) t h e r e a c t ' io n s a t A a n d C .
E ac h
accelenatjon
KINEMATICS
\PO
assume
(qgl)x = '.260;<
a
*c
b
&
lsu
|-rL
s
= e*B + e r\^c/B
3oo
= .6d{300 + ac/B^71
4-'"
u.?
a'
bC.=.trnA
, 1 5d *
.
t
6'c
ucz= * B
A+A
symme
try.
/sc
=t\rI
GT/B
Err* = i .6,{S,g0o+ i Sx SEOo
:
(aez)x .Gd + .3d)cos30o
(ae
)v
. 6 o ( ) s i n 3 0 * + ( . 3 (/r , \
a
. J l
I
779 A ->
s j n 3 0 t = 1 5 0x 4
'
K I N E T I C S : F re e B o d y B C :
.15
L M s m s (l q ) - C ( . 3 0 )= b < + ' x ( ae6) V ( . 1 5)
m g (. 1 5 )
'rf e\
tt:"'
1
r
E
m (. i l k
m g ( . 1 5 ) . 3 C = . 2 1 0 m J-
G
(t)
m(Iee)^( .26)
+ n r (. 1 5 ; . ) 1 5 - m (. 7 7 9 d) ( . 2 6 )
PROBLEM
NB/20
P. 2 of ?
DYNAMI
CS
FREE
B O D Y :TNTIRE SYSTEM
+ ) , E M n= m g ( . 1 +5. 4 5) - ' 6 c = Iot-- Ia
b
+ ( . 1 5 m a) ( . 1 . 5+ . 4 5 )
+ ( . 2 6 m e+( . 7 7 9 m o) (r. 2 6 )
. 6 m g- . 6 C = . 3 6 m&
Ar
-
Ay
.J,
t J
,t3
'15
( a ) S o lv e E q n s ( 1 ) a n d ( 2 ) s i m u tl a n e o u1sy :
g = 9_.81=
| =
3.77
4\"
2 .6
2.6
szt, ^l
d AB = 3.71 rad/
& BC = 3 . 7 7r a d / s zI A
(b)
?
U
= 2 ( 4 )( 3. 7 7 )
ZmA
C = 30.2N 'f'
zFX X= m a
.a
Yields:
A
(.260+ .779)(4)(3.77)
A
15.68 N - >
A=
8y-c
A=
4 3 .B N
X
X
f''F = m a
y
y
Yields
v
v
2 ( . 1 . 5dm)
+
- 1,
DYNAMI
CS
PROBLEM
N8/23
A u n j f o r ms l e n d e r r o d o f l e n g t h L a n d m a s sm i s r el easedfr om r est i nP. 1 o f
t h e p o s i t i o n s h o w n . D e r i v e a n e x p r e s s i o nf o r ( a ) t h e a n g ual r a c c e e
l ra t i o n o f t h e r o d , ( b ) t h e a c c e J e r a t i o no f e n d A , ( c ) t h e r e a c t i o n a t A
i mmed
a it e l y a f t e r r e l e a s e . N e Eel c t t h e m a s sand fr i cti on of the r ol I e
at A.
KINEMATICS
hl = 0
Assume01, )
p
tA=t +Eoln
6.A
F-'l +['rtl +lho {
Fo=qq=
Llza
Lu.
G
B
aN^"1
uy=(1*+
di
*A
la ) t a nB
(1)
(2)
= = - J -
sin6
+)f.Mn= *d*ffl = rA
o = #nt?ol - *dxtbl
d* =
*
(3)
Ld
S u b s t . i n t o E Q N .( 1 )
uy=t? LPt+ ,aJ
uy =? LA tan p
3
+ $p
EF = mg sinb - *u* casp - *uy sinp
S u b s ttiu t e f o r u * a n d u y f r o m E Q N S(.3 ) a n d ( 4 )
ms sinp = nr(* Ld )cosp + rrr(3L,(tanp) si np
12
g tanp = ; , V + ; L o (
tan'p
OJ
tanB
(4)
2
DYNAI'{I CS
PROBLEM
NB/z3
P. 2 of 2
( b ) F r o mE q n . ( 2 ) a n d ( 4 )
*A.
r-r
La,tan p)
si n B
sin B
=Lr l-g
6 tan 01
uA=
ts
=
tan,6
tLa\w
tan4
sIlE
Lr
3b
(c)
(t
aV
a = q t . =
6 tanzp
c
(r + 4 tanzp)sinp
R e a c t i o na t A
+ lf.Me= A(sinp ) L=
T/
2.
(AsinB)i= #mL2ls.
LL1 +
6ta
A=*ms
J
?p
4 tan
n0
(1 + 4 tan'|)
A=
tan 0
sinB
*9^'-
(1 + 4 tan'T )cosp
foA
4
PROBLEM
NBI24
CS
DYNAMI
E a c h o f t h e b a r s A B a n d B C i s o f I e n gt h l - = . 5 r na n d w e i g h t 1 : 5 k 9 . A
h o r i z o n t a I f o r c e P o f m a g nti u d e 2 0 N i s a p p li e d a t C , D e t e r m i n teh e ' a n g u l a r
a c c e lg r a t i o n o f e a c hb a r .
I CS
ri ruEMAT
Assume.
drc )
A
AB
(r\,
KINETI CS
ENTI RE SYSTEM
ryg
b7
G
e7
A
&r
mt
t
ly1At,
m5
P
L
+TEMs= pL *u2(b) =
1
Tl'
1,o'
.r
.? t
Bc
P L = .n ml'n gC+ m l L a ( ^ +
nB':
l
t-
12
1 .?
P L = m l - d/R g + * m L - 6 - ,
i
3"':- BC
itill
(1)
(1) and (2) simultaneoysly
6P 1
r4rr*l
t.5"1.5
7fm
i
2l
22.9 rad/s
A
a\
^
=-
AB
^/A B
=30
BC 7
+,
P
t'
130
mt
.
r 4 0 i (rb)
' 1
(t)
dur=
\
;$l,ld '''I /
1
2
n mmL
\d
? ? B. 6 r a d/ , 2 f
AB
+^
bc)* . m
r
2
mLl /,
+ 5 mL2'\
AE
B 6
BC
s
A
BC
il
A
o
(mar)
Lz
fL /,(n IL
onl
LTL
*
.L
A-AB + T A
2PL
S o lv i n g
L,
?PL
l2
ml L l
L AB
Ll
z^ry
(2)
DYNAMI
CS
PR0BLEM
NB/25'- F
T w ou n i f o r m s l e n d e r r o d s , e a c ho f m a s sm , a r e c o n n e c t e db y a p i n a t C. Determ'i
ne
t h e a c c e le r a t i o n o f p o in t s C a n d D im m edi
ately after the hor izontal force P has
b e e na p p li e d a t D .
KINEMATICS
UAB
t
q
c
t
A-= ?/-
-cL
Lrr
KINETICS
AB
RO-D
,
Ab
Add ( 2 ) a n d ( a ) :
,-s&c,
-
->'
c'7
=PL
o
=t6 mLa
o
7P <-
$IFx = C - *uc = 0
c=*(!x-a)
(5)
5m
"'rS''-
S u b s t i t u t e( 5 ) i n t o ( 3 ) :
RODCD
P=
2,,.F
*] ,mlo(
P=#P -
PL= *utf) + bntk
o,
P + *(!a - a) = ma
c
c
(3)
,mla(
l4ult. (3) by !t
6'
F
uD =
2 rA(
ai
2 lt
->
lod
+ a=,
5t m
6 t <uD = 16
5 m
F
PL
T:1OJ
mLa
-n1
nt?;.
(4)
Y'
h. D
i
X
p = ?na-
'Fcc rlil
(2)
J f F = p + C - ma = 0
e-4,
18 P\
/
+tfM. = PLD *utf) = rA
PL= *u(b) + rd,
1
*(, mlo{
P
L 8' m [ /\
\r T
-
7P
F-
5m
4
r 1 8 P 1 1,' :7 P
\T-m['/ 5 m
P R 0 B L ENM
B / 2 0- - .
DYNAM
I CS
T w ou n i f o r m b a r s A B a n d B C oe a c h of I ength 300mm
are welded together to form
a L- shaped ri g'id body. Knowi
ng t h a t e a c h b a r w e ig h s L . s k g, d e t e r mni e t h e
t e n s io n i n e a c hw i r e i m m e d i a t e l yafter the body has beenfel easedfrom rest.
Denoteby t,''|the weight of each bar
S i n c ev e l o c i t y a t r e l e a s ei s z e r o ,
a O : a 5 o a n d u . = a 4 5 o . W ef i n d
C
r
-rsO
I'.l3
t h a t s y s t e mi s i n . t r a n s l a t i o n w i t h
4 = 0 andaE,45o.
a
Free bo.dy
L
45o
Ti'=
B
A + C - ZWcos450 = 0
5 . 2 + C - 2 ( 1 . 5 ) ( 9 . 8 )( c o s + S o=) 0
c = 1 . 5 . 6N 4 - 4 5 0 A
b=
q50
+
\EF
L
L
Ll
z s i n 45 z\.7 A 7 )=
= 2W
co s 4 5 0
ms(
, 70 7\l
a
.35351
2ma = 0
ma
707 s
\450
r t f l M g = - A ( L ) + t^lb- hlb+ *u(f) = 0
AL =*u(!)
A=
,ma
A = m(
s)
t .707
For m = 1.5
A = 5 .2 N
{45oJ.
DYNAM
I CS
)
PROBLEM
NB/E7
A c o l l a r C of wei ght [,l^ i s r i g i d l y a t t a c h e dt o a u n i f o r ms l e n d e rr o d A B o f
I e n g t h L a n d w e i g h t h l . ' I f t h e r o d i s r e l e a s e df r o m r e s t i n t h e p o s i t i o n s h o w n ,
determine t h e r a t i o d l t f o r w h i c h t h e r e a c t i o n
l,.lenot that l^l= 0
L
F R EEB OD Y : C OL L A R
fraLe,
lcdtl
E!=
C + lvl a
cc
-mq=0
oL
hc
.1,,
l ^ e
C= Mc(1 -i_t
ua = dA
hle ng*t that C = 0 i f u a = g .
W i t h C = o , th e mo tio n o f the
r o d w i l l b e i n d e p e n d e not f W .
fs
t
F R E EB O D Y : RODAB
cl
a=la
rnt
- *ut!)= rd
EMs= ms(f)
msrll = b nr?&+ n(la) i
/ =ifl /
a =3
2
c
B u t w e se e k a = g
c
c
g
d
L
=t*s
2
3 A
DYNAMI
CS
PROBLEM
NBIZ8
A s e c t i o n o f p i p e , o f m a s s5 0 k gand radi us 250mm
rests on two corners as
s hown. Assumi
ng that/ betweent h e c o r n e r sa n d t h e p i p e i s s u f f i c i e n t t o
p r e v e n ts l i d i n g , d e t e r m i n e( a ) t h e a n g u l a ra c c e l e r a t i o no f t h e p i p e j u s t
a f t e r c o r n e r B i s r e r n o v e d(,b ) t h e c o m e s p o n d i nm
g a g n i t u d eo f t h e r e a c t i o n
at A.
R o t a t i o na b o u t c o r n e r A
KI NEIvIATI
CS
T
$s3oo
A=YA
K IN E TCI S
(b). t-6oo
mr2
r= .25m
N-hl
t-
I
y1tnSO=
(a ) +)fMo =
z
1
z
d=#=r#i
/60o
tF = mg s.in30o r F - m a =
1 mg I F = m r d =
mr(#)
A
r
=
n mgr rma Id
1 mgr=
*r2o( **rk
cos30o = 0
N=qmg
f \lco"
N
L
ZF=0
F
1
4 msf
R e a c toi n a t A
=ffi=
R= $-o,n
d , = 9 . 8 1 r a d/ s e c ?* 4
For m = 50kg
R=
4
4
(50)(e.81)
\s
R = 442NA
N o t e : t o p r e v e n ts l i d f n g a t A we must have/Z
T=
N
fr^r
5''
,4 '' '29
C-'-'"
PROtsLEM
NB/29
DYNAMI
CS
The inertia-vector diagramrepresenting
the generalrotational
of a
r i g i d b o d va b o u ta f i x e d a x i s ' i s s h o w n - t inl r 6 - i i g ; . e . - - 5 f i i l i h amotion
t the inertiaIa(a1pha)maybe e'liminated.by
movinsthe]inerti;-i;ri.-u..tors ma^r
99Yp]9
and maan"lgppint P, locateda distanc€F"o-= kg'/rn^ tiom-t[; .*i;.-oi-Gt
massof" the body. Thepo'intP is calledurthecehtUFor-perlussion-ot-ihe
body.
LMo = -toc*ut + rop*ut = Ie/
ut
t
Ig = K''G*
" ( = roG
S u b s ttiu t e
dt,
2
toc*ut - top*ut = K'oGttt;'
2
K-oG
G-+
roc= lop = rcp + toc
Thus
?
=Ib
"Gp toG